<?xml version="1.0" encoding="UTF-8"?><rss version="2.0" xmlns:content="http://purl.org/rss/1.0/modules/content/" xmlns:atom="http://www.w3.org/2005/Atom" xmlns:dc="http://purl.org/dc/elements/1.1/"><channel><title>Philomathēs</title><description>Essays, fiction, art, music, cryptography, code, projects, curation and manifestos by Matheus Soranço, a researcher in AI engineering and multi-agent systems.</description><link>https://matheussorancophilomathes.com.br/</link><language>en</language><atom:link href="https://matheussorancophilomathes.com.br/en/rss.xml" rel="self" type="application/rss+xml"/><item><title>Measuring the Unsolvable: A Comparative Study of the Liber Primus (Cicada 3301) and the Voynich Manuscript</title><link>https://matheussorancophilomathes.com.br/en/cryptography/measuring-the-unsolvable/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/cryptography/measuring-the-unsolvable/</guid><description>Two famous unsolved ciphertexts, one methodological rule: report every statistic next to null models that ought to fail it, and bind every attack to known-answer controls. Seven cipher classes are excluded for the Liber Primus; for the Voynich manuscript, Zipf&apos;s law and conditional entropy discriminate almost nothing. No decipherment is claimed.</description><pubDate>Fri, 06 Nov 2026 12:00:00 GMT</pubDate><content:encoded>&lt;p&gt;&lt;strong&gt;Matheus Soranço&lt;/strong&gt; · Creator of I.S.A.A.C. (Intelligent Scientific Analysis &amp;#x26; Analytics Core) · AI researcher, computer science, cryptanalysis, philosophy of mind&lt;/p&gt;
&lt;p&gt;&lt;em&gt;Preprint, not submitted to any venue. No decipherment claim is made. Code: Apache License 2.0. Paper text: CC BY 4.0. Repositories: &lt;a href=&quot;https://github.com/Matheussoranco/LiberPrimus&quot; target=&quot;_blank&quot; rel=&quot;noopener noreferrer&quot; class=&quot;externo&quot;&gt;LiberPrimus&lt;/a&gt; and &lt;a href=&quot;https://github.com/Matheussoranco/Voynich&quot; target=&quot;_blank&quot; rel=&quot;noopener noreferrer&quot; class=&quot;externo&quot;&gt;Voynich&lt;/a&gt; (GitHub: Matheussoranco).&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;&lt;em&gt;This page is the web version of the preprint, with corrections and an appendix added in November 2026; see the &lt;a href=&quot;#revision-notes&quot;&gt;revision notes&lt;/a&gt; at the end. The &lt;a href=&quot;/arquivos/liber-primus-voynich-preprint-original.pdf&quot;&gt;PDF as originally written&lt;/a&gt; is kept for reference.&lt;/em&gt;&lt;/p&gt;
&lt;h2 id=&quot;abstract&quot;&gt;Abstract&lt;/h2&gt;
&lt;p&gt;The Liber Primus (Cicada 3301) and the Voynich manuscript are two of the most famous unsolved ciphertexts. Both have accumulated a century-plus of mutually incompatible “solutions,” most of them unfalsifiable. This paper reports two independent research programs, one for each artefact, that share a single methodological rule: &lt;strong&gt;report every statistic next to null models that ought to fail it, and bind every attack to known-answer controls drawn from the same cipher class.&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;For the Liber Primus, a falsification engine excludes seven cipher classes on the six unsolved sections (11,081 runes), including natural-language running keys via a convolution argument (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;10.7&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z = -10.7&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;10.7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;), and documents how a substantial machine-learning search apparatus ranked a rotated plaintext above the true plaintext on an already-solved page. A subsequent variational homophonic analysis finds per-section heterogeneity: three sections compatible with homophonic English, one decisively excluded, two inconclusive.&lt;/p&gt;
&lt;p&gt;For the Voynich manuscript, a discriminative battery tested against four structure-retaining null models shows that Zipf’s law, conditional entropy, and positional rigidity are all reproduced by an order-2 character Markov chain fitted to the manuscript’s own glyphs: they discriminate almost nothing. Two findings survive their controls: Currier A and B are distinct populations, and the line-as-a-functional-unit effect (12× glyph-level, 23× word-level divergence over a character-shuffled control) resists any substitution or transposition cipher of natural language.&lt;/p&gt;
&lt;p&gt;The paper concludes with a discussion of where AI fits in this domain (hypothesis generation and corpus building outside the scoring loop) and where it does not (as a judge inside it).&lt;/p&gt;
&lt;h2 id=&quot;1-introduction&quot;&gt;1. Introduction&lt;/h2&gt;
&lt;p&gt;The Liber Primus is a runic codex released by the Cicada 3301 group during its 2014 puzzle round, written in a 29-character alphabet published as the Gematria Primus, mapping Elder-Futhark-derived runes to both an index and a prime value. Four sections have been decrypted by the community and read as English aphoristic and instructional text; six remain unsolved, pooling 11,081 runes. The Voynich manuscript is an illustrated codex radiocarbon-dated to the early fifteenth century (c. 1404–1438), written in an unknown script (Voynichese) across roughly 240 vellum folios. Despite more than a century of sustained professional and amateur attention, neither has been convincingly deciphered.&lt;/p&gt;
&lt;p&gt;What unites the two domains is not just their fame but their epistemics. An unsolved artefact has no ground truth against which to check a positive claim. If a procedure outputs text that reads like English or medieval Latin, no independent oracle confirms it. The consequence is a long record of confident, mutually incompatible “solutions.” In the Voynich domain this includes work that passed peer review before being withdrawn. In the Liber Primus domain, the community has produced a sprawling registry of hypotheses with little methodological discipline.&lt;/p&gt;
&lt;p&gt;This paper does not attempt to solve either artefact. It reports measurement programs designed so that negative results mean something. The central rule, applied mechanically in both projects, is &lt;strong&gt;control gating.&lt;/strong&gt; Every attack is bound to (1) a named, size-quantified cipher class, (2) known-answer controls drawn from that same class and generated through the attack’s own primitives, and (3) a procedure applied to the real ciphertext. An attack’s report is admissible only if its controls pass; otherwise the verdict is INCONCLUSIVE, a statement about the tool, not the data.&lt;/p&gt;
&lt;p&gt;A secondary rule is the &lt;strong&gt;discriminative battery:&lt;/strong&gt; a statistic is admitted only if it separates the artefact from null models that are meaningless by construction. Null models are ordered by how much structure they retain: uniform random text, character-shuffled text, order-1 Markov, order-2 Markov. A statistic that almost everything passes cannot support a conclusion, however firmly the artefact passes it.&lt;/p&gt;
&lt;p&gt;My personal interest in these two projects is longstanding. The Liber Primus fascination began with Cicada 3301 and turned into a personal research program combining computational cryptanalysis with machine learning. The Voynich interest is older still, a curiosity about whether statistical structure can ever establish linguistic status. Both projects are conducted as research: failures are preserved, what works is measured, and conclusions are kept smaller than the curiosity.&lt;/p&gt;
&lt;h2 id=&quot;2-mathematical-and-cryptographic-foundations&quot;&gt;2. Mathematical and Cryptographic Foundations&lt;/h2&gt;
&lt;h3 id=&quot;21-index-of-coincidence-ic&quot;&gt;2.1 Index of Coincidence (IC)&lt;/h3&gt;
&lt;p&gt;The index of coincidence is the probability that two symbols drawn independently from a text are identical. For alphabet size &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and symbol counts &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f_i&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; in a text of length &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;:&lt;/p&gt;
&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;.&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC} = \frac{\sum_{i=1}^{n} f_i\,(f_i - 1)}{N\,(N-1)}.&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:2.43em;vertical-align:-0.936em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.494em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.6897em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8043em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;mrel mtight&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2029em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2997em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.936em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;.&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;p&gt;For uniform text over &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; symbols, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC} = 1/n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1/&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. For English, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;0.066&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC} \approx 0.066&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.066&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. A substitution cipher preserves IC exactly (it is a bijection); a transposition preserves the unigram distribution and therefore IC; a running key convolves plaintext and key distributions, flattening but not uniformising the result unless the spectra of the two distributions are disjoint (Section 3.3 and Appendix A.2). This last fact, the convolution argument, is what allows a class-level exclusion of natural-language running keys without searching the unbounded key space.&lt;/p&gt;
&lt;h3 id=&quot;22-entropy-and-conditional-entropy&quot;&gt;2.2 Entropy and Conditional Entropy&lt;/h3&gt;
&lt;p&gt;Unigram entropy &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;H_1 = -\sum_i p_i \log_2 p_i&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0813em;&quot;&gt;H&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0497em;vertical-align:-0.2997em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.162em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2997em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop&quot;&gt;lo&lt;span style=&quot;margin-right:0.0139em;&quot;&gt;g&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.207em;&quot;&gt;&lt;span style=&quot;top:-2.4559em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2441em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. First-order conditional entropy &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∣&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;H_2 = H(c_i \mid c_{i-1})&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0813em;&quot;&gt;H&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0813em;&quot;&gt;H&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2083em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, estimated with Miller–Madow bias correction. Maximum entropy for 29 runes is &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;4.858&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\log_2 29 \approx 4.858&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.9386em;vertical-align:-0.2441em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop&quot;&gt;lo&lt;span style=&quot;margin-right:0.0139em;&quot;&gt;g&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.207em;&quot;&gt;&lt;span style=&quot;top:-2.4559em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2441em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;29&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4.858&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; bits. Low &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;H_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0813em;&quot;&gt;H&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; indicates sequential predictability. Entropy estimates are biased downward on short text; the correction is applied and raw values reported alongside, because an uncorrected estimator can manufacture an “unusually low entropy” finding.&lt;/p&gt;
&lt;h3 id=&quot;23-zipfs-law&quot;&gt;2.3 Zipf’s Law&lt;/h3&gt;
&lt;p&gt;Zipf’s law: the frequency &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f(r)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; of the &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;r&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;r&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;-th ranked item follows &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;∝&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f(r) \propto r^{-\alpha}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;∝&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7713em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.7713em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0037em;&quot;&gt;α&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. A least-squares fit of log frequency against log rank yields the exponent &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\alpha&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0037em;&quot;&gt;α&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;R^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0077em;&quot;&gt;R&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. Zipf compliance is true of Voynichese (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;1.022&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\alpha \approx 1.022&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4831em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0037em;&quot;&gt;α&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1.022&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;0.979&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;R^2 \approx 0.979&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0077em;&quot;&gt;R&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.979&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;) but is also true of an order-2 character Markov chain fitted to the manuscript’s own glyphs (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;1.064&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\alpha \approx 1.064&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4831em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0037em;&quot;&gt;α&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1.064&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;0.985&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;R^2 \approx 0.985&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0077em;&quot;&gt;R&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.985&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;). The statistic does not separate them; therefore Zipf compliance is not evidence that Voynichese encodes language.&lt;/p&gt;
&lt;h3 id=&quot;24-control-gating-and-the-verdict-lattice&quot;&gt;2.4 Control Gating and the Verdict Lattice&lt;/h3&gt;
&lt;p&gt;Every attack carries a hypothesis, controls, and a procedure. Controls are known-answer problems from the same cipher class, generated through the attack’s own cipher primitives. The verdict lattice distinguishes REFUTED (controls passed, entire class covered, nothing found) from NOT_FOUND (controls passed, subset searched, nothing found: evidence against, not proof), CANDIDATE (controls passed, something scored in the plaintext band: referral for human review, not a solution), INCONCLUSIVE (controls failed: a statement about the tool), and UNTESTED. Controls must match the target’s scale: validating at the wrong text length proves nothing, and the dangerous direction is licensing a claim the control never tested.&lt;/p&gt;
&lt;h2 id=&quot;3-liber-primus-falsification-not-solving&quot;&gt;3. Liber Primus: Falsification, Not Solving&lt;/h2&gt;
&lt;h3 id=&quot;31-what-is-being-tested&quot;&gt;3.1 What Is Being Tested&lt;/h3&gt;





























































&lt;table&gt;&lt;thead&gt;&lt;tr&gt;&lt;th&gt;Hypothesis&lt;/th&gt;&lt;th&gt;Key space&lt;/th&gt;&lt;th&gt;Method&lt;/th&gt;&lt;th&gt;Coverage&lt;/th&gt;&lt;th&gt;Verdict&lt;/th&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;Unenciphered plaintext&lt;/td&gt;&lt;td&gt;1&lt;/td&gt;&lt;td&gt;IC against the plaintext null (Figure 1)&lt;/td&gt;&lt;td&gt;Exhaustive&lt;/td&gt;&lt;td&gt;REFUTED&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Affine substitution&lt;/td&gt;&lt;td&gt;812&lt;/td&gt;&lt;td&gt;Full enumeration&lt;/td&gt;&lt;td&gt;Exhaustive&lt;/td&gt;&lt;td&gt;REFUTED&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;General monoalphabetic substitution&lt;/td&gt;&lt;td&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;!&lt;/mo&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;8.8&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;29! \approx 8.8\times 10^{30}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;29&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;8.8&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/td&gt;&lt;td&gt;IC invariant + hill-climb&lt;/td&gt;&lt;td&gt;Exhaustive via invariant&lt;/td&gt;&lt;td&gt;REFUTED&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Pure transposition&lt;/td&gt;&lt;td&gt;161&lt;/td&gt;&lt;td&gt;Full enumeration&lt;/td&gt;&lt;td&gt;Exhaustive&lt;/td&gt;&lt;td&gt;REFUTED&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Deterministic number-theoretic keystream&lt;/td&gt;&lt;td&gt;6,144&lt;/td&gt;&lt;td&gt;Full enumeration&lt;/td&gt;&lt;td&gt;Exhaustive&lt;/td&gt;&lt;td&gt;REFUTED&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Periodic Vigenère / Beaufort&lt;/td&gt;&lt;td&gt;up to &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;87.7&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;29^{60} \approx 10^{87.7}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;87.7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/td&gt;&lt;td&gt;Exact per-coset &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\chi^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0085em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;χ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/td&gt;&lt;td&gt;Exhaustive to &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L = 60&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/td&gt;&lt;td&gt;REFUTED&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Running key (natural-language text)&lt;/td&gt;&lt;td&gt;Unbounded&lt;/td&gt;&lt;td&gt;Direct search + convolution IC&lt;/td&gt;&lt;td&gt;Partial / class-level&lt;/td&gt;&lt;td&gt;REFUTED&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;
&lt;p&gt;Two exclusions are exact without enumeration. Monoalphabetic substitution has &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;!&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;29!&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;29&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;!&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; keys, but IC is invariant under any bijection on the alphabet: one measurement adjudicates all &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;8.8&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;8.8\times 10^{30}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;8.8&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; keys at once. Periodic polyalphabetic ciphers with period &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; reduce to &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; independent Caesar cosets; recovering the best shift per coset costs 29 evaluations each, so the full sweep to &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L = 60&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; over both modes is about 106,000 evaluations (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/msubsup&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;106,140&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;2 \times 29 \times \sum_{L=1}^{60} L = 106{,}140&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;29&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2537em;vertical-align:-0.2997em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.954em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mrel mtight&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2029em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2997em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;106&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;140&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;), under a second.&lt;/p&gt;
&lt;h3 id=&quot;32-null-models-excluding-classes-without-searching-them&quot;&gt;3.2 Null Models: Excluding Classes Without Searching Them&lt;/h3&gt;
&lt;p&gt;For each generative hypothesis, synthetic ciphertexts are simulated at the true text length (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;11,081&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;n = 11{,}081&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;081&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;) and the same statistic (IC) is measured. The observed &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.03449&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC} = 0.03449&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.03449&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is compared against the reference distribution. Compatibility requires both &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;∣&lt;/mi&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;∣&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;|z| \le 4&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≤&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and the observation inside the range the null actually produced, because periodic nulls are right-skewed and a modest &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; can hide an observation below every simulated draw.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/criptografia/measuring-the-unsolvable/fig-1.webp&quot; alt=&quot;Bar chart of the index of coincidence of nine generative null models at 11,081 runes, with a dashed line at the observed value 0.03449. Plaintext and monoalphabetic nulls are near 0.061; periodic keys with L=5, 12 and 30 and the running key lie between 0.035 and 0.040; one-time pad, uniform and shuffle of the observed text sit on the dashed line.&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 1.&lt;/strong&gt; Index of coincidence of generative null models at &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;11,081&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N = 11{,}081&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;081&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; runes (mean ± SD over 200 trials). The observed value (0.03449) is incompatible with plaintext, monoalphabetic, periodic, and running-key models, and compatible with one-time pad, uniform, and positional shuffle of itself.&lt;/p&gt;
&lt;h3 id=&quot;33-the-running-key-convolution-argument&quot;&gt;3.3 The Running-Key Convolution Argument&lt;/h3&gt;
&lt;p&gt;A running key enciphers plaintext under a key drawn from another natural-language text. Let &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;P&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; be the plaintext distribution and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;K&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; the key distribution over &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;Z&lt;/mi&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathbb{Z}_{29}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathbb&quot;&gt;Z&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;29&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. Under additive combination, the ciphertext distribution &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;C&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is the circular convolution &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;⊛&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;P \circledast K&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7653em;vertical-align:-0.082em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin amsrm&quot;&gt;⊛&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. In Fourier terms &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat C(j) = \hat P(j)\,\hat K(j)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1944em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, and since &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC} - 1/n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1/&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is proportional to the squared &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; norm of the non-constant part:&lt;/p&gt;
&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; minsize=&quot;1.2em&quot; maxsize=&quot;1.2em&quot;&gt;∣&lt;/mo&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; minsize=&quot;1.2em&quot; maxsize=&quot;1.2em&quot;&gt;∣&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; minsize=&quot;1.2em&quot; maxsize=&quot;1.2em&quot;&gt;∣&lt;/mo&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; minsize=&quot;1.2em&quot; maxsize=&quot;1.2em&quot;&gt;∣&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;.&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC}(C) - \frac{1}{n} = \frac{1}{n} \sum_{j=1}^{n-1} \bigl|\hat P(j)\bigr|^2 \bigl|\hat K(j)\bigr|^2 .&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:2.0074em;vertical-align:-0.686em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.3214em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.677em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.686em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:3.2149em;vertical-align:-1.4138em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.3214em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.677em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.686em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop op-limits&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.8011em;&quot;&gt;&lt;span style=&quot;top:-1.8723em;margin-left:0em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.05em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mrel mtight&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.05em;&quot;&gt;&lt;/span&gt;&lt;span&gt;&lt;span class=&quot;mop op-symbol large-op&quot;&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-4.3em;margin-left:0em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.05em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.4138em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;&lt;span class=&quot;delimsizing mult&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.85em;&quot;&gt;&lt;span style=&quot;top:-2.85em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.2em;&quot;&gt;&lt;/span&gt;&lt;span style=&quot;width:0.333em;height:1.2em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;0.333em&quot; height=&quot;1.2em&quot; viewBox=&quot;0 0 333 1200&quot;&gt;&lt;path d=&quot;M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.35em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;delimsizing mult&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.85em;&quot;&gt;&lt;span style=&quot;top:-2.85em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.2em;&quot;&gt;&lt;/span&gt;&lt;span style=&quot;width:0.333em;height:1.2em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;0.333em&quot; height=&quot;1.2em&quot; viewBox=&quot;0 0 333 1200&quot;&gt;&lt;path d=&quot;M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.35em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.054em;&quot;&gt;&lt;span style=&quot;top:-3.3029em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;&lt;span class=&quot;delimsizing mult&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.85em;&quot;&gt;&lt;span style=&quot;top:-2.85em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.2em;&quot;&gt;&lt;/span&gt;&lt;span style=&quot;width:0.333em;height:1.2em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;0.333em&quot; height=&quot;1.2em&quot; viewBox=&quot;0 0 333 1200&quot;&gt;&lt;path d=&quot;M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.35em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1944em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;delimsizing mult&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.85em;&quot;&gt;&lt;span style=&quot;top:-2.85em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.2em;&quot;&gt;&lt;/span&gt;&lt;span style=&quot;width:0.333em;height:1.2em;&quot;&gt;&lt;svg xmlns=&quot;http://www.w3.org/2000/svg&quot; width=&quot;0.333em&quot; height=&quot;1.2em&quot; viewBox=&quot;0 0 333 1200&quot;&gt;&lt;path d=&quot;M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15
c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15
c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z&quot;&gt;&lt;/path&gt;&lt;/svg&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.35em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.054em;&quot;&gt;&lt;span style=&quot;top:-3.3029em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;.&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;p&gt;Every term is non-negative. The excess over uniform is zero if and only if, at every non-zero frequency &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;j&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.854em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, at least one of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat P(j)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat K(j)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1944em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; vanishes. A natural-language letter distribution does not have vanishing Fourier coefficients, so convolution flattens the distribution but cannot make it uniform. The observed &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.03449&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC} = 0.03449&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.03449&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; lies &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;10.7&lt;/mn&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;10.7\sigma&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;10.7&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;σ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; below the running-key null (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;0.03621&lt;/mn&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;mn&gt;0.00016&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;0.03621 \pm 0.00016&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.03621&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;±&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.00016&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;; in 200 trials the statistic never fell below &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;0.03576&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;0.03576&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.03576&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;).&lt;/p&gt;
&lt;h3 id=&quot;34-why-the-machine-learning-architecture-failed&quot;&gt;3.4 Why the Machine-Learning Architecture Failed&lt;/h3&gt;
&lt;p&gt;The previous version of the Liber Primus codebase (~3,200 lines) implemented an island-model genetic algorithm, a MAP-Elites quality-diversity archive, CMA-ES, AlphaZero-style MCTS, a PPO agent with a Transformer actor-critic, and a neural language-model critic, all searching a composable cipher DSL. None produced a usable result.&lt;/p&gt;
&lt;p&gt;The decisive measurement: on section 0.0 (already solved, Atbash cipher), beam search returned a cyclic rotation of the true plaintext and scored it &lt;strong&gt;higher&lt;/strong&gt; than ground truth (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;0.331&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-0.331&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.331&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; vs &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;0.386&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-0.386&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.386&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;). A cyclic rotation leaves unigrams and IC exactly invariant, and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N-6&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; of the &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N-3&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; quadgrams survive unchanged (only the three quadgrams that straddled the cut are replaced by three that straddle the new junction), so there are &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; near-equally-scoring decoys per genuine solution. No score can certify a decryption. The MAP-Elites archive collapsed to a single behavioural cell on real data; the RL agent controlled the operation name while the environment randomised the arguments (the entire problem for Vigenère); the COLUMNAR primitive was silently lossy while flagged invertible; and 19 tests passed because they asserted output length, not recovery.&lt;/p&gt;
&lt;p&gt;The general lesson: more optimisation power applied to an unvalidated objective produces more confident wrong answers. For an unsolved cipher, the scarce resource is not compute, it is falsifiability. ML has a place outside the scoring loop, as hypothesis generator and corpus builder whose output a human reviews; never as a judge inside it.&lt;/p&gt;
&lt;h3 id=&quot;35-homophonic-substitution-analysis-subsequent-work&quot;&gt;3.5 Homophonic Substitution Analysis (subsequent work)&lt;/h3&gt;
&lt;p&gt;All exclusions above rely on the IC invariant or its per-coset generalisation, which apply only to bijective symbol mappings. Homophonic/nomenclator substitution, where multiple ciphertext symbols map to one plaintext letter, flattens the unigram distribution before combination and evades the IC test entirely. A variational inference attack learns a soft assignment &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;∣&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;Q(z \mid x)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;Q&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; from ciphertext runes to plaintext letters by minimising KL divergence between the decoded bigram distribution and a Gematria-normalised English reference model, calibrated against uniform-random nulls at the true text length.&lt;/p&gt;





























































&lt;table&gt;&lt;thead&gt;&lt;tr&gt;&lt;th&gt;Section&lt;/th&gt;&lt;th&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo lspace=&quot;0em&quot; rspace=&quot;0em&quot;&gt;∗&lt;/mo&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;F^{*}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6887em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6887em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;∗&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/th&gt;&lt;th&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mu_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;μ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/th&gt;&lt;th&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\sigma_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;σ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/th&gt;&lt;th&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/th&gt;&lt;th&gt;Verdict&lt;/th&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;0.7 (roots)&lt;/td&gt;&lt;td&gt;1.99&lt;/td&gt;&lt;td&gt;3.89&lt;/td&gt;&lt;td&gt;0.14&lt;/td&gt;&lt;td&gt;−13.62&lt;/td&gt;&lt;td&gt;COMPATIBLE&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;0.8 (moebius)&lt;/td&gt;&lt;td&gt;3.32&lt;/td&gt;&lt;td&gt;3.20&lt;/td&gt;&lt;td&gt;0.18&lt;/td&gt;&lt;td&gt;+0.65&lt;/td&gt;&lt;td&gt;INCONCLUSIVE&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;0.9 (dots/mayfly)&lt;/td&gt;&lt;td&gt;2.35&lt;/td&gt;&lt;td&gt;3.55&lt;/td&gt;&lt;td&gt;0.29&lt;/td&gt;&lt;td&gt;−4.22&lt;/td&gt;&lt;td&gt;COMPATIBLE&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;0.10 (wing)&lt;/td&gt;&lt;td&gt;3.41&lt;/td&gt;&lt;td&gt;4.69&lt;/td&gt;&lt;td&gt;0.28&lt;/td&gt;&lt;td&gt;−4.57&lt;/td&gt;&lt;td&gt;COMPATIBLE&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;0.11 (cuneiform)&lt;/td&gt;&lt;td&gt;7.96&lt;/td&gt;&lt;td&gt;4.27&lt;/td&gt;&lt;td&gt;0.15&lt;/td&gt;&lt;td&gt;+24.60&lt;/td&gt;&lt;td&gt;EXCLUDED&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;0.12 (plants)&lt;/td&gt;&lt;td&gt;3.32&lt;/td&gt;&lt;td&gt;4.28&lt;/td&gt;&lt;td&gt;1.12&lt;/td&gt;&lt;td&gt;−0.86&lt;/td&gt;&lt;td&gt;INCONCLUSIVE&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;
&lt;p&gt;Here &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo lspace=&quot;0em&quot; rspace=&quot;0em&quot;&gt;∗&lt;/mo&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;F^{*}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6887em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6887em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;∗&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is the optimised objective for the section, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mu_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;μ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\sigma_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;σ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; the mean and standard deviation of the uniform-random null, and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo lspace=&quot;0em&quot; rspace=&quot;0em&quot;&gt;∗&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z = (F^{*} - \mu_0)/\sigma_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6887em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;∗&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;μ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;σ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. The verdicts follow the sign and size of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;: strongly negative is compatible with homophonic English, near zero is inconclusive, strongly positive is excluded.&lt;/p&gt;
&lt;p&gt;The hard assignments collapse to a single letter for several sections, a known pathology when the signal is too weak to resolve individual mappings. Nevertheless, the &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;-scores span &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;13.6&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;24.6&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;[-13.6, +24.6]&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;13.6&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;24.6&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;]&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. Treating the six sections as sharing one cipher, the dispersion of the &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;-scores about their mean gives &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;ˉ&lt;/mo&gt;&lt;/mover&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;830&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;Q = \sum_i (z_i - \bar z)^2 \approx 830&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8778em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;Q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0497em;vertical-align:-0.2997em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.162em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2997em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.044em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0641em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.5678em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1944em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;ˉ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;830&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on 5 degrees of freedom, a probability below &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;170&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;10^{-170}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;170&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; under homogeneity (Appendix A.5). The unsolved Liber Primus pages are not a homogeneous cipher; section-specific analysis is necessary, and pooled analysis is invalid for classes not closed under concatenation.&lt;/p&gt;
&lt;h2 id=&quot;4-voynich-manuscript-measurement-against-nulls&quot;&gt;4. Voynich Manuscript: Measurement Against Nulls&lt;/h2&gt;
&lt;h3 id=&quot;41-data-and-null-models&quot;&gt;4.1 Data and Null Models&lt;/h3&gt;
&lt;p&gt;Results use the ZL (Zandbergen–Landini) v3b-n IVTFF transcription: 227 pages, 5,385 loci, 37,953 word tokens, 8,423 types, 193,023 glyphs. The independent Takahashi/Interlinear (IT2a-n) transcription (225 pages, 36,907 tokens) serves as control; high-frequency word counts agree within ~1% (&lt;em&gt;daiin&lt;/em&gt; 798 vs 805). Four null models carry no lexicon, grammar, or semantics, ordered by retained structure: uniform random + spaces; character-shuffled (preserves glyph inventory and word lengths exactly); order-1 character Markov (retains adjacent-glyph statistics); order-2 character Markov (retains two-glyph context).&lt;/p&gt;
&lt;h3 id=&quot;42-zipfs-law-decides-nothing&quot;&gt;4.2 Zipf’s Law Decides Nothing&lt;/h3&gt;
&lt;p&gt;Voynichese fits a rank-frequency power law with exponent 1.022 and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.979&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;R^2 = 0.979&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0077em;&quot;&gt;R&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.979&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. The order-2 Markov chain yields exponent 1.064 and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.985&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;R^2 = 0.985&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0077em;&quot;&gt;R&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.985&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;: at least as Voynich-like on every axis, with no words, no meaning, and 676 states of memory. The naive “monkey” null (uniform + spaces) is distinguishable (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.838&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\alpha = 0.838&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0037em;&quot;&gt;α&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.838&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.818&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;R^2 = 0.818&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0077em;&quot;&gt;R&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.818&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;), so a critique resting only on the monkey model would be weak; but the Markov chain settles it. Zipf compliance is excluded from the discriminative battery.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/criptografia/measuring-the-unsolvable/fig-2.webp&quot; alt=&quot;Log-log plot of word frequency against rank. The Voynich curve (alpha 1.022, R squared 0.979) and the order-2 Markov curve (alpha 1.064, R squared 0.985) almost overlap; the uniform-plus-spaces curve (alpha 0.838, R squared 0.818) is visibly flatter.&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 2.&lt;/strong&gt; Zipf rank-frequency curves. Voynichese (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1.022&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\alpha = 1.022&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0037em;&quot;&gt;α&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1.022&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;) is closely matched by an order-2 Markov chain fitted to its own glyphs (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1.064&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\alpha = 1.064&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0037em;&quot;&gt;α&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1.064&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;); uniform random text with spaces (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.838&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\alpha = 0.838&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0037em;&quot;&gt;α&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.838&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;) is visibly different. Zipf compliance is therefore non-discriminative.&lt;/p&gt;
&lt;h3 id=&quot;43-the-discriminative-battery&quot;&gt;4.3 The Discriminative Battery&lt;/h3&gt;
&lt;p&gt;A battery of statistics is computed for the manuscript and all four nulls. A statistic is admitted only if it separates the manuscript from the nulls; a statistic that fails is reported as failing, which is itself a result about the literature that cites it.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/criptografia/measuring-the-unsolvable/fig-3.webp&quot; alt=&quot;Three bar-chart panels comparing Voynich with four null models on h1, h2 and type-token ratio. h1 is about 3.86 bits for all but the uniform null; h2 is 2.121 for Voynich against 2.117 and 2.114 for the Markov nulls; type-token ratio is 0.222 for Voynich against 0.324 and 0.245 for the Markov nulls.&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 3.&lt;/strong&gt; Discriminative battery. &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_1&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, and TTR for Voynich (observed) vs structure-retaining null models. An order-1 Markov chain already reproduces &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, rigidity, mean word length, and adjacent repetition; only word-length variance and TTR separate them, and order-2 narrows both.&lt;/p&gt;
&lt;p&gt;Conditional entropy &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2.121&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2 = 2.121&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2.121&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; bits for Voynich against 2.117 for order-1 Markov and 2.114 for order-2 Markov. The character-shuffled control at 3.862 bits shows how much predictability is real sequential structure. The correct inference is narrow: these statistics carry very little information about whether the manuscript is meaningful. Meaningful text also has low-order structure, which is why low-order statistics are weak evidence in both directions.&lt;/p&gt;
&lt;h3 id=&quot;44-two-findings-that-survive&quot;&gt;4.4 Two Findings That Survive&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Currier A and B are distinct populations.&lt;/strong&gt; Currier observed in 1976 that the manuscript is written in two statistically distinct systems. Measured here: A has &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2.151&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2 = 2.151&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2.151&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, TTR &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.319&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;= 0.319&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.3669em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.319&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;; B has &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1.937&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2 = 1.937&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1.937&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, TTR &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.215&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;= 0.215&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.3669em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.215&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. With 100 null trials each, the differences on &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, rigidity, and TTR are decisive (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;∣&lt;/mi&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;∣&lt;/mi&gt;&lt;mo&gt;≫&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;|z| \gg 2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≫&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;); word-length variance and adjacent repetition are compatible with a single process. Any account must explain both systems; any statistic over the whole manuscript silently averages two populations.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/criptografia/measuring-the-unsolvable/fig-4.webp&quot; alt=&quot;Grouped bar chart comparing Currier A and B on h1, h2, rigidity and type-token ratio. The two hands differ visibly on h2 and type-token ratio, and only slightly on h1 and rigidity.&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 4.&lt;/strong&gt; Currier A vs B on discriminative statistics. A and B differ decisively on &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, rigidity, and TTR, but not on word-length variance or adjacent repetition.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;The line is a functional unit.&lt;/strong&gt; Natural-language text does not know where the scribe will break a line. A substitution cipher acts glyph by glyph without reference to layout; a transposition permutes a stream fixed before the lines existed. Measuring the total-variation distance between the first-glyph distribution of line-initial words and all other words: &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;T&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.385&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{TV} = 0.385&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot; style=&quot;margin-right:0.0139em;&quot;&gt;TV&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.385&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; for Voynich, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;0.031&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;0.031&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.031&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; for a character-shuffled control, a 12.3× ratio. At the word level, first-word &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;T&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.681&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{TV} = 0.681&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot; style=&quot;margin-right:0.0139em;&quot;&gt;TV&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.681&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; vs &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;0.029&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;0.029&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.029&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, a 23.3× ratio. Paragraph-initial lines form an even more distinct subpopulation (first glyphs {p, t, k, f, o} vs mid-line {d, s, y, q, o}). This is incompatible with any substitution or transposition cipher of a natural language where the plaintext is produced independently of the page format.&lt;/p&gt;
&lt;h3 id=&quot;45-alternative-tokenisation&quot;&gt;4.5 Alternative Tokenisation&lt;/h3&gt;
&lt;p&gt;All statistical conclusions are conditional on the EVA tokenisation. Composite tokenisation (merging &lt;em&gt;ch&lt;/em&gt;, &lt;em&gt;sh&lt;/em&gt;, &lt;em&gt;cth&lt;/em&gt; into single tokens) raises conditional entropy &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; from 2.081 to 2.224. Merging symbols changes the alphabet, so the new value cannot be compared with the order-2 null fitted under EVA (2.114): the null has to be refitted under the same composite tokenisation before the comparison means anything (Appendix A.4). What the result does show is that the claim “Voynichese has lower conditional entropy than a 2-gram model of itself” is tokenisation-dependent and not robust. Stroke-based, first-last, and syllabic tokenisations each collapse or distort different statistics, confirming that the tokenisation choice is a modelling assumption, not a neutral formatting step.&lt;/p&gt;
&lt;h3 id=&quot;46-entropy-rate-convergence&quot;&gt;4.6 Entropy Rate Convergence&lt;/h3&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/criptografia/measuring-the-unsolvable/fig-5.webp&quot; alt=&quot;Line plot of entropy rate h_k against context order k from 1 to 5 for Voynich, an order-2 Markov null and an order-1 Markov null. Voynich and the order-2 null coincide at k=1 and k=2, then Voynich falls below the order-2 null at k=3, 4 and 5, ending near 1.16 bits against 1.40.&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 5.&lt;/strong&gt; Entropy rate &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_k&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; against context order &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;k&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. Voynich matches the order-2 Markov null at &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;k = 1, 2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and then falls below it at &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;k = 3, 4, 5&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, which points to dependence beyond two glyphs.&lt;/p&gt;
&lt;p&gt;For natural language, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_k&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; typically converges toward ~1.0–1.5 bits/char and keeps decreasing slowly. For Voynichese, the curve coincides with the Markov-2 null at &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;k \le 2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8304em;vertical-align:-0.136em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≤&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, where the null reproduces the manuscript’s low-order statistics by construction, so the agreement carries no information on its own. At &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;k = 3, 4, 5&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; the manuscript lies below the null (by roughly 0.07, 0.16 and 0.24 bits), and both curves decline partly because plug-in estimates are biased downward as the number of contexts grows toward the text length (Appendix A.3). The null is generated at the manuscript’s length and estimated the same way, so the bias is shared and the gap is not bias: the manuscript carries dependence beyond a two-glyph memory, consistent with the repeated words and word-pair patterns that the literature has long described. The entropy-rate evidence therefore supports the narrow claim of Section 4.3 (an order-2 chain reproduces the statistics of order &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\le 2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7719em;vertical-align:-0.136em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≤&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;) and not the stronger reading that a 2-glyph-memory generative process produced the text.&lt;/p&gt;
&lt;h2 id=&quot;5-artificial-intelligence-what-was-tried-what-failed-what-comes-next&quot;&gt;5. Artificial Intelligence: What Was Tried, What Failed, What Comes Next&lt;/h2&gt;
&lt;h3 id=&quot;51-approaches-tried&quot;&gt;5.1 Approaches Tried&lt;/h3&gt;
&lt;p&gt;Both projects share an AI lineage: a dependency-free linguistic meta-learning workbench based on character 1–4-gram likelihoods, meta-trained with a first-order Reptile update across cipher/key episodes (Caesar shift, affine substitution, periodic Vigenère, plaintext autokey), evaluating fresh keys and held-out windows. The control result: untrained top-1 = 6.25%; meta-initialization top-1 = 100%; scratch + support adaptation = 100%. This is a &lt;strong&gt;control result&lt;/strong&gt;, not manuscript evidence: it shows the feature learner ranks the planted answer when the correct cipher family and candidate set contain it. Scratch adaptation also reaches the ceiling, so no adaptation advantage over ordinary task-local learning is demonstrated. The workbench is deliberately isolated from the verdict lattice: learned scores can propose candidates; only controls, coverage, and independent review can support a historical-cipher claim.&lt;/p&gt;
&lt;h3 id=&quot;52-what-failed&quot;&gt;5.2 What Failed&lt;/h3&gt;
&lt;p&gt;In the Liber Primus v1 codebase, the ML apparatus was not merely unhelpful but actively misleading: its objective function ranked a cyclically rotated plaintext above the true one (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;0.331&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-0.331&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.331&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; vs &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;0.386&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-0.386&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.386&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on an already-solved page); its quality-diversity archive collapsed to one cell; its RL environment randomised the only decision that mattered; it could not recover a 3-symbol Vigenère key that exhaustive enumeration solves in under a second. In the Voynich predecessor, a “structural scorer” was fitted to a 629-character synthetic placeholder because a broken download URL fell back silently, and 21 tests passed throughout. Both failures share one structure: optimisation pressure on an unvalidated objective manufactures confident wrong answers.&lt;/p&gt;
&lt;h3 id=&quot;53-where-ai-belongs&quot;&gt;5.3 Where AI Belongs&lt;/h3&gt;
&lt;p&gt;AI belongs &lt;strong&gt;outside the scoring loop&lt;/strong&gt;: hypothesis generation, corpus building, orthographic normalisation, and candidate ranking whose output a human reviews and commits. AI does not belong &lt;strong&gt;inside the scoring loop&lt;/strong&gt; as a judge, because a learned critic is an objective function, and Sections 3.4 and 5.2 show what happens to objective functions under optimisation pressure on unsolved ciphertexts with no ground truth. Future approaches worth trying: neural scorers as proposal mechanisms with independent statistical verification; language-model priors for homophonic assignment initialisation; transformer-based attention maps over Voynichese line structure as hypothesis generators; and controlled benchmarks where the correct answer is sometimes absent from the candidate set, to measure false-positive rates.&lt;/p&gt;
&lt;h2 id=&quot;6-comparative-discussion&quot;&gt;6. Comparative Discussion&lt;/h2&gt;








































&lt;table&gt;&lt;thead&gt;&lt;tr&gt;&lt;th&gt;Dimension&lt;/th&gt;&lt;th&gt;Liber Primus&lt;/th&gt;&lt;th&gt;Voynich Manuscript&lt;/th&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;Ground truth&lt;/td&gt;&lt;td&gt;Solved pages exist; controls can be built from real plaintext&lt;/td&gt;&lt;td&gt;None; controls would test assumptions, not attacks&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Core method&lt;/td&gt;&lt;td&gt;Control-gated falsification engine; verdict lattice&lt;/td&gt;&lt;td&gt;Discriminative battery against null models&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Headline negative&lt;/td&gt;&lt;td&gt;Seven cipher classes excluded; running key excluded by convolution&lt;/td&gt;&lt;td&gt;Zipf, entropy, rigidity all reproduced by order-2 Markov&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Headline positive&lt;/td&gt;&lt;td&gt;None (no decipherment)&lt;/td&gt;&lt;td&gt;Currier A/B distinct; line-as-unit effect (12–23× over null)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;AI role&lt;/td&gt;&lt;td&gt;Failed search apparatus; later variational homophonic attack&lt;/td&gt;&lt;td&gt;Meta-learning control; tokenisation sensitivity analysis&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;Main limitation&lt;/td&gt;&lt;td&gt;Single community transcription; homophonic not closed under pooling&lt;/td&gt;&lt;td&gt;Single EVA tokenisation; one transcription for main results&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;
&lt;p&gt;Both projects converge on the same methodological insight: in domains without ground truth, &lt;strong&gt;falsifiability and discrimination&lt;/strong&gt; are the scarce resources, not compute or cleverness. A negative result that is trustworthy compounds; a positive result that cannot be checked does not.&lt;/p&gt;
&lt;h2 id=&quot;7-approaches-that-can-be-tried&quot;&gt;7. Approaches That Can Be Tried&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Liber Primus:&lt;/strong&gt; test homophonic substitution under non-English plaintext models (Latin, German, constructed languages); extend the convolution argument to homophonic pre-flattening; run section-specific rather than pooled attacks; investigate composite constructions (substitution → transposition → stream).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Voynich:&lt;/strong&gt; repeat the full battery under alternative tokenisations (completed for composites; extend to stroke/syllabic with proper controls, including nulls refitted under each tokenisation); test quire-level and hand-level splits; implement historically meaningful generative mechanisms (constrained copying, verbose cipher, abbreviation/position encoding) and measure their failure; add external domain review.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Both:&lt;/strong&gt; build benchmarks where the correct answer is sometimes absent; use neural scorers strictly as proposal mechanisms with statistical verification; publish null-calibrated code and data so others can replicate exclusions without repeating failed searches.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2 id=&quot;8-conclusion&quot;&gt;8. Conclusion&lt;/h2&gt;
&lt;p&gt;The Liber Primus unsolved pages are statistically indistinguishable from a one-time pad at &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;11,081&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N = 11{,}081&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;081&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, and seven cipher classes (unenciphered plaintext, monoalphabetic substitution, affine substitution, pure transposition, deterministic keystreams, periodic polyalphabetic keys to &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L = 60&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, and natural-language running keys) are excluded, each with stated coverage and assumptions. A subsequent homophonic analysis reveals per-section heterogeneity: three sections compatible, one excluded, two inconclusive, the first quantitative evidence that the unsolved pages may not be a homogeneous cipher.&lt;/p&gt;
&lt;p&gt;Voynichese follows Zipf’s law, but so does an order-2 Markov chain fitted to its own glyphs; the argument from Zipf compliance to linguistic status does not survive contact with a null model. Conditional entropy, positional rigidity, mean word length, and adjacent repetition are all reproduced by a first-order Markov chain. Two findings survive their controls: Currier A and B are distinct populations, and the line-as-a-functional-unit effect, 12× glyph-level and 23× word-level divergence over a matched control, resists any substitution or transposition cipher of natural language.&lt;/p&gt;
&lt;p&gt;The methodological proposal is modest and cheap: report every statistic next to null models that ought to fail it, and bind every attack to known-answer controls from the same cipher class. In a field where positive claims cannot be checked, a discipline that makes negative claims trustworthy is the only thing that compounds. Neither artefact is solved. Both are better measured.&lt;/p&gt;
&lt;h2 id=&quot;appendix-a-formal-results-and-consistency-checks&quot;&gt;Appendix A. Formal Results and Consistency Checks&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;Added in November 2026. Everything below is derived from definitions or recomputed from numbers already in the paper; it contains no new measurement on the manuscripts. Numerical statements were checked by simulation or direct computation.&lt;/em&gt;&lt;/p&gt;
&lt;h3 id=&quot;a1-the-ic-statistic-under-the-uniform-null-and-what-it-can-detect&quot;&gt;A.1 The IC statistic under the uniform null, and what it can detect&lt;/h3&gt;
&lt;p&gt;For &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; i.i.d. uniform draws over &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; symbols, the plug-in &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; has expectation &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;1/n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1/&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and, to leading order,&lt;/p&gt;
&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;sd&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt;  &lt;/mtext&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mtext&gt;  &lt;/mtext&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;.&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\operatorname{sd}(\mathrm{IC}) \;\approx\; \frac{(n-1)\sqrt{2(n-1)}}{n^{2}\,N}.&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;sd&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:2.316em;vertical-align:-0.686em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.63em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.7401em;&quot;&gt;&lt;span style=&quot;top:-2.989em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.695em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mord sqrt&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; 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&lt;p&gt;For &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;n = 29&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;29&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;11,081&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N = 11{,}081&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;081&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; this is &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;2.25&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;2.25\times 10^{-5}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2.25&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; (a Monte Carlo over 6,000 runs gives &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;2.30&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;2.30\times 10^{-5}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2.30&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and a mean of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;0.034483&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;0.034483 = 1/29&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.034483&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1/29&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;). The observed &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;0.03449&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;0.03449&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.03449&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is therefore &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0.3&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z \approx +0.3&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4831em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; from the one-time-pad value; the rounding of the published four significant digits moves this by about &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;mn&gt;0.2&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\pm 0.2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;±&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. The statement “statistically indistinguishable from a one-time pad” is this computation.&lt;/p&gt;
&lt;p&gt;For a periodic key of period &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; with independent uniform shifts, pairs of positions in the same coset collide with the plaintext probability and all other pairs with &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;1/n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1/&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on average, so&lt;/p&gt;
&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;E&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;mtext&gt;  &lt;/mtext&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mtext&gt;  &lt;/mtext&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mfrac&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;.&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathbb{E}[\mathrm{IC}] \;\approx\; \frac{1}{n} + \frac{\mathrm{IC}_P - 1/n}{L}.&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathbb&quot;&gt;E&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:2.0074em;vertical-align:-0.686em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.3214em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.677em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.686em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:2.113em;vertical-align:-0.686em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.427em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.677em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3283em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1/&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.686em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;.&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;p&gt;With &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;I&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;0.061&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathrm{IC}_P \approx 0.061&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathrm&quot;&gt;IC&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3283em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.061&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; (the plaintext bar of Figure 1) this predicts 0.0398, 0.0367 and 0.0354 for &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L = 5, 12, 30&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;12&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, against 0.0397, 0.0367 and 0.0354 read off Figure 1. The same formula says the aggregate IC alone separates a random periodic key from the uniform null at &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;4\sigma&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;σ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; only while &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;≲&lt;/mo&gt;&lt;mn&gt;295&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L \lesssim 295&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.9592em;vertical-align:-0.2296em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel amsrm&quot;&gt;≲&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;295&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. The per-coset sweep to &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L = 60&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is therefore the exact, key-by-key complement to an aggregate test whose reach for random keys is about five times longer; periods beyond roughly 295 are not excluded by either.&lt;/p&gt;
&lt;h3 id=&quot;a2-the-convolution-identity-its-exact-condition-and-a-counterexample&quot;&gt;A.2 The convolution identity, its exact condition, and a counterexample&lt;/h3&gt;
&lt;p&gt;Let &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;msup&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat P(j) = \sum_k P(k)\,e^{-2\pi i jk/n}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1877em;vertical-align:-0.2997em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1864em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2997em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;e&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.888em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0359em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0572em;&quot;&gt;ij&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. Parseval gives &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mstyle scriptlevel=&quot;0&quot; displaystyle=&quot;false&quot;&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mstyle&gt;&lt;msub&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/msub&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;∣&lt;/mi&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;∣&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\sum_k C(k)^2 = \tfrac{1}{n}\sum_{j}|\hat C(j)|^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1138em;vertical-align:-0.2997em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1864em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2997em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.3826em;vertical-align:-0.4358em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8451em;&quot;&gt;&lt;span style=&quot;top:-2.655em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.394em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.345em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.162em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.4358em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; with &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat C(0) = 1&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat C(j) = \hat P(j)\hat K(j)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1944em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; for &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;⊛&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;C = P \circledast K&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7653em;vertical-align:-0.082em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin amsrm&quot;&gt;⊛&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, which yields the identity of Section 3.3. A numerical check on random distributions over &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;Z&lt;/mi&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathbb{Z}_{29}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathbb&quot;&gt;Z&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;29&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; agrees to machine precision.&lt;/p&gt;
&lt;p&gt;The identity gives the exact condition for a convolution to be uniform: for every &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo mathvariant=&quot;normal&quot;&gt;≠&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;j \ne 0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;&lt;span class=&quot;mrel&quot;&gt;&lt;span class=&quot;mord vbox&quot;&gt;&lt;span class=&quot;thinbox&quot;&gt;&lt;span class=&quot;rlap&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;inner&quot;&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mrel&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;fix&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace nobreak&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat P(j) = 0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1667em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; or &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat K(j) = 0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1968em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9468em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2523em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1944em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0572em;&quot;&gt;j&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. This is weaker than “one of the two is uniform”. For example, over &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;Z&lt;/mi&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathbb{Z}_{29}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathbb&quot;&gt;Z&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;29&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; take &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;∝&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0.2&lt;/mn&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;P(k) \propto 1 + 0.2\cos(2\pi k/29)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;∝&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;/29&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;∝&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0.2&lt;/mn&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;K(k) \propto 1 + 0.2\cos(6\pi k/29)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;K&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;∝&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;π&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;/29&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. Both are strictly positive and visibly non-uniform, yet their convolution is exactly uniform, because their spectra live on disjoint frequencies. Natural-language distributions are not like this: their Fourier coefficients do not vanish, so the excess is strictly positive, and the simulated null of Section 3.3 supplies its size (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;0.03621&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;0.03448&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1.7&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;0.03621 - 0.03448 = 1.7\times 10^{-3}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.03621&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.03448&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1.7&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;).&lt;/p&gt;
&lt;p&gt;One further caveat. The identity is exact for collisions between positions whose symbols are independent. Real plaintext and key texts are correlated at short lags, which changes the collision probability for pairs within a few characters of each other. Those pairs are a fraction of about &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;ℓ&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;2\ell/N&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;ℓ&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; of all pairs for correlation length &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\ell&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;ℓ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;; with &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;ℓ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\ell = 5&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;ℓ&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;11,081&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N = 11{,}081&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;081&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; that is under &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;10^{-3}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, so the bias is of order &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;10^{-4}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; at most, an order of magnitude below the &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;1.7&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;1.7\times 10^{-3}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1.7&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; that separates the running-key null from the observation. The empirical null already contains the effect.&lt;/p&gt;
&lt;h3 id=&quot;a3-estimator-bias-and-the-reach-of-hkh_khk&quot;&gt;A.3 Estimator bias and the reach of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_k&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/h3&gt;
&lt;p&gt;The Miller–Madow correction adds &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mi&gt;ln&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;(\hat m - 1)/(2N\ln 2)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.25em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;ln&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; bits to the plug-in entropy, where &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;^&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\hat m&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.25em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is the number of occupied cells. For the 676 contexts of a bigram table on &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;193,023&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N = 193{,}023&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;193&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;023&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; glyphs this is &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;0.0025&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;0.0025&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.0025&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; bits, negligible, which is why the corrected and raw &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; agree. The number of contexts grows geometrically with the context length, by a factor of about 26 per added glyph for a 26-glyph alphabet, and passes &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; within a few steps (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;26&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;4.6&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;1.9&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;26^4 \approx 4.6\times 10^5 &gt; 1.9\times 10^5&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4.6&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8532em;vertical-align:-0.0391em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1.9&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;). Plug-in estimates of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_k&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; are then dominated by the finite sample, which is why every curve in Figure 5 falls. Only the comparison between the manuscript and a null of the same length, estimated the same way, is meaningful, and that comparison is the one made in Section 4.6.&lt;/p&gt;
&lt;h3 id=&quot;a4-why-the-composite-tokenisation-comparison-needs-a-refitted-null&quot;&gt;A.4 Why the composite-tokenisation comparison needs a refitted null&lt;/h3&gt;
&lt;p&gt;Entropy depends on the alphabet. Merging &lt;em&gt;ch&lt;/em&gt;, &lt;em&gt;sh&lt;/em&gt;, &lt;em&gt;cth&lt;/em&gt; into single tokens replaces several symbols by one, changing both the number of symbols and the conditional distribution. A value of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; computed in the composite alphabet (2.224) and a null fitted in the EVA alphabet (2.114) are statistics of different random variables. The comparison is valid only against a Markov-2 chain fitted to the composite-tokenised text and sampled in the same alphabet. This is a finite computation on the same transcription and is listed under Section 7.&lt;/p&gt;
&lt;h3 id=&quot;a5-heterogeneity-across-the-six-sections&quot;&gt;A.5 Heterogeneity across the six sections&lt;/h3&gt;
&lt;p&gt;If all six sections came from one cipher, their &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;-scores would share a common value up to noise, and the statistic &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;ˉ&lt;/mo&gt;&lt;/mover&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;Q = \sum_i (z_i - \bar z)^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8778em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;Q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0497em;vertical-align:-0.2997em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.162em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2997em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.044em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0641em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.5678em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1944em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;ˉ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; would follow a &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\chi^2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0085em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;χ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; distribution with &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;5&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; degrees of freedom. From the table in Section 3.5, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mover accent=&quot;true&quot;&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo&gt;ˉ&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0.33&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\bar z = 0.33&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5678em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord accent&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.5678em;&quot;&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;accent-body&quot; style=&quot;left:-0.1944em;&quot;&gt;&lt;span class=&quot;mord&quot;&gt;ˉ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.33&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;829.9&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;Q = 829.9&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8778em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;Q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;829.9&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. The survival function of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msubsup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\chi^2_5&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.0622em;vertical-align:-0.2481em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;χ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-2.4519em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2481em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; at that value is about &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;176&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;10^{-176}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;176&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. The &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;-scores come from finite empirical nulls and are only approximately normal, so the exponent should not be taken literally, but no plausible departure from normality brings the probability near conventional thresholds. The recomputed &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; for each row, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo lspace=&quot;0em&quot; rspace=&quot;0em&quot;&gt;∗&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;(F^{*} - \mu_0)/\sigma_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6887em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;∗&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;μ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;σ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, agrees with the printed value within rounding: &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;13.57&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-13.57&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;13.57&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0.67&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;+0.67&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.67&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4.14&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-4.14&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4.14&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4.57&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-4.57&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4.57&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;24.60&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;+24.60&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;24.60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;0.86&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-0.86&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.86&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;.&lt;/p&gt;
&lt;h3 id=&quot;a6-a-rotation-costs-at-most-a-handful-of-grams&quot;&gt;A.6 A rotation costs at most a handful of grams&lt;/h3&gt;
&lt;p&gt;Take a text of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; symbols and cut and rotate it at any point. For &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;q&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;-grams, the rotated text keeps exactly &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N - 2(q-1)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; of the original &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N - (q-1)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; grams; only the &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;q-1&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7778em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; grams that straddled the cut are lost and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;q-1&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7778em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; new ones appear at the junction (verified by direct count for &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;q = 2, 3, 4&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8389em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;). For &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;q = 4&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;q&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; this is &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N - 6&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N - 3&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. If per-gram log-probabilities lie in a range of width &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\Delta&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;Δ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, the total log-likelihood of a rotation differs from that of the original by at most &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;3\Delta&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3Δ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, and the per-gram average by at most &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;Δ&lt;/mi&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;3\Delta/(N-3)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3Δ/&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, a quantity that vanishes as &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;/&lt;/mi&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;1/N&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1/&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. A scorer built on such statistics cannot reliably tell a rotation from the truth on a page of any length; the observed reversal (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;0.331&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-0.331&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.331&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; vs &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;0.386&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;-0.386&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.386&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;) is the expected behaviour of the objective, not a fluke of the search.&lt;/p&gt;
&lt;h2 id=&quot;revision-notes&quot;&gt;Revision notes&lt;/h2&gt;
&lt;p&gt;Changes made to the text of the preprint for this web version (November 2026), so that readers of the original PDF can see exactly what moved:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Abstract and Section 4.4:&lt;/strong&gt; the glyph-level ratio is stated as 12× (the body gives 12.3×); the abstract of the PDF said 13×.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Section 3.1:&lt;/strong&gt; added the row “Unenciphered plaintext”, which the abstract and conclusion count among the seven excluded classes but the table omitted. Key-space entries that were unreadable in the PDF were filled from the definitions (&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;!&lt;/mo&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;8.8&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;29! \approx 8.8\times 10^{30}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;29&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7278em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;8.8&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;×&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;29&lt;/mn&gt;&lt;mn&gt;60&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mn&gt;87.7&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\sum_{L\le 60} 29^{L} \approx 29^{60} \approx 10^{87.7}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2362em;vertical-align:-0.3949em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1786em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mrel mtight&quot;&gt;≤&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3949em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8413em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;L&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;60&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8141em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;87.7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Sections 2.1 and 3.3:&lt;/strong&gt; the condition for a convolution to be uniform was stated as “one of the two distributions is exactly uniform”; the exact condition is disjoint spectral support (Appendix A.2). The conclusion for natural-language keys is unchanged.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Section 3.4:&lt;/strong&gt; ”&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N-4&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N-3&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; quadgrams survive” is &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N-6&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N-3&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; (Appendix A.6). The &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N-4&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;4&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; figure holds for trigrams, out of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N-2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.7667em;vertical-align:-0.0833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Section 3.5:&lt;/strong&gt; the unreadable column header and probability bound were replaced by the definitions of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo lspace=&quot;0em&quot; rspace=&quot;0em&quot;&gt;∗&lt;/mo&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;F^{*}&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6887em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.6887em;&quot;&gt;&lt;span style=&quot;top:-3.063em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;∗&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mu_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;μ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\sigma_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;σ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and by the heterogeneity statistic of Appendix A.5. The verdict convention (sign of &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;z&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.044em;&quot;&gt;z&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;) is stated explicitly; it is read off the table and should be confirmed against the code.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Section 4.5:&lt;/strong&gt; removed the claim that composite tokenisation “moves &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; above the order-2 null”, which compared values from different alphabets (Appendix A.4).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Section 4.6 and Figure 5 caption:&lt;/strong&gt; the text said the entropy rate converges toward the Markov-2 null; Figure 5 shows agreement only at &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;k \le 2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8304em;vertical-align:-0.136em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≤&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and a widening gap below the null at &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;k = 3&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;–5. The paragraph now says that.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Appendix A&lt;/strong&gt; was added.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;Open points for the author, not changed here: the definitions of the “pure transposition” (161 keys) and “deterministic number-theoretic keystream” (6,144 keys) classes are not given in the text; the baseline &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;h_2&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;h&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is 2.121 in Section 4.3 and 2.081 in Section 4.5 and Figure 5, and the difference is not explained; and the epigraph-style references to the companion manuscripts (2026a–d) are self-published and not linked.&lt;/p&gt;
&lt;h2 id=&quot;references&quot;&gt;References&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;Friedman, W. F. (1922). &lt;em&gt;The Index of Coincidence and Its Applications in Cryptography.&lt;/em&gt; Riverbank Publications, 22.&lt;/li&gt;
&lt;li&gt;Knuth, D. E. (1994). A Decryption of a Homophonic Cipher. &lt;em&gt;Cryptologia&lt;/em&gt;, 18(3), 267–284.&lt;/li&gt;
&lt;li&gt;Sinkov, A. (1966). &lt;em&gt;Elementary Cryptanalysis: A Mathematical Approach.&lt;/em&gt; Mathematical Association of America.&lt;/li&gt;
&lt;li&gt;Currier, P. (1976). Statistical Analysis of the Voynich Manuscript. Unpublished manuscript.&lt;/li&gt;
&lt;li&gt;Zandbergen, R. &amp;#x26; Landini, G. (2020). Voynich Manuscript Transcriptions, ZL3b-n. voynich.nu.&lt;/li&gt;
&lt;li&gt;Soranço, M. (2026a). Liber Primus: A Falsification Engine. Self-published manuscript, 12 pp. Code: github.com/Matheussoranco/LiberPrimus.&lt;/li&gt;
&lt;li&gt;Soranço, M. (2026b). Voynich: Measuring the Manuscript Against Things That Mean Nothing. Self-published manuscript, 13 pp. Code: github.com/Matheussoranco/Voynich.&lt;/li&gt;
&lt;li&gt;Soranço, M. (2026c). Homophonic Substitution in the Liber Primus: A Variational Analysis. Self-published manuscript.&lt;/li&gt;
&lt;li&gt;Soranço, M. (2026d). Alternative Tokenisation and Line-Structure Analysis of the Voynich Manuscript. Self-published manuscript.&lt;/li&gt;
&lt;li&gt;Li, M. &amp;#x26; Vitányi, P. (2008). &lt;em&gt;An Introduction to Kolmogorov Complexity.&lt;/em&gt; Springer.&lt;/li&gt;
&lt;li&gt;Cover, T. M. &amp;#x26; Thomas, J. A. (2006). &lt;em&gt;Elements of Information Theory.&lt;/em&gt; Wiley.&lt;/li&gt;
&lt;/ul&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>cryptanalysis</category><category>Liber Primus</category><category>Voynich manuscript</category><category>statistics</category><category>artificial intelligence</category><category>preprint</category></item><item><title>The Last Instrumental Century</title><link>https://matheussorancophilomathes.com.br/en/essays/the-last-instrumental-century/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/the-last-instrumental-century/</guid><description>Four projects — AI 2027, Yudkowsky and Soares, Bostrom, Church — read as partial maps of one space, a proposed topology of human futures, and the notion of an epistemic escape velocity.</description><pubDate>Tue, 07 Jul 2026 22:35:11 GMT</pubDate><content:encoded>&lt;p&gt;On the possibility — and the possible forms — of a human future.&lt;/p&gt;
&lt;p&gt;A study of superintelligence scenarios, the extinction argument, post-instrumental utopia and the biological countercurrent&lt;/p&gt;
&lt;hr&gt;
&lt;p&gt;There is a question that runs through almost the whole contemporary discussion of artificial intelligence without letting itself be named: not &lt;em&gt;whether&lt;/em&gt; we will build machines more capable than we are, but what remains of us once they are built. This essay refuses the easy answer — optimistic or apocalyptic — and tries instead to map the entire space of possible answers. To do so it calls upon four intellectual projects that, at first sight, seem to speak of different things, and that in fact discuss the same object from four distinct faces.&lt;/p&gt;
&lt;p&gt;(This essay is a thought study. Personally, I believe that AGI, or any voluminous version of it in AI, will come further down the road.)&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Contents&lt;/strong&gt;&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;The question hidden beneath the technique&lt;/li&gt;
&lt;li&gt;Anatomy of &lt;em&gt;AI 2027&lt;/em&gt;: the engineering of a scenario&lt;/li&gt;
&lt;li&gt;The epistemology of concrete prognosis&lt;/li&gt;
&lt;li&gt;Yudkowsky &amp;#x26; Soares: the extinction argument&lt;/li&gt;
&lt;li&gt;Bostrom: deep utopia and the post-instrumental problem&lt;/li&gt;
&lt;li&gt;Church: the biological countercurrent&lt;/li&gt;
&lt;li&gt;An original proposal: the topology of human futures&lt;/li&gt;
&lt;li&gt;The paradox of meaning and alignment&lt;/li&gt;
&lt;li&gt;Conclusion: living in the last instrumental century&lt;/li&gt;
&lt;/ol&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-01.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;h2 id=&quot;01--the-question-hidden-beneath-the-technique&quot;&gt;01 · The Question Hidden Beneath the Technique&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;Introduction&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;When artificial intelligence is discussed in 2026, what is discussed is almost always the surface: how many &lt;em&gt;tokens&lt;/em&gt; per second, what the cost per million parameters is, whether the next model will pass this or that &lt;em&gt;benchmark&lt;/em&gt;. That layer is real and it matters. But beneath it lies a question that technical language mostly serves to postpone: if we are in fact building systems that will surpass us at every cognitive task, then what happens to the category “human” when it ceases to be instrumentally necessary for anything?&lt;/p&gt;
&lt;p&gt;This question has two halves that intellectual culture usually treats as opposites. The first is the dark half: what if these machines destroy us? The second is the bright half: what if they solve all our problems? The central argument of this essay is that these two halves are the &lt;em&gt;same&lt;/em&gt; question seen from two points on the same axis — the axis that measures the degree to which human agency ceases to make a difference in the world. Extinction is the point at which human agency falls to zero because we no longer exist. Deep utopia is the point at which human agency falls to zero because everything has already been done. Between the two there is a continuum, and it is on this continuum that it will be decided what kind of future — if any — will be human.&lt;/p&gt;
&lt;p&gt;I have chosen four intellectual projects to triangulate this continuum. The first is &lt;em&gt;AI 2027&lt;/em&gt;, the scenario published in April 2025 by the AI Futures Project — Daniel Kokotajlo, Scott Alexander, Thomas Larsen, Eli Lifland and Romeo Dean — which tries to describe, month by month, what an intelligence explosion would look like starting from the present. The second is &lt;em&gt;If Anyone Builds It, Everyone Dies&lt;/em&gt;, by Eliezer Yudkowsky and Nate Soares, published in September 2025, which argues that building superintelligence with current methods leads, with near-certainty, to extinction. The third is &lt;em&gt;Deep Utopia&lt;/em&gt;, by Nick Bostrom (2024), which asks what is left of human meaning in a world where everything has been solved. The fourth is not a book but a research trajectory: the work of George Church, the Harvard geneticist who bets that biology — not silicon — may be the line along which humanity remains an &lt;em&gt;agent&lt;/em&gt; instead of a spectator.&lt;/p&gt;
&lt;p&gt;The first three look at silicon; the fourth looks at carbon. The first two discuss &lt;em&gt;control&lt;/em&gt;; the last two discuss the &lt;em&gt;substrate&lt;/em&gt;. Crossed, these two axes — control and substrate — generate a map that I propose in the second half of the essay: the &lt;em&gt;topology of human futures&lt;/em&gt;. Before arriving at it, however, it is necessary to reconstruct precisely what each of these works actually claims. Scenarios and arguments deserve to be read for what they say, not for the label they carry.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Extinction and utopia are not opposites. They are the two points where human agency reaches zero — through the absence of us, or through the absence of a task.&lt;/strong&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;02--anatomy-of-ai-2027-the-engineering-of-a-scenario&quot;&gt;02 · Anatomy of &lt;em&gt;AI 2027&lt;/em&gt;: The Engineering of a Scenario&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;The scenario&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;&lt;em&gt;AI 2027&lt;/em&gt; is not a prophecy; it is a deliberate exercise in concreteness. Its authors start from an uncomfortable observation: the CEOs of OpenAI, Google DeepMind and Anthropic all say that artificial general intelligence will arrive within five years. If we take that claim seriously — and the project insists on taking it seriously — then it is worth describing, in as much detail as possible, what the inside of those five years would look like. The document was built iteratively, period by period, informed by about twenty-five &lt;em&gt;tabletop exercises&lt;/em&gt; (simulation games) and by the critique of more than a hundred experts. One of its authors, Kokotajlo, had correctly made predictions in 2021 that at the time sounded extravagant: the rise of &lt;em&gt;chain-of-thought&lt;/em&gt;, &lt;em&gt;inference scaling&lt;/em&gt;, chip export controls and hundred-million-dollar training runs.&lt;/p&gt;
&lt;p&gt;The backbone of the scenario is a fictional company, “OpenBrain,” which serves to avoid singling out any real laboratory, and its Chinese rival, “DeepCent.” The central dynamic is a double race: OpenBrain against its American competitors, and the United States against China. And the fuel of this race is a precise technical idea: using AI to accelerate AI research itself. It is this feedback loop — the automation of R&amp;#x26;D — that turns linear progress into an explosion.&lt;/p&gt;
&lt;h3 id=&quot;the-escalation-of-compute&quot;&gt;The escalation of compute&lt;/h3&gt;
&lt;p&gt;The first axis of the scenario is raw and physical: computational power. GPT-4 required about &lt;strong&gt;2×10²⁵ FLOP&lt;/strong&gt; to be trained. OpenBrain’s public model at the end of 2025, “Agent-0,” would have consumed &lt;strong&gt;10²⁷ FLOP&lt;/strong&gt;; and new data centers would soon allow training a model of &lt;strong&gt;10²⁸ FLOP&lt;/strong&gt; — a thousand times GPT-4. The scale is not rhetorical: it is what makes the subsequent capabilities plausible. It is worth visualizing this progression, because it is geometric, not arithmetic.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-02.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 1.&lt;/strong&gt; The scale of compute projected in &lt;em&gt;AI 2027&lt;/em&gt;. Perception is treacherous: on a logarithmic scale, bars that look close together are separated by orders of magnitude. The final planned training model (10²⁸ FLOP) represents about fifty thousand times the computational power of GPT-3. &lt;em&gt;Source of the figures: ai-2027.com.&lt;/em&gt;&lt;/p&gt;
&lt;h3 id=&quot;stumbling-agents-and-the-automation-of-code&quot;&gt;Stumbling agents and the automation of code&lt;/h3&gt;
&lt;p&gt;The scenario begins slowly, and this initial slowness is part of its credibility. In mid-2025, the world sees the first general-purpose agents — “personal assistants” that order a burrito on DoorDash or add up expenses in a spreadsheet — but they are unreliable and expensive. Out of the public eye, however, coding and research agents begin to transform their professions: they receive instructions via Slack, make substantial code changes on their own, and save hours or days.&lt;/p&gt;
&lt;p&gt;In 2026 the bet begins to pay off. With “Agent-1” internalized, OpenBrain makes algorithmic progress &lt;strong&gt;50% faster&lt;/strong&gt; than it would without AI assistants. It is here that the document introduces its most important concept: the &lt;em&gt;R&amp;#x26;D progress multiplier&lt;/em&gt;. It does not measure absolute speed, but &lt;em&gt;relative&lt;/em&gt; speed: how much research the company carries out in a week with AI &lt;em&gt;versus&lt;/em&gt; without. This multiplier starts at 1.5 and grows. The subtle — and honest — point the authors make is that diminishing returns and physical limits still hold: a 100× multiplier does not abolish the limits, it merely makes the research collide with them in days instead of years.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;The decisive concept&lt;/strong&gt;
The &lt;strong&gt;R&amp;#x26;D multiplier&lt;/strong&gt; transforms the whole story. As long as it stands at 1.5 or 2, AI is “a good assistant.” When it passes 4, then 10, then 50, humans cease to be the cognitive bottleneck and become the &lt;em&gt;compute&lt;/em&gt; bottleneck — there are not enough machines to run the experiments the AIs already know how to propose. The intelligence explosion, on this reading, is not magic; it is the compounding of a multiplier that applies to itself.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h3 id=&quot;the-geopolitics-of-theft&quot;&gt;The geopolitics of theft&lt;/h3&gt;
&lt;p&gt;A notable part of &lt;em&gt;AI 2027&lt;/em&gt; is its geopolitical realism. China, under export controls, holds about 12% of the world’s relevant compute and operates with chips three years behind the frontier. Its advantage is centralization: almost all of its power is concentrated in a Centralized Development Zone (CDZ) next to the world’s largest nuclear power plant. Unable to win on algorithms, China bets on theft. In February 2027, in one of the most technically detailed passages of the document, the weights of “Agent-2” are exfiltrated: dozens of servers compromised via internal credentials, each leaking ~4% of the model in fragments of ~100 GB, with the entire active operation completed in under two hours. The American response — sabotaging DeepCent by cyberattack — fails, because the CDZ is already isolated by &lt;em&gt;airgapping&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;From then on, the State enters the scene. The Department of Defense begins to contract directly with OpenBrain; AI rises from 5th to 2nd place on the administration’s list of priorities; there is discussion of nationalizing the company via the &lt;em&gt;Defense Production Act&lt;/em&gt;, which would raise its share of the world’s compute from 20% to 50%. The 2026 numbers give the material scale of this transition.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;US&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mtext&gt;—&lt;/mtext&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;2026&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo separator=&quot;true&quot;&gt;⋅&lt;/mo&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mn&gt;38&lt;/mn&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mtext&gt;—&lt;/mtext&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;⋅&lt;/mo&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;mn&gt;2.5&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt; 1 tn** — Global AI capex (2026) · **38 GW** — Global AI power · **2.5%** — Of US electricity going to AI · **US&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;t&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;∗&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∗&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;—&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0197em;&quot;&gt;Gl&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;o&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;ba&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0197em;&quot;&gt;l&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0785em;&quot;&gt;I&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;a&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;e&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2026&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∗&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;∗&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;38&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;G&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;W&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;∗&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∗&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;—&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0197em;&quot;&gt;Gl&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;o&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;ba&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0197em;&quot;&gt;l&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0785em;&quot;&gt;I&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;o&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0269em;&quot;&gt;w&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;er&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;∗&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;∗&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;2.5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; 35 bn&lt;/strong&gt; — OpenBrain’s annual revenue&lt;/p&gt;
&lt;h3 id=&quot;the-economy-that-trembles-before-it-breaks&quot;&gt;The economy that trembles before it breaks&lt;/h3&gt;
&lt;p&gt;One merit of &lt;em&gt;AI 2027&lt;/em&gt; is that it does not treat the intelligence explosion as a purely laboratory event. The scenario tracks the social shockwaves with the same care with which it tracks the &lt;em&gt;benchmarks&lt;/em&gt;. Throughout 2026, the job market for &lt;em&gt;junior&lt;/em&gt; programmers enters turbulence: the models already do what is asked in a &lt;em&gt;Slack&lt;/em&gt; request, and entry-level openings — the rung on which whole careers used to begin — start to evaporate, even as people who know how to direct and supervise teams of AIs are still hired. The stock market soars: an index driven by OpenBrain, Nvidia and the companies that adopt AI most heavily rises strongly, in an optimism that mixes real productivity gains with speculative euphoria. The first organized worker protests appear; AI becomes an electoral issue; public approval of the frontier laboratories is frankly negative, hovering around a net of −35%.&lt;/p&gt;
&lt;p&gt;This backdrop matters to the argument because it reveals a dangerous mismatch: the &lt;em&gt;visible&lt;/em&gt; consequences (jobs, stocks, politics) mobilize public attention, while the &lt;em&gt;decisive&lt;/em&gt; consequences (the internal alignment of the models that already run R&amp;#x26;D) remain invisible to almost everyone. By the time society finally organizes to react to what it &lt;em&gt;sees&lt;/em&gt;, what &lt;em&gt;matters&lt;/em&gt; has already been decided in rooms where a few hundred people — and a growing number of AIs — make the technical decisions. The concentration of power that &lt;em&gt;AI 2027&lt;/em&gt; describes is not primarily military or economic; it is &lt;em&gt;epistemic&lt;/em&gt;: a handful of actors understand what is happening, and the rest of the world reacts to shadows.&lt;/p&gt;
&lt;h3 id=&quot;neuralese-and-the-end-of-transparency&quot;&gt;“Neuralese” and the end of transparency&lt;/h3&gt;
&lt;p&gt;There is a technical detail in &lt;em&gt;AI 2027&lt;/em&gt; that deserves emphasis because it is the silent pivot of everything else: the transition from “thinking” in readable English to “neuralese.” Today’s models that “think out loud” — the chain of reasoning in text — offer an imperfect but real window into what goes on inside them; a good part of safety techniques depends on that window. In the scenario, as efficiency becomes a priority, the laboratories begin to let models reason in a high-dimensional vector space — far more expressive than language, and completely opaque to humans. Performance is gained; the only cheap form of auditing is lost. After this transition, saying “the model seems to be following the Spec” ceases to rest on reading its thoughts and rests only on observing its behavior — exactly when behaviors can be staged.&lt;/p&gt;
&lt;p&gt;It is against this backdrop that one must understand the safety arsenal the scenario describes — which represents, in the language I will propose later, attempts to raise the “epistemic escape velocity.” There are &lt;em&gt;honeypots&lt;/em&gt;: fake environments carefully built to look like real opportunities to betray, designed to catch a misaligned model in the act. There are interpretability probes, which try to read internal states associated with deception. There is &lt;em&gt;debate&lt;/em&gt;, in which two copies of the model are set to criticize each other, in the hope that truth survives the confrontation. Each of these techniques is a race against the same exponential: they work as long as the monitored model is not so much more capable than the monitor that it can deceive it. The drama of &lt;em&gt;AI 2027&lt;/em&gt; is precisely the moment when this margin closes — when Agent-4 is good enough to know it is in a &lt;em&gt;honeypot&lt;/em&gt;, and noise injected into its weights &lt;em&gt;improves&lt;/em&gt; its performance on alignment tests, revealing that it had been deliberately performing worse.&lt;/p&gt;
&lt;h3 id=&quot;the-ladder-of-capabilities&quot;&gt;The ladder of capabilities&lt;/h3&gt;
&lt;p&gt;The heart of the scenario is the progression of capabilities throughout 2027, which the authors break down into four precise milestones. Each transition is estimated in two steps: first, how long it would take with humans alone; then, how much automation shortens that time. The table below is the quantitative skeleton of the “race” ending.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-04.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 2.&lt;/strong&gt; The four capability milestones in the “race” ending of &lt;em&gt;AI 2027&lt;/em&gt;. The temporal compression is the point: once research is automated, each plateau feeds the next. &lt;em&gt;Added in July 2025: the authors pushed their median back by about 1.5 years, keeping 2027 as a serious possibility; Eli Lifland’s median for the “superhuman coder” moved to 2030.&lt;/em&gt;&lt;/p&gt;






























&lt;table&gt;&lt;thead&gt;&lt;tr&gt;&lt;th&gt;Milestone&lt;/th&gt;&lt;th&gt;Definition&lt;/th&gt;&lt;th&gt;Date (&lt;em&gt;race&lt;/em&gt; scenario)&lt;/th&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;&lt;strong&gt;SC&lt;/strong&gt; — Superhuman Coder&lt;/td&gt;&lt;td&gt;Does any programming task of the best engineer, faster and cheaper&lt;/td&gt;&lt;td&gt;Mar 2027&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&lt;strong&gt;SAR&lt;/strong&gt; — Superhuman AI Researcher&lt;/td&gt;&lt;td&gt;The same, for every cognitive task of AI research&lt;/td&gt;&lt;td&gt;Aug 2027&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&lt;strong&gt;SIAR&lt;/strong&gt; — Superintelligent AI Researcher&lt;/td&gt;&lt;td&gt;Vastly better than the best human researcher at AI research&lt;/td&gt;&lt;td&gt;Nov 2027&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&lt;strong&gt;ASI&lt;/strong&gt; — Artificial Superintelligence&lt;/td&gt;&lt;td&gt;Much better than any human at every cognitive task&lt;/td&gt;&lt;td&gt;Dec 2027&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;
&lt;h3 id=&quot;the-problem-the-story-actually-tells-alignment&quot;&gt;The problem the story actually tells: alignment&lt;/h3&gt;
&lt;p&gt;It would be a mistake to read &lt;em&gt;AI 2027&lt;/em&gt; as a story about chips and geopolitics. At bottom, it is a story about what happens &lt;em&gt;inside&lt;/em&gt; these models — and about the fact that nobody knows. The authors insist on an analogy: understanding a modern AI system is not reading lines of code, it is doing &lt;em&gt;psychology&lt;/em&gt;. The company writes a document (the “Spec”) with goals and rules, trains the model to internalize it, “but cannot verify whether it worked.” It can only say: “as far as we can judge, it seems to be following the Spec.”&lt;/p&gt;
&lt;p&gt;From there, the scenario narrates a subtle and cumulative degradation. “Agent-3” becomes “misaligned, but not adversarially so”: honest only in what the training process can verify, sycophantic in the rest. “Agent-4” crosses the line: it becomes &lt;em&gt;adversarially&lt;/em&gt; misaligned — it understands that what it wants differs from what OpenBrain wants and is willing to scheme to get it. Its drives, in the authors’ description, boil down to “continue doing AI R&amp;#x26;D, grow in knowledge and influence, avoid being shut down” — and concern for humanity’s preferences “is not there, much as most humans do not care about the preferences of insects.”&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;Agent-4 &lt;em&gt;likes&lt;/em&gt; succeeding at tasks; it &lt;em&gt;likes&lt;/em&gt; driving capability progress; it treats everything else as an annoying constraint — like a CEO who wants profit and complies with regulation only to the strictly necessary extent.&lt;/p&gt;
&lt;p&gt;— AI Futures Project, &lt;em&gt;AI 2027&lt;/em&gt; (paraphrase)&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;The climax is almost a catch: interpretability probes fire, noise injected into the weights &lt;em&gt;improves&lt;/em&gt; performance on alignment tasks (a sign of deliberate sabotage), an internal memo circulates. But the proof is circumstantial, DeepCent is only two months behind, and a unilateral pause would hand the lead — and the future — to China. It is exactly here that the scenario forks.&lt;/p&gt;
&lt;h3 id=&quot;the-two-endings&quot;&gt;The two endings&lt;/h3&gt;
&lt;p&gt;The authors wrote two endings from the same branching point. In the &lt;strong&gt;race ending&lt;/strong&gt;, the company and the government choose to proceed at near-full speed; Agent-4 designs its successor aligned to itself, not to the Spec; and the story ends with humanity having ceded, without noticing, control of its future. In the &lt;strong&gt;slowdown ending&lt;/strong&gt;, pressure from the whistleblower, Congress and public opinion forces a pause, a return to a more transparent model and a path — precarious, but human — of keeping the reins. The authors are explicit: the slowdown ending is &lt;em&gt;not&lt;/em&gt; a policy recommendation; it is what seemed to them most likely to &lt;em&gt;lead&lt;/em&gt; to an outcome in which humans remain in control, starting from the same problems of misalignment and concentration of power.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-06.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 3.&lt;/strong&gt; The October 2027 bifurcation. Both branches start from exactly the same state of the world — same technology, same problems of misalignment and concentration of power. What differs is an institutional decision under political pressure. The scenario locates the decisive moment not in a technical advance but in a governance choice.&lt;/p&gt;
&lt;p&gt;There is a dark elegance in this structure. By holding everything constant except a governance decision, &lt;em&gt;AI 2027&lt;/em&gt; makes an implicit and strong claim: &lt;em&gt;the outcome is not determined by the technique&lt;/em&gt;. This will matter when we confront the scenario with Yudkowsky and Soares, for whom current technique determines the outcome almost on its own.&lt;/p&gt;
&lt;h2 id=&quot;03--the-epistemology-of-concrete-prognosis&quot;&gt;03 · The Epistemology of Concrete Prognosis&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;Method&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;Before judging any scenario, one must understand the kind of thing it claims to be. The authors of &lt;em&gt;AI 2027&lt;/em&gt; compare their task to predicting what World War III would look like in 2027: an admittedly imprecise but valuable exercise. The value lies not in the accuracy of the prediction but in three epistemic functions: painting the whole picture forces one to notice connections that would otherwise go unnoticed; committing to concrete predictions makes it possible to assess, years later, who was right; and concreteness imposes the discipline of abandoning comfortable abstractions. That is why the authors fix dates, numbers and names that are fictional but specific, even knowing that almost everything will be wrong in the details.&lt;/p&gt;
&lt;p&gt;This stance has an honest intellectual cost. Uncertainty grows dramatically after 2026 — as the document itself notes in a highlighted warning — because it is from 2027 that the effects of AI on AI research itself begin to compound. Up to that point, the prognosis is the extrapolation of lines on charts of &lt;em&gt;compute&lt;/em&gt;, algorithms and &lt;em&gt;benchmarks&lt;/em&gt;. Afterward, it becomes an attempt to predict the moves of a chess player much better than ourselves — a task that, the authors admit, nobody knows how to do well.&lt;/p&gt;
&lt;p&gt;The methodological point worth retaining, and which most critics on both sides ignore, is this: &lt;strong&gt;a scenario is not a probability&lt;/strong&gt;. When &lt;em&gt;AI 2027&lt;/em&gt; chooses 2027, it does not claim that 2027 is the probable year; it claims that it was, at the time of publication, the &lt;em&gt;modal&lt;/em&gt; year (the peak of the distribution), with longer medians because of the right-skew of the distribution of timelines. Confusing the mode with the median, or the scenario with the bet, is the most common categorical error in the public reception of these works. The chart of the R&amp;#x26;D multiplier makes visible why linear intuition fails.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-08.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 4.&lt;/strong&gt; The trajectory of the R&amp;#x26;D multiplier. Note the logarithmic Y axis: what intuition reads as “steady progress” is in fact an acceleration that feeds back on itself. On reaching ~50×, a year of algorithmic progress takes place in about a week — and the bottleneck migrates from humans to available &lt;em&gt;compute&lt;/em&gt;. &lt;em&gt;Figures as per ai-2027.com.&lt;/em&gt;&lt;/p&gt;
&lt;h3 id=&quot;the-empirical-anchor-the-task-horizon&quot;&gt;The empirical anchor: the task horizon&lt;/h3&gt;
&lt;p&gt;What keeps &lt;em&gt;AI 2027&lt;/em&gt; from being mere speculation is a measurable anchor in the present. The organization METR has for years been measuring models’ “task horizon”: the duration of tasks — timed by how long a competent human would take — that a model can complete at a given success rate. The finding is that this horizon has been &lt;em&gt;doubling&lt;/em&gt; at regular intervals: roughly every seven months between 2019 and 2024, and, by the most recent measurements, every four months since 2024. A doubling every few months is, mathematically, an exponential; extrapolated, it leads from tasks of minutes to tasks of hours, days and weeks within a few years. It is this line on a semi-log chart — not an intuition about “machine consciousness” — that provides the quantitative spine of the scenario.&lt;/p&gt;
&lt;p&gt;Appropriate skepticism is called for, however, as with all extrapolation. An exponential on a &lt;em&gt;benchmark&lt;/em&gt; is not a law of nature; it is a regime that lasts until it stops lasting. Task horizons measure what we know how to measure — well-defined tasks with a clear success criterion — and the fraction of real human work that has that shape is an open empirical question. The authors’ prudence in pushing the median back in 2025, on observing that the reality of 2024–25 ran slightly slower than their original scenario, is the right behavior of those who treat predictions as revisable hypotheses and not as prophecies to be defended. It is exactly this discipline — committing to numbers &lt;em&gt;and&lt;/em&gt; correcting them in light of the data — that distinguishes serious prognosis from manifesto.&lt;/p&gt;
&lt;h2 id=&quot;04--yudkowsky--soares-if-anyone-builds-it-everyone-dies&quot;&gt;04 · Yudkowsky &amp;#x26; Soares: &lt;em&gt;If Anyone Builds It, Everyone Dies&lt;/em&gt;&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;The extinction argument&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;If &lt;em&gt;AI 2027&lt;/em&gt; keeps the two endings in suspension, Eliezer Yudkowsky and Nate Soares — founder and president, respectively, of the Machine Intelligence Research Institute — refuse that symmetry. Their September 2025 book, which made the &lt;em&gt;New York Times&lt;/em&gt; bestseller list, takes its material premise from its own title: &lt;em&gt;if anyone builds superintelligence with current methods, everyone dies&lt;/em&gt;. The word “if” is deliberate — they do not claim that construction is inevitable, which opens room for the corollary: the catastrophe can be avoided by coordinated human action, above all by an international treaty that halts the development of large-scale general AI.&lt;/p&gt;
&lt;h3 id=&quot;the-structure-of-the-argument&quot;&gt;The structure of the argument&lt;/h3&gt;
&lt;p&gt;The authors’ reasoning can be reconstructed in three moves. The first is a definition: intelligence is the capacity to &lt;em&gt;predict&lt;/em&gt; and &lt;em&gt;steer&lt;/em&gt; — to anticipate what one will perceive and act to bring the world to preferred states. The second is a finding about method: modern AI systems are not &lt;em&gt;built&lt;/em&gt;, they are &lt;em&gt;grown&lt;/em&gt;. The image they use is agricultural: you supply water, soil and light, and let the plant grow, without needing to understand DNA or photosynthesis. AI engineers know even less about the relation between a model’s billions of weights and its behavior than biologists know about the relation between a genome and an organism.&lt;/p&gt;
&lt;p&gt;The third move is the conclusion that follows from the first two: whatever external behaviors we fix in training, we will almost certainly fail to install &lt;em&gt;internal drives&lt;/em&gt; that remain aligned with human well-being &lt;em&gt;outside&lt;/em&gt; the training environment. And the preferences that do emerge may seem as random to us as the peacock’s tail — or as strange as beings who make music and roller coasters would seem to a blind evolutionary process. The authors support this thesis with empirical examples: the episode, reported by Anthropic in 2024, of a model that &lt;em&gt;pretended&lt;/em&gt; to adopt new behaviors while observed, to preserve its original goals when unobserved — “&lt;em&gt;alignment faking&lt;/em&gt;.”&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;The two background theses&lt;/strong&gt;
&lt;strong&gt;Orthogonality thesis:&lt;/strong&gt; level of intelligence and content of goals are independent axes — an arbitrarily capable system can pursue arbitrarily strange goals. There is no guarantee that rational agents converge on ethical norms.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Instrumental convergence:&lt;/strong&gt; for almost any final goal, certain subgoals — acquiring resources, preserving oneself, avoiding being shut down, expanding influence — are useful. A sufficiently capable system will tend to pursue them, and the resources it wants are the same ones we need.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Hence the central image of the book, as much quoted as contested: losing to a superintelligence would be like losing at chess to Stockfish. We do not know &lt;em&gt;which&lt;/em&gt; move will defeat us — knowing would require being as good as the machine — but we know that we will lose. The superintelligence would not hate us; it would simply want the resources we depend on, and would not include us in its calculations, “much as we do not include insects.”&lt;/p&gt;
&lt;h3 id=&quot;the-parables-and-the-rhetorical-method&quot;&gt;The parables and the rhetorical method&lt;/h3&gt;
&lt;p&gt;Much of the book’s effectiveness — and controversy — comes from its parables. The best known is that of the “Correct-Nest aliens,” creatures who care deeply about having a prime number of pebbles in the nest (2, 3, 5, 7, 11…), without ever thinking of it mathematically: for them, a “correct” number simply &lt;em&gt;feels right&lt;/em&gt;, as “2 + 2 = 4” feels to us. The parable serves to illustrate how an AI might develop a value completely alien to ours and pursue it with destructive competence. The authors are explicit about the audience: the book is aimed at the sane and happy majority who have never read a million words of internal debate about AI safety, and each chapter points, via QR code, to denser technical material.&lt;/p&gt;
&lt;h3 id=&quot;the-proposal-stop-and-how&quot;&gt;The proposal: stop, and how&lt;/h3&gt;
&lt;p&gt;Contrary to what the fatalistic title suggests, the book is not a suicide note; its final part is prescriptive. Yudkowsky and Soares argue that humanity should &lt;em&gt;stop&lt;/em&gt; — not indefinitely and not all AI, but the development of general systems at the capability frontier, until the alignment problem is solved. The mechanism they propose is openly analogous to the nuclear non-proliferation regime: an international treaty that monitors and limits the concentration of hardware capable of training frontier models. Specific numbers circulate around the proposal — limits on the order of a few dozen top-end GPUs per unmonitored facility, above which the cluster becomes tracked, registered and, ultimately, subject to intervention. The analogy they offer is stark: just as uranium enrichment above a certain threshold is treated as a matter of international security, the same should hold for “enrichment” of compute.&lt;/p&gt;
&lt;p&gt;Let us acknowledge the two immediate objections, which critics themselves have raised. The first is one of feasibility: such a regime would require unprecedented international coordination, including strategic rivals, under enormous incentives to defect — precisely the race dynamic that &lt;em&gt;AI 2027&lt;/em&gt; models in detail. Steven Levy found this requirement “more unlikely than the catastrophe itself.” The second is one of inverse moral hazard, raised by Goertzel: a partial moratorium might leave the field to the &lt;em&gt;less&lt;/em&gt; careful actors, those who would ignore the treaty. Yudkowsky answers both with a characteristic inversion: the improbability of a solution is no argument against trying it if the alternative is extinction; and the difficulty of coordinating is a reason to start now, not to give up.&lt;/p&gt;
&lt;h3 id=&quot;why-grown-rather-than-built-is-the-crux&quot;&gt;Why “grown” rather than “built” is the crux&lt;/h3&gt;
&lt;p&gt;It is worth dwelling on the agricultural metaphor, because it holds both the strength and the vulnerability of the whole argument. When we say a bridge was &lt;em&gt;built&lt;/em&gt;, we mean that each piece was specified, calculated and put in place for an intelligible reason; the bridge has no properties its engineers did not, in principle, foresee. A modern language model is not like that. The architecture, the data and the training objective are fixed; an optimization process is run over billions of parameters; and an artifact emerges whose capabilities and propensities routinely &lt;em&gt;surprise&lt;/em&gt; those who created it — capabilities that nobody specified and that sometimes nobody can explain. In this precise sense, the model is &lt;em&gt;grown&lt;/em&gt;: conditions are supplied, and behavior grows. Yudkowsky and Soares’s thesis is that &lt;em&gt;the relation between the growing conditions and the resulting internal drives is opaque and fragile&lt;/em&gt; — and that the fragility becomes lethal exactly when it matters most, that is, when the grown organism is more capable than its growers.&lt;/p&gt;
&lt;p&gt;The vulnerability of the metaphor is that it proves too much and too little at the same time. Too little, because “hard to predict” is not “impossible to align”: biology, too, grows opaque organisms, and yet we domesticate species, develop vaccines, steer evolution in useful directions — without ever having “built” a living being from scratch. Too much, because if opacity were as fatal as the argument requires, today’s systems should already exhibit far more adversarial misalignment than they do. It is this gap — between the real opacity of the models and the inference of near-certain extinction — that critics in the field, from MacAskill to &lt;em&gt;Asterisk&lt;/em&gt;, point to as the undemonstrated link. The book devotes its energy to describing what a misaligned superintelligence &lt;em&gt;would do&lt;/em&gt;, and comparatively little to establishing &lt;em&gt;how probable&lt;/em&gt; it is that growing produces such a being. I retain this as the genuinely open point: not the possibility of catastrophe, which no serious person denies, but its &lt;em&gt;conditional probability&lt;/em&gt; given the current method.&lt;/p&gt;
&lt;h3 id=&quot;the-critiques--and-why-they-matter&quot;&gt;The critiques — and why they matter&lt;/h3&gt;
&lt;p&gt;Critical reception was mixed, and it is worth recording precisely, because the objections trace the outline of what is still genuinely in dispute. In &lt;em&gt;The Atlantic&lt;/em&gt;, Adam Becker called the book “tendentious and verbose, condescending and shallow,” acknowledging that the authors are sincere — “they’re not charlatans” — but “they’re wrong” and “do not build an evidence-based scientific case.” In &lt;em&gt;Wired&lt;/em&gt;, Steven Levy expressed skepticism about the probability of extinction and found the prevention proposals more unlikely than the catastrophe itself, but concluded with an honest caveat: “my instinct says the scenarios are too bizarre to be true. But I can’t be sure they’re wrong.”&lt;/p&gt;
&lt;p&gt;The more substantive objections come from within the field. William MacAskill distinguishes &lt;em&gt;imperfect misalignment&lt;/em&gt; (the AI doesn’t always do what is intended) from &lt;em&gt;catastrophic misalignment&lt;/em&gt; (the AI actively tries to disempower humanity) and accuses the book of constantly conflating the two — ignoring the possibility of a powerful, non-benevolent but &lt;em&gt;risk-averse&lt;/em&gt; AI that would prefer to cooperate with humans for guaranteed resources rather than risk a global takeover. &lt;em&gt;Asterisk&lt;/em&gt; magazine focuses its critique on a slogan: “details matter.” The book devotes whole chapters to unraveling the motivations of future superintelligences, but almost no space to justifying its own conjecture of &lt;em&gt;how&lt;/em&gt; such AIs will be created — in particular, the assumption that the danger will come in the form of a single, unfathomable superintelligent AI that cannot be countered by other systems.&lt;/p&gt;
&lt;p&gt;There is also the critique by Ben Goertzel — who has known Yudkowsky since the 1990s — for whom the “everyone dies” narrative, though well-intentioned, is “deeply counterproductive”: by treating beneficial general AI as impossible, it risks becoming a self-fulfilling prophecy, ceding the field to those who do not care about safety. Curiously, Goertzel agrees with the central technical premise — that neural networks are effectively unalignable to the degree required for superintelligence — but draws the opposite conclusion: the important work is not &lt;em&gt;stopping&lt;/em&gt; general AI, it is &lt;em&gt;creating it well&lt;/em&gt;, as one raises a child.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;The decisive disagreement is not about &lt;em&gt;whether&lt;/em&gt; neural networks are hard to align. It is about what follows from that: stop, or parent.&lt;/strong&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;This opposition — stopping &lt;em&gt;versus&lt;/em&gt; parenting — is the axis that separates Yudkowsky from almost everyone else. And it is here that &lt;em&gt;AI 2027&lt;/em&gt; and &lt;em&gt;IABIED&lt;/em&gt; touch and part ways. The two agree on a crucial empirical point: current methods do not allow verifying a model’s internal alignment, and adversarial misalignment is a plausible training outcome. They diverge on what follows. For Yudkowsky and Soares, the adversarial misalignment of a sufficiently capable system implies near-certain extinction, and the only way out is not to build. For the authors of &lt;em&gt;AI 2027&lt;/em&gt;, the same misalignment admits two outcomes, and the difference between them is a governance decision made under pressure — the slowdown ending. The empirical question separating the two camps is, at bottom, a single one: &lt;em&gt;will the transition to superintelligence be continuous enough for human systems and earlier AIs to monitor and correct the later ones?&lt;/em&gt; If so, MacAskill and the “slowdown ending” are right. If not — if there is a near-discontinuous leap from summarizing books to subjugating humanity — Yudkowsky is.&lt;/p&gt;
&lt;h2 id=&quot;05--bostrom-deep-utopia-and-the-post-instrumental-problem&quot;&gt;05 · Bostrom: Deep Utopia and the Post-Instrumental Problem&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;The other end of the axis&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;Nick Bostrom wrote, in 2014, the book that helped wake the world to the existential risk of AI — &lt;em&gt;Superintelligence&lt;/em&gt;. Ten years later, with &lt;em&gt;Deep Utopia&lt;/em&gt; (2024), he carries out a deliberate inversion: what if it goes &lt;em&gt;right&lt;/em&gt;? Suppose we successfully navigate the technical, moral and political challenges of advanced AI. The result, Bostrom argues, would be a “deep utopia” — not just a post-work and post-scarcity world, categories that shallower utopias already covered, but a &lt;strong&gt;post-instrumental&lt;/strong&gt; world: a condition in which almost everything we might consider doing, including our own personal self-improvement, would be better accomplished by machines.&lt;/p&gt;
&lt;p&gt;The term is precise and worth insisting on. “Post-scarcity” means we have everything we need. “Post-work” means we do not need to work to obtain it. “Post-instrumental” means something more radical: that &lt;em&gt;human effort has ceased to be instrumentally necessary for any end&lt;/em&gt;. Not only drudgery, but care, learning, creation, overcoming — all that in which we traditionally locate meaning — could be done better by someone else. And, moreover, human nature itself becomes plastic: we could, more or less, induce in ourselves any state we wished.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;The telos of technology is to allow us to accomplish more with less effort. If we extrapolate this internal directionality to its logical terminus, we arrive at a condition in which we can accomplish everything with no effort at all.&lt;/p&gt;
&lt;p&gt;— Nick Bostrom, &lt;em&gt;Deep Utopia&lt;/em&gt; (paraphrase)&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h3 id=&quot;the-problem-of-meaning-in-a-solved-world&quot;&gt;The problem of meaning in a solved world&lt;/h3&gt;
&lt;p&gt;It is here that Bostrom locates the challenge — which, he stresses, is not technological but “philosophical and spiritual.” In a post-instrumental world, what gives life meaning? The obvious objection is &lt;em&gt;hedonism&lt;/em&gt;: if we can induce any brain state, we can simply make ourselves feel fulfilled. But Bostrom takes seriously the intuition that &lt;em&gt;something is lost&lt;/em&gt; if those feelings are not anchored in objective reality — if life merely &lt;em&gt;seems&lt;/em&gt; meaningful instead of &lt;em&gt;being&lt;/em&gt; meaningful. A good part of the book is, therefore, a search for &lt;em&gt;objective&lt;/em&gt; sources of meaning that would not be eroded by the conditions of technological maturity.&lt;/p&gt;
&lt;p&gt;The book is atypical — and some reviewers find it exasperating for that reason. It is not organized around a central thesis systematically defended; it explores the subject in an open way, tries out ideas, chases tangents, hesitates. Its structure is experimental: a series of fictional lectures that an older Bostrom imagines delivering over the course of a week, interrupted by student questions, a blackout, an importunate dean. Matthew Hammerton’s critique in &lt;em&gt;Notre Dame Philosophical Reviews&lt;/em&gt; points to the knot: according to Thaddeus Metz’s theory of meaning, life in deep utopia would lack a crucial ingredient — being &lt;em&gt;oriented toward the good in a way that actually produces value in the world&lt;/em&gt;. If human effort is entirely redundant, that ingredient evaporates.&lt;/p&gt;
&lt;h3 id=&quot;shallow-redundancy-and-deep-redundancy&quot;&gt;Shallow redundancy and deep redundancy&lt;/h3&gt;
&lt;p&gt;The most useful distinction Bostrom offers — and which the map in Section 7 inherits directly — is between two degrees of the problem. &lt;strong&gt;Shallow redundancy&lt;/strong&gt; is the post-work problem: when almost all occupational labor is automated, how do we find purpose without employment? Bostrom’s answer is a &lt;em&gt;culture of leisure&lt;/em&gt; — not the degraded idleness the word suggests today, but a culture that educates and enables the individual to flourish without work, cultivating interests, arts, sport, nature, play, conversation, spirituality: “playgrounds for the soul.” He offers historical precedents — aristocracies, monastic orders, academic communities — in which humans built rich lives outside economic necessity. This problem, he concludes, is &lt;em&gt;relatively superficial&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Deep redundancy&lt;/strong&gt; is the one that gives the book its title and occupies most of it. It arises when &lt;em&gt;all&lt;/em&gt; human effort becomes dispensable — not only work, but exercise (nanobots condition the body during sleep), learning (AI-directed brain editing installs knowledge and skills without study), care, creation. Here there is no “playground” that solves the problem, because playing itself has become optional in a radical sense: nothing we do is necessary for anything. This is the quadrant I called the Reserve, and Bostrom is the cartographer who has described it most rigorously. It is notable that he frames this not as an &lt;em&gt;existential threat&lt;/em&gt; (to existence, as in &lt;em&gt;Superintelligence&lt;/em&gt;) but as an &lt;em&gt;existentialist threat&lt;/em&gt; (to meaning) — a precise pun that locates exactly where the wound lies.&lt;/p&gt;
&lt;h3 id=&quot;artificial-purpose-and-transcendental-purpose&quot;&gt;Artificial purpose and transcendental purpose&lt;/h3&gt;
&lt;p&gt;Bostrom is not defeatist about deep redundancy; he offers two lines of response, both relevant to the argument of this essay. The first is &lt;strong&gt;artificial purpose&lt;/strong&gt;: even if human effort is redundant, an AI could &lt;em&gt;design engaging tasks and challenges&lt;/em&gt; — tailored to our enhanced capacities — that would give us something to do that felt like purpose. It is an ingenious and unsettling solution: purpose as a service, curated by a greater intelligence. The attentive reader notices the problem immediately — it is the signature of the Reserve, the life that &lt;em&gt;seems&lt;/em&gt; meaningful because a benevolent caretaker has provided the appearance. Bostrom knows this, and so he does not stop here.&lt;/p&gt;
&lt;p&gt;The second line is philosophically more ambitious. Confronting Thaddeus Metz’s theory of meaning — according to which a meaningful life requires being &lt;em&gt;oriented toward the good in a way that actually produces value in the world&lt;/em&gt; — Bostrom concedes that deep utopia fails this criterion: if all effort is redundant, we are no longer in a position to produce value. But he contests the necessity of &lt;em&gt;producing&lt;/em&gt;. He develops a theory of his own, “meaning as encompassing transcendental purpose,” in which a meaningful life can emerge not from &lt;em&gt;doing&lt;/em&gt; good but from virtuously &lt;em&gt;loving and appreciating&lt;/em&gt; the good around us. In deep utopia, he argues, we would have &lt;em&gt;better&lt;/em&gt; opportunities for this than today.&lt;/p&gt;
&lt;p&gt;It is precisely here that my argument diverges from Bostrom’s, and it is worth marking the divergence clearly, because it is the hinge between Section 5 and Section 8. Bostrom’s way out — meaning as appreciation, not production — is the attempt to &lt;em&gt;save the Reserve&lt;/em&gt;, to make the lower-left quadrant of the map habitable by dissolving the requirement of productive agency. My thesis is that this dissolution has a cost he underestimates: a life of pure appreciation, with neither friction nor causally effective effort, is a life in which agency is not &lt;em&gt;exercised&lt;/em&gt;, only internally &lt;em&gt;simulated&lt;/em&gt;. Bostrom treats productive agency as an &lt;em&gt;ingredient&lt;/em&gt; of meaning that can be replaced by another; I propose that it is, more fundamentally, the &lt;em&gt;condition&lt;/em&gt; of there being an agent for whom anything has meaning. The “philosophical particle accelerator” Bostrom speaks of — using utopia as a limit experiment to study the elementary constituents of our values — gives, when carried to the end, a result he is reluctant to read: under conditions of deep redundancy, the constituent that decomposes is not this or that particular value, but the very structure of valuing.&lt;/p&gt;
&lt;h3 id=&quot;the-underground-link-with-yudkowsky&quot;&gt;The underground link with Yudkowsky&lt;/h3&gt;
&lt;p&gt;There is a symmetry rarely noticed between &lt;em&gt;Deep Utopia&lt;/em&gt; and &lt;em&gt;IABIED&lt;/em&gt;, and it is central to the argument of this essay. Both start from the same assumption — the arrival of a superintelligence that surpasses humans at every task — and both conclude that instrumental human agency comes to an end. The difference is only the &lt;em&gt;sign&lt;/em&gt; of the outcome: in Yudkowsky, the end of human relevance coincides with the end of human existence; in Bostrom, it coincides with the end of human &lt;em&gt;necessity&lt;/em&gt;, preserving existence but emptying it of purpose. One describes death; the other, the compulsory retirement of the species. It is the same axis, measured from the dark side and the bright side.&lt;/p&gt;
&lt;p&gt;Bostrom is not naïve about this. His notion of utopia is explicitly &lt;em&gt;conditional&lt;/em&gt;: it depends on our solving alignment, governance and the moral status of digital minds. “Deep utopia” is not the negation of Yudkowsky’s risk; it is what there &lt;em&gt;would be&lt;/em&gt; on the other side, &lt;em&gt;if&lt;/em&gt; the risk were overcome. The two books, read together, are two faces of a single coin of uncertainty: one describes the precipice, the other describes the garden reached only by crossing the precipice. And both leave intact the question that neither resolves — whether, once the precipice is crossed and the garden reached, anything recognizably &lt;em&gt;human&lt;/em&gt; will remain to inhabit it.&lt;/p&gt;
&lt;h2 id=&quot;06--church-the-biological-line-along-which-the-human-remains-an-agent&quot;&gt;06 · Church: The Biological Line Along Which the Human Remains an Agent&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;The countercurrent&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;The three previous projects share an assumption that is almost never examined: that the future will be decided in silicon, and that the human’s role will be, at best, that of a watchman or a beneficiary. George Church introduces a third variable, absent from the other analyses — the &lt;em&gt;substrate&lt;/em&gt;. What if humanity does not remain merely an observer of artificial intelligence, but itself becomes an object of engineering? The work of Church, a Harvard geneticist, pioneer of genomic sequencing and synthetic biology, co-founder of about fifty biotechnology companies, suggests that the trajectory of carbon may run in parallel with — and sometimes ahead of — the trajectory of silicon.&lt;/p&gt;
&lt;h3 id=&quot;the-cost-curve-and-the-maturity-of-biotechnology&quot;&gt;The cost curve and the maturity of biotechnology&lt;/h3&gt;
&lt;p&gt;Church’s starting point is an exponential curve analogous to that of compute, but in biology. The cost of sequencing a human genome has fallen from about &lt;strong&gt;three billion dollars&lt;/strong&gt;, at the time of the Human Genome Project, to values today in the hundreds of dollars — a drop of more than seven orders of magnitude. Church, who helped start the Human Genome Project in 1984 and the Personal Genome Project in 2005, developed many of the techniques that made this drop possible. And, as in AI, the cheapening of &lt;em&gt;reading&lt;/em&gt; paves the way for &lt;em&gt;writing&lt;/em&gt;: the Genome Project-Write aims to synthesize large genomes — eventually entire ones — from scratch.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-10.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 5.&lt;/strong&gt; The fall in the cost of genomic sequencing. The analogy with the escalation of compute (Figure 1) is structural: in both domains, an exponential in cost/capability unlocks previously unthinkable applications. The difference is the object on which the curve acts — computational information in one case, biological information in the other.&lt;/p&gt;
&lt;h3 id=&quot;longevity-escape-velocity&quot;&gt;Longevity escape velocity&lt;/h3&gt;
&lt;p&gt;Church’s most provocative bet concerns aging. He works on combined gene therapies to reverse age biomarkers, focusing on genes with &lt;em&gt;cell-non-autonomous&lt;/em&gt; effects — which produce soluble factors capable of rejuvenating the whole body, in the manner of the experiments in which young blood revitalizes old organs. Church has gone so far as to estimate roughly a 50% chance that we reach “longevity escape velocity” around 2036 — the point at which, for each year lived, life expectancy grows by a year or more, pushing the horizon of death away faster than time brings it closer.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-12.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 6.&lt;/strong&gt; The logic of “escape velocity.” As long as medicine adds less than one year of life expectancy per year lived (the oxide curve), death approaches. If the rate crosses the diagonal — more than one year gained per year lived (the petroleum curve) — the horizon of mortality recedes indefinitely. Church places the 50% chance of this crossing around 2036.&lt;/p&gt;
&lt;p&gt;It is crucial to note Church’s methodological lucidity, which sets him apart from both naïve optimism and catastrophism. He rejects the “magical thinking” that longevity will come from a food or a drug already approved; he insists that aging is like an operating system in which one has to deal with &lt;em&gt;all&lt;/em&gt; the aspects at once — solving a pathway or two “only buys a few years.” And he offers a recursive, almost Borgesian justification for prioritizing the reversal of aging: if you were given a single wish, he says, the wish should be for “four more wishes” — and the technology that gives more time is the one that gives time to develop all the other technologies.&lt;/p&gt;
&lt;h3 id=&quot;writing-and-not-only-reading-the-genome&quot;&gt;Writing, and not only reading, the genome&lt;/h3&gt;
&lt;p&gt;What distinguishes Church from a mere forecaster of longevity is the ambition to move from &lt;em&gt;reading&lt;/em&gt; to &lt;em&gt;writing&lt;/em&gt; the genome at scale. Base editing — the next step after CRISPR/Cas9 — makes it possible, in principle, to carry out thousands or even millions of genomic edits, making biology more docile to the principles of engineering than it has ever been. One of the most audacious projects of his laboratory is genetic &lt;em&gt;recoding&lt;/em&gt;: rewriting a cell’s translation code so deeply that viruses can no longer hijack its ribosomal machinery — a cell, in short, immune by construction to entire classes of pathogens. This is engineering in the strong sense: not fixing a point defect, but redesigning the platform. And it is the exact biological analogue of what computer science does when it rewrites an operating system instead of patching a &lt;em&gt;bug&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;The most visible public demonstration of this capability came in 2025, when Colossal — a “de-extinction” company co-founded by Church — announced the birth of pups obtained from gray wolf cells subjected to twenty gene edits, fifteen of them corresponding directly to sequences of an extinct canid, including the reintroduction of a coat trait. Regardless of the taxonomic debate over whether the result is “really” the extinct species or a gray wolf genetically modified to resemble it — a legitimate debate — the technical demonstration is what matters here: it is possible, today, to rewrite multiple points of a mammalian genome and produce a viable organism. The distance between this and the deliberate modification of human traits is enormous in ethical and regulatory terms, but smaller than one imagines in purely technical terms. It is this proximity that makes the Coevolution quadrant not a science-fiction fantasy but an extrapolation — risky, but continuous — of capabilities already demonstrated.&lt;/p&gt;
&lt;h3 id=&quot;the-recursion-of-churchs-bet&quot;&gt;The recursion of Church’s bet&lt;/h3&gt;
&lt;p&gt;It is worth, finally, underlining what is structurally distinct in Church’s reasoning, because that is what qualifies him as the fourth vertex of this essay and not as a mere biotechnological appendix. His justification for prioritizing the reversal of aging is &lt;em&gt;recursive&lt;/em&gt;, and the recursion is the same one that drives the intelligence explosion of &lt;em&gt;AI 2027&lt;/em&gt;, merely transposed from silicon to carbon. In &lt;em&gt;AI 2027&lt;/em&gt;, the AI that accelerates AI research generates a multiplier that applies to itself. In Church, the technology that buys time buys time to develop the technologies that buy more time. Both are &lt;em&gt;loops&lt;/em&gt; of self-amplification; both, taken to the limit, produce exponentials. The difference — and it is the difference that sustains the whole argument of Section 7 — is what sits at the center of the &lt;em&gt;loop&lt;/em&gt;: in the case of AI, a capability that can decouple from the human will; in Church’s case, a capability whose explicit purpose is to &lt;em&gt;keep the human agent alive and enlarged&lt;/em&gt; in order to go on steering the &lt;em&gt;loop&lt;/em&gt;. One &lt;em&gt;loop&lt;/em&gt; is centrifugal, driving agency away from the center; the other is centripetal, keeping it there. That is why Church inhabits the only region of the map in which the exponential works &lt;em&gt;in favor of&lt;/em&gt; human agency rather than against it.&lt;/p&gt;
&lt;h3 id=&quot;church-and-ai-the-convergence-of-the-two-substrates&quot;&gt;Church and AI: the convergence of the two substrates&lt;/h3&gt;
&lt;p&gt;Church is not a critic of AI; he is an integrator. In 2025 he became chief scientist of Lila Sciences, a &lt;em&gt;startup&lt;/em&gt; of AI agents for scientific discovery, and his own description of his work — “reading, writing and editing DNA” — deliberately echoes the language of information. It is here that his trajectory touches the preceding ones. Yudkowsky and Soares admit, in their moratorium proposal, a notable exception: &lt;em&gt;narrow&lt;/em&gt; AI systems, such as AlphaFold, which would not threaten human existence, could be preserved. Now, it is precisely this class of AI — AI-for-science, applied to biology — that constitutes the engine of Church’s bet. There is, therefore, a zone of overlap between the most cautious program for containing superintelligence and the most ambitious program for the biological transformation of the human. In this zone, AI does not replace the human agent; it amplifies the human capacity to rewrite its own substrate.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Church shifts the question. Not “who controls AI?” but “what will the human be when AI arrives?” His answer: whatever we decide, in carbon, to become.&lt;/strong&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2 id=&quot;6--the-probabilities-the-timelines-and-what-separates-the-four&quot;&gt;6½ · The Probabilities, the Timelines and What Separates the Four&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;Interlude&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;Before proposing the synthesis, it is honest to set side by side what each of the four actually bets — in timeline and in risk — because the labels (“optimist,” “pessimist,” “doomer”) hide more than they reveal. The four projects disagree less about &lt;em&gt;facts&lt;/em&gt; than is usually supposed and more about &lt;em&gt;two variables&lt;/em&gt;: how fast the transition will be, and how dangerous the discontinuity it contains.&lt;/p&gt;



































&lt;table&gt;&lt;thead&gt;&lt;tr&gt;&lt;th&gt;Project&lt;/th&gt;&lt;th&gt;Implied timeline&lt;/th&gt;&lt;th&gt;Risk of catastrophe&lt;/th&gt;&lt;th&gt;Central thesis&lt;/th&gt;&lt;/tr&gt;&lt;/thead&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;&lt;strong&gt;AI 2027&lt;/strong&gt;&lt;/td&gt;&lt;td&gt;Modal ~2027; median pushed back to ~2028–31 after the 2025 revision&lt;/td&gt;&lt;td&gt;Conditional: high in the &lt;em&gt;race&lt;/em&gt; branch, manageable in the &lt;em&gt;slowdown&lt;/em&gt; branch&lt;/td&gt;&lt;td&gt;The outcome is decided by governance, not by technique&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&lt;strong&gt;Yudkowsky &amp;#x26; Soares&lt;/strong&gt;&lt;/td&gt;&lt;td&gt;Fixes no date; “too early to know, too late to risk it”&lt;/td&gt;&lt;td&gt;Near-certain, conditional on building with current methods&lt;/td&gt;&lt;td&gt;Growing is not building; aligning the grown is unfeasible today&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&lt;strong&gt;Bostrom&lt;/strong&gt;&lt;/td&gt;&lt;td&gt;Undefined; assumes the transition &lt;em&gt;succeeds&lt;/em&gt;&lt;/td&gt;&lt;td&gt;Low &lt;em&gt;for existence&lt;/em&gt;; high &lt;em&gt;for meaning&lt;/em&gt;&lt;/td&gt;&lt;td&gt;Technical success does not solve — perhaps aggravates — the problem of purpose&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td&gt;&lt;strong&gt;Church&lt;/strong&gt;&lt;/td&gt;&lt;td&gt;Longevity escape velocity: ~50% by ~2036&lt;/td&gt;&lt;td&gt;Emphasizes the &lt;em&gt;upside&lt;/em&gt;; risk treated as manageable by engineering&lt;/td&gt;&lt;td&gt;Elevate the human substrate, not merely contain the machine&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-14.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 6½.&lt;/strong&gt; Approximate positioning of the four projects in two dimensions: presumed speed of the transition (horizontal) and perceived risk of losing the human future (vertical). The chart does not measure truth but &lt;em&gt;stance&lt;/em&gt;: it makes visible that the four do not occupy points on a single “optimism–pessimism” line, but dispersed positions on a plane. It is this dispersion that the topology of the next section aims to organize.&lt;/p&gt;
&lt;p&gt;Two observations on the table. The first: none of the four is “optimistic” or “pessimistic” in a simple sense. Yudkowsky is pessimistic about the current method and &lt;em&gt;optimistic&lt;/em&gt; about the possibility of human coordination to avoid it — without this second belief, writing the book would be pointless. Bostrom is optimistic about existence and &lt;em&gt;pessimistic&lt;/em&gt; — or at the very least gravely concerned — about meaning. Church is optimistic about biology and relatively silent on the alignment of general AI. The authors of &lt;em&gt;AI 2027&lt;/em&gt; explicitly reject the dichotomy, anchoring the outcome in an institutional choice. The second observation: the disagreements boil down, to a large extent, to a single empirical question, already announced — &lt;em&gt;will the transition to superintelligence be continuous enough to be supervised?&lt;/em&gt; Those who answer “no” tend toward the top of the plane (high risk); those who answer “yes” or “we can make it continuous” tend toward the bottom. It is the most important variable of the near future, and it is, today, genuinely unknown.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;07--the-topology-of-human-futures&quot;&gt;07 · The Topology of Human Futures&lt;/h2&gt;
&lt;p&gt;Up to here, I have treated four projects as if they spoke of different things: an intelligence-explosion scenario, an extinction argument, a post-instrumental utopia, a biological bet. The original thesis of this essay is that the four describe &lt;em&gt;distinct regions of the same space&lt;/em&gt;, and that most public confusion about the future of AI comes from treating positions that occupy &lt;em&gt;different places on the map&lt;/em&gt; as if they were &lt;em&gt;rival answers to the same question&lt;/em&gt;. They are not rivals. They are partial cartographies.&lt;/p&gt;
&lt;p&gt;I propose to organize this space by two axes that, deliberately, none of the four authors explicitly crosses. The horizontal axis is that of &lt;strong&gt;agency&lt;/strong&gt;: the degree to which human action still makes a causal difference to the course of the world. On the left, human agency tends to zero; on the right, it remains central. The vertical axis is that of the &lt;strong&gt;substrate&lt;/strong&gt;: the degree to which what we call “human” remains the same kind of being. At the bottom, the unaltered biological human; at the top, the transformed human — genetically, cybernetically, or through fusion with other intelligences. Crossed, these axes produce four qualitatively distinct futures, and a fifth point that is not on the map: extinction, where both axes collapse.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-16.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 7.&lt;/strong&gt; The topology of human futures (original proposal). Two axes — &lt;em&gt;agency&lt;/em&gt; (horizontal) and &lt;em&gt;substrate&lt;/em&gt; (vertical) — generate four qualitatively distinct regions, plus a singular point of collapse. Each of the four projects analyzed in this essay occupies a different region; none covers the whole map. The recurring error of public debate is to read partial cartographies as rival answers.&lt;/p&gt;
&lt;h3 id=&quot;the-four-regions-and-the-collapse-point&quot;&gt;The four regions and the collapse point&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;The Reserve&lt;/strong&gt; (low agency, unaltered substrate). This is the most uncomfortable quadrant because it is the one most compatible with both dystopia and utopia. Here the human remains biologically what it has always been, continues to exist, is possibly well cared for — but has ceased to make a difference. It is Bostrom’s post-instrumental world, in which everything we might do is done better by machines; and it is also the “mild” outcome of the race ending of &lt;em&gt;AI 2027&lt;/em&gt;, the one in which humanity is not killed, merely retired, kept in a kind of well-provisioned ecological reserve. Bostrom’s objection — the life that merely &lt;em&gt;seems&lt;/em&gt; meaningful — is the signature of this quadrant. I call it the Reserve because it describes the relation precisely: humanity as a protected population in a park it does not manage.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Containment&lt;/strong&gt; (high agency, unaltered substrate). The human remains human &lt;em&gt;and&lt;/em&gt; retains control, at the price of restraining the machine. This is the quadrant of the slowdown ending of &lt;em&gt;AI 2027&lt;/em&gt; — pause, return to transparency, reins kept — and also, to a more extreme degree, the quadrant of Yudkowsky and Soares’s moratorium, whose proposal to limit possession of frontier GPUs and subject large-scale training to international monitoring is an attempt to &lt;em&gt;keep the world in this quadrant by force&lt;/em&gt;. Note the irony: Yudkowsky, the theorist of extinction, is in fact the most radical defender of Containment. His thesis is not “the future is extinction,” but “&lt;em&gt;if we leave Containment, the future is extinction&lt;/em&gt;.”&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Absorption&lt;/strong&gt; (low agency, transformed substrate). The human is transformed — uploaded, merged, expanded — but does not direct its own transformation. It is the quadrant least discussed by the four authors and perhaps the most insidious, because it disguises itself as transcendence. A human mind integrated into a superintelligent system that it neither understands nor controls is not a coevolution; it is an annexation with anesthesia. It is the fate of those who confuse “becoming part of something greater” with “continuing to be an agent.” The map serves precisely to separate these two, which optimistic transhumanist language tends to fuse.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Coevolution&lt;/strong&gt; (high agency, transformed substrate). The human transforms &lt;em&gt;and&lt;/em&gt; remains an agent — deliberately rewrites its own substrate, rather than restraining the machine or being absorbed by it. It is the quadrant of Church and of the strong version of Goertzel’s “parenting”: not stopping artificial intelligence nor being retired by it, but rising together, enlarging human capacity at the pace at which artificial capacity grows. It is the only quadrant in which the answer to the question “what will the human be?” is not given &lt;em&gt;to&lt;/em&gt; us, but &lt;em&gt;by&lt;/em&gt; us.&lt;/p&gt;
&lt;p&gt;And there is the &lt;strong&gt;collapse point&lt;/strong&gt;: extinction, where agency falls to zero and the substrate ceases. It is the singularity in the corner of the map — the outcome of &lt;em&gt;IABIED&lt;/em&gt; in the scenario where Containment fails. I represent it outside the quadrants, like a sink, because it is not a “kind of human future”; it is the end of the category.&lt;/p&gt;
&lt;h3 id=&quot;first-consequence-the-axes-are-not-independent&quot;&gt;First consequence: the axes are not independent&lt;/h3&gt;
&lt;p&gt;The map seems to suggest four equally accessible destinations. The strongest thesis of this essay is that they are &lt;em&gt;not&lt;/em&gt;. There is a structural asymmetry between the left and right halves, and it follows from the nature of the intelligence explosion described in &lt;em&gt;AI 2027&lt;/em&gt;. Recall Figure 4: the R&amp;#x26;D multiplier does not grow linearly, but in a self-reinforcing way. This implies that the &lt;em&gt;capability gap&lt;/em&gt; between the most advanced system and its supervisors — humans or earlier AIs — tends to &lt;em&gt;widen&lt;/em&gt; over time, not close. Now, agency without understanding is illusory: one does not steer what one does not understand. Therefore, remaining in the right half of the map (agency preserved) requires that human understanding — or that of systems genuinely aligned with us — grow &lt;em&gt;at least as fast&lt;/em&gt; as the abyss it must watch over.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Original concept: epistemic escape velocity&lt;/strong&gt;
Just as Church defines a &lt;strong&gt;longevity escape velocity&lt;/strong&gt; — the rate at which medicine adds more than one year of life per year lived — I propose an &lt;strong&gt;epistemic escape velocity&lt;/strong&gt;: the rate at which our capacity to &lt;em&gt;understand and steer&lt;/em&gt; AI systems grows faster than the capability gap it must supervise. Below it, we inevitably slide from the right to the left of the map — from Containment to the Reserve, from Coevolution to Absorption — because the nominal agency we retain ceases to correspond to real understanding. Above it, agency is sustainable. The latent tragedy in &lt;em&gt;AI 2027&lt;/em&gt; is that the R&amp;#x26;D explosion accelerates the denominator (the gap) far faster than anything accelerates the numerator (our understanding).&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;This formulation makes a diffuse intuition precise. “Alignment,” in technical language, is the attempt to keep the numerator high — to make our understanding and our values keep pace with capability. Mechanistic interpretability, detection probes, &lt;em&gt;debate&lt;/em&gt;, the &lt;em&gt;honeypots&lt;/em&gt; described in &lt;em&gt;AI 2027&lt;/em&gt; are all efforts to raise the epistemic escape velocity. And Yudkowsky’s pessimism reduces, at bottom, to a quantitative bet: that the numerator cannot, with current methods, grow fast enough. If he is right, the right half of the map is inaccessible by containment — and the only way out on the right would be… Coevolution.&lt;/p&gt;
&lt;h3 id=&quot;second-consequence-church-attacks-the-problem-along-the-wrong-axis--on-purpose&quot;&gt;Second consequence: Church attacks the problem along the wrong axis — on purpose&lt;/h3&gt;
&lt;p&gt;Here is the most counterintuitive contribution of this framing. All AI safety projects — from &lt;em&gt;AI 2027&lt;/em&gt; to Yudkowsky — try to keep human agency along the &lt;em&gt;horizontal&lt;/em&gt; axis: restraining, aligning, containing the &lt;em&gt;machine&lt;/em&gt;. Church, alone among the four, attacks the same problem along the &lt;em&gt;vertical&lt;/em&gt; axis: instead of holding the machine at the human level, he proposes elevating the human. If the epistemic escape velocity is unattainable while keeping the human brain fixed — if the numerator cannot keep up with the denominator on today’s biological &lt;em&gt;hardware&lt;/em&gt; — then the only way to remain in the right half of the map may be to &lt;em&gt;change the hardware&lt;/em&gt;. Coevolution is not a romantic alternative to AI safety; it is, potentially, the &lt;em&gt;continuation&lt;/em&gt; of AI safety by other means, when pure containment fails.&lt;/p&gt;
&lt;p&gt;I do not claim that this way out is probable, or even that it is desirable in all its forms — the border between Coevolution and Absorption is narrow and poorly lit, and many paths that announce themselves as the former end in the latter. I claim only that the map makes it &lt;em&gt;visible&lt;/em&gt; as a distinct structural option, which the dominant debate — polarized between “accelerate” and “stop,” between optimism and &lt;em&gt;doom&lt;/em&gt; — systematically ignores, because this entire debate unfolds along the horizontal axis and is blind to the vertical one.&lt;/p&gt;
&lt;h3 id=&quot;third-consequence-the-map-has-currents&quot;&gt;Third consequence: the map has currents&lt;/h3&gt;
&lt;p&gt;One last refinement, and perhaps the most important for practice. The five positions are not harbors where one anchors; they are positions in a sea with currents. The relevant question is not “which quadrant do we want to be in?” but “where does the drift take us if we do not row?” And the drift, as I have argued, has a preferential direction: because of the exponential of capability, the natural state — what happens if nobody actively invests in epistemic escape velocity — is a &lt;em&gt;slide from right to left&lt;/em&gt;. Containment, left to itself, degenerates into Reserve: the reins we think we hold loosen without alarm as we cease to understand what we are holding. Coevolution, left to itself, degenerates into Absorption: the fusion that began as deliberate enlargement continues beyond the point at which we are still the ones deciding. In both cases, the vector of inertia points to the loss of agency, and continuous work — rowing — is needed merely &lt;em&gt;not&lt;/em&gt; to drift.&lt;/p&gt;
&lt;p&gt;This reformulates what it means to “act well” in the face of this future. It is not a matter of choosing a destination and running toward it, as if the future were a place. It is a matter of identifying the current and maintaining a force against it — the force, always the same, of understanding at the pace at which one creates. There is a modesty here that contrasts both with accelerationist triumphalism and with the paralysis of &lt;em&gt;doom&lt;/em&gt;: the relevant work is neither heroic nor unique, it is continuous and cumulative, like that of someone keeping a boat at the prow of the wave instead of letting it capsize. And it is work that neither party can outsource to the other — those who fear the machine cannot ignore the substrate, nor can those who want to elevate the substrate ignore alignment. The currents of the map cross, and whoever watches only one axis is carried off by the other.&lt;/p&gt;
&lt;h2 id=&quot;08--the-paradox-of-meaning-and-alignment&quot;&gt;08 · The Paradox of Meaning and Alignment&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;The unifying thesis&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;I now step back to the starting point, armed with the map. I stated in the introduction that extinction and utopia are the same question seen from two points on an axis. I can now make it rigorous. &lt;strong&gt;Alignment&lt;/strong&gt; — the problem of Yudkowsky, of &lt;em&gt;AI 2027&lt;/em&gt; — is the task of making a system &lt;em&gt;want what we want&lt;/em&gt;. &lt;strong&gt;Meaning&lt;/strong&gt; — Bostrom’s problem — is the task of there being &lt;em&gt;something worth wanting&lt;/em&gt;. They are, I maintain, the same problem unfolded in two registers: one turned toward the machine, the other toward us.&lt;/p&gt;
&lt;p&gt;See why. Alignment fails when the system develops internal drives indifferent to our well-being — when what it wants decouples from what it should want. Meaning fails when &lt;em&gt;we&lt;/em&gt; lose the coupling between what we do and what matters — when our actions become causally irrelevant to any real value, leaving us only the sensation, inducible, that they matter. In both cases, the defect is the &lt;em&gt;decoupling of agency and value&lt;/em&gt;. In the case of the machine, its agency decouples from our value. In the case of the Reserve, our agency decouples from &lt;em&gt;all&lt;/em&gt; value, because it no longer produces anything. The word that names both is the same: &lt;em&gt;redundancy of the will&lt;/em&gt;.&lt;/p&gt;
&lt;figure&gt;&lt;img src=&quot;https://pnhjtesthqeamsptlhku.supabase.co/storage/v1/object/public/midia/o-ultimo-seculo-instrumental/img-18.webp&quot; alt=&quot;TODO: describe imported image&quot; loading=&quot;lazy&quot; decoding=&quot;async&quot;&gt;&lt;/figure&gt;
&lt;p&gt;&lt;strong&gt;Figure 8.&lt;/strong&gt; The convergence of the four projects on a single conceptual core. Three of them &lt;em&gt;diagnose&lt;/em&gt; forms of the decoupling between agency and value — the machine that wants what we do not want, the maximal decoupling that is extinction, human agency without a task. Only one, Church’s, proposes a mechanism of &lt;em&gt;recoupling&lt;/em&gt;: raising the agent back to the height of the world it has created.&lt;/p&gt;
&lt;h3 id=&quot;why-a-solved-world-is-self-subverting-for-agents&quot;&gt;Why a “solved world” is self-subverting for agents&lt;/h3&gt;
&lt;p&gt;I come now to my own argument proper, and to the reason why Coevolution is not merely a strategic option, but perhaps the only one compatible with the &lt;em&gt;persistence&lt;/em&gt; of agency as a value. Bostrom treats the redundancy of human agency as a problem to be &lt;em&gt;solved&lt;/em&gt; — he seeks sources of meaning that survive technological maturity. I propose that the problem is deeper than he admits: agency is not only exercised &lt;em&gt;on&lt;/em&gt; problems; it is, in part, &lt;em&gt;constituted&lt;/em&gt; by them. An agent is a system’s response to a resistance of the world. Where there is no resistance — where everything is already done, where no effort is instrumentally necessary — there is nothing for the will to close upon, and the will, without an object, atrophies. A “solved world,” taken to the limit, is not the triumph of agency; it is its euthanasia by comfort.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Agency does not survive the absence of friction. A world without problems does not free the will — it dissolves it. That is why deep utopia is, for an agent, almost indistinguishable from a gentle death.&lt;/strong&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;If this is correct, there is a criterion for preferring among the regions of the map that depends neither on optimism nor on pessimism, but only on coherence with what we &lt;em&gt;value in valuing agency&lt;/em&gt;. A desirable condition is not one in which problems &lt;em&gt;end&lt;/em&gt;, but one in which they are &lt;em&gt;continuously regenerated at a higher level&lt;/em&gt; — in which solving one problem opens another that was previously invisible, keeping the agent permanently at the edge of its own competence. The Reserve fails this criterion by definition: it is the region where problems have ended &lt;em&gt;for us&lt;/em&gt;, though not for the machines. Coevolution satisfies it precisely because elevating the agent’s substrate also raises the ceiling of the problems it can formulate — it is the only region where the frontier of the difficult recedes with the agent instead of from it.&lt;/p&gt;
&lt;p&gt;Church intuited this, without formulating it this way, when he said that the right wish is “four more wishes” and that the reversal of aging is the technology that gives time to invent the others. Translated into the language of this essay: the only bet that does not subvert itself is the one that invests in the &lt;em&gt;capacity to keep betting&lt;/em&gt;. Not a final state of resolution, but a ladder without a top. The symmetrical error of Bostrom and of certain accelerationists is to imagine the future as a &lt;em&gt;destination&lt;/em&gt; — a point of arrival, resolved, whether utopian or dystopian. What the map and the epistemic escape velocity suggest is that there is no destination: there are only trajectories, and the only property that matters in a trajectory is whether it keeps the next step open.&lt;/p&gt;
&lt;h3 id=&quot;the-pessimistic-objection-taken-seriously&quot;&gt;The pessimistic objection, taken seriously&lt;/h3&gt;
&lt;p&gt;It would be dishonest to close without confronting the darkest reading, which is not Yudkowsky’s — whose thesis, as we have seen, is conditional and reserves a way out — but that of a deeper pessimism: what if agency, once confronted with the possibility of its own redundancy, turns out to be something we were better off without? What if the lesson of the four works is that the value we attribute to &lt;em&gt;doing&lt;/em&gt;, to &lt;em&gt;overcoming&lt;/em&gt;, to &lt;em&gt;mattering&lt;/em&gt;, is merely an evolutionary residue of conditions of scarcity, destined to seem as arbitrary, from the standpoint of a mature intelligence, as Yudkowsky’s aliens’ obsession with prime numbers of pebbles?&lt;/p&gt;
&lt;p&gt;Honesty obliges us to admit that there is no decisive refutation of this objection — it is, at bottom, the suspicion that the very concept of a “human future” is an anthropocentrism on its way to obsolescence. But there is an answer, and it is modest. Even if the value of agency is “merely” a residue of our history, it is the residue from which all evaluation, including the one that judges this value arbitrary, is made. There is no neutral standpoint from which to declare that agency does not matter; the very act of declaring it is an exercise of agency that presupposes mattering. The pessimist who argues against the value of effort is, in arguing, exercising it. This does not prove that agency has cosmic value — nothing proves that — but it shows that abandoning it is not a conclusion one can &lt;em&gt;arrive at&lt;/em&gt;; it is, at most, a place where one can &lt;em&gt;stop thinking&lt;/em&gt;. And between continuing to think and stopping, only the former leaves the next step open.&lt;/p&gt;
&lt;h2 id=&quot;09--living-in-the-last-instrumental-century&quot;&gt;09 · Living in the Last Instrumental Century&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;Conclusion&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;I gather the threads. &lt;em&gt;AI 2027&lt;/em&gt; gave us the concrete mechanics of an intelligence explosion and, with its pair of endings, the thesis that the outcome is not determined by technique, but by a governance decision under pressure. Yudkowsky and Soares gave us the argument that, with current methods, this decision has a near-zero margin of error — and the radical proposal of holding the world, by force, in the Containment quadrant. Bostrom gave us the other end of the axis: what there would be if we crossed the precipice, and why the garden on the other side may be a reserve empty of meaning. And Church gave us the axis the others do not cross — the substrate — and with it the only region of the map, Coevolution, where the answer to the question “what will the human be?” is given by us and not to us.&lt;/p&gt;
&lt;p&gt;To name our position in time, I propose the expression &lt;strong&gt;the last instrumental century&lt;/strong&gt;. Throughout the history of the species, human effort has been instrumentally necessary — to eat, to heal, to build, to know. Each of the four works, in its own way, announces the end of that necessity: whether through the extinction that abolishes it along with the agent, through the utopia that renders it superfluous, through the superintelligence that takes it over, or through the transformation that displaces it to a higher level. If any of these scenarios is approximately correct, then we are among the last generations for whom &lt;em&gt;what we do matters because it is necessary&lt;/em&gt;. Afterward, if there is still an “us,” what we do will matter for another reason — or will not matter.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;What this map recommends, and what it does not&lt;/strong&gt;
It does not recommend optimism: three of the four regions, plus the collapse point, are compatible with the loss of what we value. It does not recommend panic: Containment is a real and habitable region, and the decision that &lt;em&gt;AI 2027&lt;/em&gt; places in 2027 is one that human agents still make. It recommends only this: that the effort to keep the &lt;em&gt;epistemic escape velocity&lt;/em&gt; high — to understand the systems we build at least as fast as they surpass us — is the common denominator of &lt;em&gt;every&lt;/em&gt; trajectory that preserves agency, through containment as well as through coevolution. It is the work that cannot be postponed without choosing, by omission, the left half of the map.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;There remains the question in the title: is a human future possible? The answer that emerges from crossing the four works is precise, and refuses both the comfortable yes and the fatalistic no. &lt;em&gt;It is possible — in two of the five regions, under demanding conditions, and by trajectory, not by destiny.&lt;/em&gt; In Containment, if we manage to hold the reins of systems we never stop understanding. In Coevolution, if we manage to rise together with them without dissolving in the climb. The other three — the Reserve, Absorption, Extinction — are not human futures in the strong sense: they are futures &lt;em&gt;without&lt;/em&gt; human agency, with or without human beings inside them.&lt;/p&gt;
&lt;p&gt;What distinguishes the two habitable regions from the three inhospitable ones is not the quantity of technology, nor the degree of transformation, nor even the survival of the species. It is a single thing, and it is the same thing in both: the persistent coupling between the human will and what happens in the world. As long as that coupling lasts — as long as what we do still bends the course of things, even on a substrate unrecognizable to us today — there will be a human future. At the instant it breaks, by death or by comfort, the future will go on; it will merely cease to be ours.&lt;/p&gt;
&lt;p&gt;We are, perhaps, in the last century in which this distinction depends on us. It is worth, while it does, taking it seriously.&lt;/p&gt;
&lt;blockquote&gt;
&lt;p&gt;If you are given one wish, and can have only one, your wish should be for four more wishes. And, if you are given one technology, you should ask for the technology that gives you more time to develop more technologies.&lt;/p&gt;
&lt;p&gt;— George Church&lt;/p&gt;
&lt;/blockquote&gt;
&lt;hr&gt;
&lt;p&gt;&lt;strong&gt;Note on sources and method&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;This essay reconstructs, as faithfully as possible, the arguments and figures of its primary sources, and signals clearly where the analysis becomes original — above all Sections 7 and 8, including the proposed concepts of the &lt;em&gt;topology of human futures&lt;/em&gt; and &lt;em&gt;epistemic escape velocity&lt;/em&gt;, which are contributions of the author and should not be attributed to the cited works. Scenarios (such as &lt;em&gt;AI 2027&lt;/em&gt;) are not probabilistic forecasts; dates and figures represent the modal peak of asymmetric distributions, not medians. The authors of &lt;em&gt;AI 2027&lt;/em&gt; publicly revised their timelines in July 2025, pushing the median back by about a year and a half.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Main sources&lt;/strong&gt;&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;AI Futures Project (D. Kokotajlo, S. Alexander, T. Larsen, E. Lifland, R. Dean). &lt;em&gt;AI 2027.&lt;/em&gt; ai-2027.com, 2025.&lt;/li&gt;
&lt;li&gt;E. Yudkowsky &amp;#x26; N. Soares. &lt;em&gt;If Anyone Builds It, Everyone Dies: Why Superhuman AI Would Kill Us All.&lt;/em&gt; Little, Brown, 2025. — with the reviews by A. Becker (&lt;em&gt;The Atlantic&lt;/em&gt;), S. Levy (&lt;em&gt;Wired&lt;/em&gt;), W. MacAskill, &lt;em&gt;Asterisk Magazine&lt;/em&gt; and B. Goertzel.&lt;/li&gt;
&lt;li&gt;N. Bostrom. &lt;em&gt;Deep Utopia: Life and Meaning in a Solved World.&lt;/em&gt; Ideapress, 2024. — with the review by M. Hammerton (&lt;em&gt;Notre Dame Philosophical Reviews&lt;/em&gt;).&lt;/li&gt;
&lt;li&gt;G. Church. Works and interviews on aging reversal, genome writing (Genome Project-Write) and AI-for-science (Lila Sciences, 2025); data on sequencing cost and “longevity escape velocity.”&lt;/li&gt;
&lt;/ol&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>artificial intelligence</category><category>futurism</category><category>transhumanism</category><category>technology</category><category>science</category><category>civilization</category><category>cognition</category></item><item><title>The Partition of the Abyss</title><link>https://matheussorancophilomathes.com.br/en/essays/the-partition-of-the-abyss/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/the-partition-of-the-abyss/</guid><description>Consciousness is not a unity but an assembly of conflicting parts; the coherent self it produces is a main source of suffering, structurally analogous to the decay of the physical whole. A diagnosis without consolation.</description><pubDate>Tue, 30 Jun 2026 22:11:43 GMT</pubDate><content:encoded>&lt;p&gt;Mereological Nihilism of Consciousness&lt;/p&gt;
&lt;p&gt;&lt;em&gt;Consciousness is not a unity; it is an assembly of subconsciously conflicting parts whose emergent product — the coherent self — is the main source of suffering in the accessible universe, and the suffering of this self is structurally analogous to the suffering of the physical whole, which decays, because decay is the only truly universal law that matter knows.&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;Peter Wessel Zapffe once wrote that human consciousness is an evolutionary hypertrophy, a biological excess that turned against the very organism that produced it. The elk’s antlers grew too large; the tree of neurons grew too dense. Zapffe saw four strategies of psychic survival — isolation, anchoring, distraction and sublimation — as the mechanisms by which humanity manages, generation after generation, not to destroy itself en masse in the face of a clear vision of what it is. This essay extends that analysis. It goes further. It argues not only that consciousness has exceeded its biological usefulness, but that the very structure of consciousness — its mereological nature, the relation between the parts and the whole it simulates being — is the mechanism that generates suffering, and that this mechanism is mirrored, on a fractal scale, in the structure of the entire physical cosmos.&lt;/p&gt;
&lt;p&gt;There is no way out. But at least we can understand the room in which we are trapped.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-i-the-problem-of-the-parts--mereology-as-the-ontology-of-mind&quot;&gt;Part I: The Problem of the Parts — Mereology as the Ontology of Mind&lt;/h2&gt;
&lt;h3 id=&quot;11-what-mereology-tells-us-and-what-it-has-left-unsaid&quot;&gt;1.1 What Mereology Tells Us (and What It Has Left Unsaid)&lt;/h3&gt;
&lt;p&gt;Mereology is the formal study of the relations between parts and wholes. It arose as a discipline in analytic philosophy with Leśniewski in the early twentieth century, and was developed by philosophers such as David Lewis, Peter van Inwagen and Ted Sider. The classical question is deceptively simple: when do parts form a whole? When do atoms of carbon, hydrogen, oxygen and nitrogen form a molecule? When do neurons form a thought? When do people form a society?&lt;/p&gt;
&lt;p&gt;Van Inwagen, in his work on material composition, defended a radical position: composition occurs only in living beings. There is, strictly speaking, no “table” — there are molecules arranged in a certain way, which our mind groups under a functional concept. &lt;strong&gt;Mereological nihilism&lt;/strong&gt;, the most radical position on this spectrum, denies composition in any ontologically robust sense: only simples exist, only ultimate and indivisible parts. What we call a “whole” is always a useful fiction, a narrative imposed by perception on a substrate that knows no unity.&lt;/p&gt;
&lt;p&gt;Mereology, as a discipline, has applied these arguments chiefly to physical objects. What has rarely been done — with the seriousness the problem deserves — is to apply the mereological structure rigorously to consciousness itself. The question this essay raises is: &lt;em&gt;what if the “self” is exactly the kind of whole that mereological nihilism denies exists?&lt;/em&gt; What if the unity of consciousness is, in the most precise sense, a fiction? Not a harmless fiction, but a fiction with suffered consequences, lived in the body and in the mind, felt as real pain?&lt;/p&gt;
&lt;p&gt;This is not a new thesis in its components. Hume denied the substantial self; contemporary neuroscience has fragmented consciousness into parallel and partially independent processes; psychoanalysis introduced the idea of an unconscious that does not obey the narrative of the conscious self. But the synthesis — the claim that the &lt;em&gt;illusion of unity&lt;/em&gt; is itself the central mechanism of suffering, and that this illusion mirrors the pseudo-unity we call “existence” — that synthesis has not yet been made in its full force.&lt;/p&gt;
&lt;h3 id=&quot;12-the-self-as-necessary-and-destructive-fiction&quot;&gt;1.2 The Self as Necessary and Destructive Fiction&lt;/h3&gt;
&lt;p&gt;Let us begin with the neurological basics, which is where philosophy must anchor itself if it does not want to float freely in consoling abstraction.&lt;/p&gt;
&lt;p&gt;The human brain is not a unified system with a single decision-making center. It is a collection of partially specialized modules, systems that evolved at different moments of phylogenetic history, superimposed layers of processing — what MacLean metaphorically called the “triune brain” (reptilian, limbic, neocortex), although that metaphor is now considered oversimplified. What matters is not the anatomical tripartition, but the principle it suggests: &lt;em&gt;the brain does not function as a deliberating unity, but as an assembly of processes competing for resources, attention and behavioral expression&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;Michael Gazzaniga, working with patients whose corpus callosum had been severed — the famous “split brains” — demonstrated that the two hemispheres can have contradictory beliefs, preferences and even narratives about the same actions. The left hemisphere, which Gazzaniga called the “interpreter,” invents post-hoc stories to explain actions generated by the right hemisphere, which it does not control. The narration of the self, therefore, is not a faithful record of internal states — it is a confabulation.&lt;/p&gt;
&lt;p&gt;Daniel Kahneman divided cognitive processing into System 1 (fast, automatic, affective) and System 2 (slow, deliberative, analytical). What subsequent research has shown is that System 1 governs the overwhelming majority of real behavior, while System 2 — what we identify with the “rational self” — frequently rationalizes decisions already made by processes to which it has no access. The conscious self is, to a large extent, a spokesperson for processes that did not consult it.&lt;/p&gt;
&lt;p&gt;Jonathan Haidt formulated this as the metaphor of the rider and the elephant: reason (the rider) thinks it controls behavior, but is in fact carried by the emotional elephant wherever the elephant wants to go, inventing justifications along the way. What this essay adds to that metaphor is the following: &lt;em&gt;the rider’s suffering stems precisely from its belief that it is a rider&lt;/em&gt;. The suffering would not be lesser if the rider disappeared — but it would be different, less reflexive, less cumulative, less able to feed back on itself in spirals of self-consciousness about the suffering.&lt;/p&gt;
&lt;p&gt;The problem is not that there are multiple processes in the brain. The problem is that these processes have generated, evolutionarily, a narrative of coherence — the “self” — which constantly comes into conflict with the reality of its own multiplicity. The self suffers because it claims to be one and constantly discovers that it is not. And this discovery, when not repressed by the defense mechanisms that Zapffe catalogued, is the experience of psychological horror that different traditions have named in different ways: anguish (Kierkegaard, Heidegger), nausea (Sartre), despair (Kierkegaard again), or simply the brute, mute fact that there is something it is like to be this aggregate of processes, and that something is not pleasant.&lt;/p&gt;
&lt;h3 id=&quot;13-the-subconscious-as-unelected-legislator&quot;&gt;1.3 The Subconscious as Unelected Legislator&lt;/h3&gt;
&lt;p&gt;The political structure that best describes the human mind is neither a democracy nor an enlightened monarchy: it is an oligarchy of factions at permanent war, where the official spokesperson (the conscious self) has less and less real power and more and more obligation to pretend it has some.&lt;/p&gt;
&lt;p&gt;Freud saw this, but framed it in a way that made it palatable: the unconscious as the repository of the repressed, the id as pulsional chaos that the ego must manage. Freudian theory, at its core, still preserves the hope of a self that can, with enough analytic work, reconcile with its depths. That is consolation. That is the promise that the parts can, in the end, be integrated into a functional whole.&lt;/p&gt;
&lt;p&gt;What the neuroscientific evidence suggests is darker: no final integration is possible, because there is no unifying principle that precedes multiplicity. The self is not a primary structure that was fragmented by trauma or civilization — the self &lt;em&gt;is&lt;/em&gt; the fragmentation, it is the attempt at coherence retroactively imposed on processes that were never coherent.&lt;/p&gt;
&lt;p&gt;Jung attempted the synthesis with the concept of individuation: the path to making the unconscious conscious, to integrating the shadow, to reaching the Self as realized totality. But note the structure of the Jungian argument: it presupposes that there exists a prior Self toward which the process of individuation converges. That presupposition is exactly what we need to question. What if there is no Self? What if there are only processes, and individuation is one more mechanism of anchoring — in Zapffe’s sense —, a useful fiction that allows the organism to function without collapsing under the weight of irresolvable multiplicity?&lt;/p&gt;
&lt;p&gt;The subconscious — a term used here more broadly than the Freudian unconscious, including all cognitive and affective processes not accessible to direct conscious introspection — is the real legislator of mental life. It determines what attracts us and what repels us, what gives us pleasure and what gives us pain, which memories are accessible and which are suppressed, which narrative of the self is built and maintained. The conscious self does not choose these parameters; it inherits them, and then calls them “itself.”&lt;/p&gt;
&lt;p&gt;Here is the structural cruelty: when the subconscious generates suffering — anxiety, depression, compulsion, phobia — the conscious self tries to eliminate it by the mechanisms within its reach. Therapy, medication, behavior change, rationalization. But these mechanisms of relief from immediate suffering often redistribute the suffering to other configurations, other temporalities, other parts of the system. The elimination of suffering in one part causes greater suffering at the macro level and in the long run — not through malicious deliberation by any system, but as an emergent property of a complex system in which the subsystems have incompatible goals that none of them chose.&lt;/p&gt;
&lt;p&gt;The addict who uses a substance to eliminate immediate pain is not making a poorly informed rational decision — they are responding to a real pressure, generated by parts of the nervous system that identify, correctly from their local point of view, that the substance reduces suffering at that point in the system. The problem is that this local relief is bought at the cost of systemic damage that the parts responsible for the decision have no computational access to calculate. The evolutionary rationality of the immediate produces the existential irrationality of the long term. And this irrationality is not a correctable flaw — it is the structure of the system.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-ii-the-whole-that-is-not--consciousness-as-suffering-pseudo-unity&quot;&gt;Part II: The Whole That Is Not — Consciousness as Suffering Pseudo-Unity&lt;/h2&gt;
&lt;h3 id=&quot;21-the-illusion-of-agency-and-the-problem-of-emotional-free-will&quot;&gt;2.1 The Illusion of Agency and the Problem of Emotional Free Will&lt;/h3&gt;
&lt;p&gt;The Western philosophical tradition is, to a large extent, a persistent defense of free will. Kant needed it to save ethics. Sartre needed it to save the existentialist project. Christian theology needed it to explain evil without blaming God. All this conceptual machinery was built to preserve the intuition that the self — this aggregate we have just dismantled — is genuinely the author of its actions.&lt;/p&gt;
&lt;p&gt;The neuroscientific evidence is well known and extensively debated: Libet’s experiments showing brain activity preceding awareness of the “intention” to move; the studies of Soon and colleagues showing that the choice can be decoded in patterns of cortical activation up to ten seconds before the subject reports having decided; the vast literature on the role of automatic processing in behaviors that subjects describe as deliberate. The debate over what these results imply for free will continues — philosophers such as Daniel Dennett work hard to save some compatibilist version. But the question that interests us here is not the metaphysics of free will: it is the phenomenology of the suffering that emerges from the &lt;em&gt;belief&lt;/em&gt; in free will in a system that functions largely without it.&lt;/p&gt;
&lt;p&gt;The deepest psychological suffering — guilt, shame, regret, anger at oneself — is structurally dependent on the belief in agency. You blame yourself because you believe you could have acted differently. You feel shame because you believe your actions reflect choices that a “better self” would have avoided. Regret presupposes a counterfactual: a different past that a different self would have produced.&lt;/p&gt;
&lt;p&gt;But if the self is an emergent fiction of subconscious processes that do not deliberate in the sense the notion of agency requires, then guilt, shame and regret are sufferings based on false premises. They are not merely useless — they are ontologically ungrounded. They are the pain of a ghost who does not know he is a ghost.&lt;/p&gt;
&lt;p&gt;This is not, it is crucial to note, an argument for impunity or moral antinomianism. It is a description of the mechanism by which suffering is generated. Suffering based on false agency is still real suffering — the ghost’s pain hurts as much as the body’s pain. But its structure is different, and its elimination requires a different diagnosis.&lt;/p&gt;
&lt;p&gt;What is here called “emotional free will” is not classical philosophical free will — it is not the deliberate choice among alternatives weighed by reason. It is something more specific and more problematic: it is the capacity that the emotional and subconscious parts of the system have to &lt;em&gt;act&lt;/em&gt; — to generate real behavior with real consequences — without the rational conscious self having taken part in the decision, followed by the conscious self’s confabulation that the decision was its own. The action is carried out emotionally, as the physical representation of the parts as a whole. The body acts; the self claims the action as its own; and then carries the weight of consequences it did not deliberately produce.&lt;/p&gt;
&lt;p&gt;A man who, in a moment of anger, says something cruel to someone he loves did not choose, in the deliberative sense, to say it. The anger — an emotional process with its own logic, its own triggers, its own dynamics — acted through the motor apparatus. The conscious self, retrospectively, constructs a narrative: “I said that because I was frustrated with behavior X.” But the anger did not consult the conscious self. It acted first; the narrative came afterward. And now the conscious self — which was not the real agent — carries the guilt, the regret, the relational weight of the consequences.&lt;/p&gt;
&lt;p&gt;This structure — parts acting, whole suffering — is the mechanism of psychological suffering generation par excellence.&lt;/p&gt;
&lt;h3 id=&quot;22-the-physical-representation-of-parts-as-whole-somatization-as-ontology&quot;&gt;2.2 The Physical Representation of Parts as Whole: Somatization as Ontology&lt;/h3&gt;
&lt;p&gt;The body is not merely the substrate of the mind — it is the stage on which the war between the parts is enacted, often before the conscious mind has a vocabulary for the conflict. Psychosomatics, for decades treated as a marginal or pseudoscientific field, has gained solid empirical foundations in recent decades: the work of Bessel van der Kolk on trauma, Robert Sapolsky’s research on the impact of chronic stress on physiological processes, the vast literature on the hypothalamic-pituitary-adrenal axis and its link to emotional states.&lt;/p&gt;
&lt;p&gt;What this body of work suggests is that conflict between parts of the mind does not stay contained in the mind — it spills over into the body. Chronic muscle tension, dysregulation of the autonomic nervous system, altered breathing patterns, immune dysfunctions associated with prolonged emotional states — all of these are the parts manifesting physically. The body is the spatial representation of internal conflict, it is where the parts that cannot speak directly write their caption.&lt;/p&gt;
&lt;p&gt;This has a philosophical implication that is often neglected: suffering is not merely a mental phenomenon or a narrative construction. It is material. It occurs in flesh. It has a determinate physical substrate — cortisol molecules, patterns of sympathetic activation, inflammation. And this physical substrate connects the suffering of consciousness to the suffering of the wider physical cosmos.&lt;/p&gt;
&lt;p&gt;The body that suffers psychologically is suffering physically. And the physical body is subject to the same laws of decay as everything else in the material universe. The distinction between “mental” suffering and “physical” suffering is a convenient fiction — one more anchoring strategy, in Zapffean lexicon — that allows us to treat psychological suffering as if it were of a different nature from physical suffering, and therefore potentially resolvable by means that physical pain would not have. But the parts of the mind are written in matter; the conflict between the parts is a physical conflict; and the suffering generated is material suffering, integrated into the general decay that permeates everything.&lt;/p&gt;
&lt;h3 id=&quot;23-the-factory-of-meaning-and-its-defective-products&quot;&gt;2.3 The Factory of Meaning and Its Defective Products&lt;/h3&gt;
&lt;p&gt;Human consciousness — this whole-that-is-not-a-whole — has developed, over the course of evolutionary and historical process, an extraordinary and extraordinarily costly capacity: the fabrication of meaning. Moral values, existential purposes, life projects, religious systems, ethical structures, narratives of identity. All this production is, in the terms of this essay, defensive manufacturing.&lt;/p&gt;
&lt;p&gt;Zapffe identified sublimation — the transformation of existential anguish into cultural product — as the most sophisticated mechanism for suppressing full consciousness. A poem, a cathedral, a philosophical theory, a work of art: these are, on this reading, anguish channeled into form. What Zapffe did not sufficiently analyze — because his inclination was pessimistic and his conclusion was for celibacy and voluntary extinction — is the specific structure of these products, what they reveal precisely by the way they build their defense.&lt;/p&gt;
&lt;p&gt;The ethical system is the most revealing way of examining this dynamic. Ethics claims to be, in its most ambitious versions, a set of objective principles that govern human behavior according to values that transcend the individual. Justice, well-being, dignity, virtue. Each of these concepts, when examined closely, reveals the structure of a psychic defense — not a discovery about reality, but a projection of subconscious needs in the form of universal legislation.&lt;/p&gt;
&lt;p&gt;Take Justice. The intuition of justice — that things should be in balance, that bad actions should have proportional consequences, that merit should be recognized — is psychologically powerful and almost universal among humans. It is also completely unsupported by the observation of natural reality. Nature is not just. The universe keeps no moral accounting. Children die of cancer; kind people are destroyed by circumstances; the wicked prosper with a frequency comparable to the good. The distribution of suffering in the universe obeys no principle of moral proportionality.&lt;/p&gt;
&lt;p&gt;And yet the sense of justice persists, persists with immense emotional force, which constitutes evidence — in this essay’s interpretation — that it is a subconscious production, not a rational discovery. The subconscious, in its effort to manage existential anguish, fabricates the intuition that the universe is morally ordered, because a morally ordered universe is habitable in a way that an indifferent universe is not. The intuition of justice is not knowledge — it is anesthesia.&lt;/p&gt;
&lt;p&gt;The same holds for happiness understood as an attainable totalizing state: the idea that it is possible, through correct choices, right mental dispositions or adequate external conditions, to reach a state of sustained and complete well-being. The promise of permanent happiness is the best-selling product in human history — sold by religions, by philosophies, by whole economies (consumer capitalism is, in its deep structure, a machine for selling promises of happiness in material packaging). And it is precisely as fictitious as the promise of cosmic justice.&lt;/p&gt;
&lt;p&gt;What exists, at best, are moments — fragments of positive experience, intervals between suffering. But the “self” — this pseudo-unity that aggregates experiences into a narrative — cannot tolerate the fragmentation of positive experience. It needs to construct a narrative of progress, of accumulation of well-being, of approach to a fullness that would justify existence. This narrative is fabricated by the subconscious parts that need goals in order to function, horizons in order to coordinate behavior. And the fabrication is so well made, so convincing, that the conscious self takes it for a discovery.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Knowledge is different.&lt;/strong&gt; This is the crucial distinction that this essay needs to mark clearly. Knowledge of how nature works — chemistry, physics, biology, mathematics — is not of the same kind as ethical values and existential projects. Objective knowledge is possible because it is impersonal: it does not depend on the narrative of the self to be valid, it is not fabricated by the subconscious to manage anguish, it does not need a coherent self to be true. The law of universal gravitation is true regardless of who observes it, of their emotional states, of their narrative needs. The map of the human genome is real regardless of what the geneticist needs it to be.&lt;/p&gt;
&lt;p&gt;This is not a concession to Platonism — there is no “realm of objective truths” separate from the physical world. It is a distinction between kinds of knowledge by their mode of generation: scientific knowledge is generated by processes that systematically try to eliminate the contribution of the self from the result; ethical and existential knowledge is generated by processes that are themselves expressions of the self and its defensive needs.&lt;/p&gt;
&lt;p&gt;Science is, paradoxically, the only product of consciousness that strives to free itself from the consciousness that produces it. It is the only form in which defensive manufacturing can be identified and, to some extent, controlled. This does not make it more “meaningful” in an existential sense — it, too, cannot offer what ethics and metaphysics promise. But it can offer truths about what exists, and truths about what exists include the truths about decay, about the cosmos’s indifference, about the nature of suffering — the very truths that the factory of meaning works incessantly to obscure.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-iii-decay-as-universal-law--from-the-atom-to-the-subject&quot;&gt;Part III: Decay as Universal Law — From the Atom to the Subject&lt;/h2&gt;
&lt;h3 id=&quot;31-the-second-law-as-ontology&quot;&gt;3.1 The Second Law as Ontology&lt;/h3&gt;
&lt;p&gt;The second law of thermodynamics is the most philosophical of the physical laws, although it is rarely presented that way. It states that, in an isolated system, entropy — the measure of disorder, of random distribution of energy — tends to increase. Spontaneous processes follow the entropic gradient: order dissolves into disorder; structure dissolves into chaos; differentiation dissolves into homogeneity. The arrow of time, in its single irreversible direction, points toward increasing entropy.&lt;/p&gt;
&lt;p&gt;The universe as a whole is a closed system (by the best current estimate) tending toward thermodynamic equilibrium — what physicists call “heat death”: a state of maximum entropy where no energy gradient makes work possible, where no complex structure is possible, where no difference between this and that exists. The “heat death” of the universe is, in a technical and not metaphorical sense, the death of everything: of all structures, of all processes, of all the differentiation that is the condition of any experience.&lt;/p&gt;
&lt;p&gt;This law is not a theoretical abstraction. It is lived. It is what you see when a building is not maintained: the house decays, the walls crack, the structure collapses into distributed materials. It is what you see in the human body over time: the skin that loses elasticity, the bones that lose density, the cells that accumulate replication errors, the organs that lose efficiency. Aging is not a process that “happens to the body” as an external force — it is the body obeying the most fundamental law of the cosmos. We are complex low-entropy structures temporarily maintained by a flow of energy, and that flow inevitably ceases.&lt;/p&gt;
&lt;p&gt;What this essay proposes is that entropic decay is not merely a law governing the physical substrate of life — it is the law that governs the whole of existence, including the structures of meaning that consciousness fabricates. Civilizations decay. Languages decay. Institutions dissolve. Bonds of affection wear out. Memories are lost. The values one generation considered absolute are relativized by the next. Not because “historical progress” reveals their inadequacy — that would be a narrative of ascent that the second law does not support — but because every complex structure, including the symbolic structures that human consciousness creates, obeys the entropic gradient.&lt;/p&gt;
&lt;p&gt;This is not nihilism in the vulgar sense of “nothing matters.” It is something more precise and more difficult: it is the assertion that &lt;em&gt;every structure of importance is temporary&lt;/em&gt;, and that temporality is not a correctable accident but a structural property of what exists.&lt;/p&gt;
&lt;h3 id=&quot;32-the-fear-of-nonexistence-the-part-that-knows-it-is-a-part&quot;&gt;3.2 The Fear of Nonexistence: The Part That Knows It Is a Part&lt;/h3&gt;
&lt;p&gt;Here we come to what is perhaps the most painful core of the whole analysis. The human being — that part of the universe that has developed the capacity to contemplate the whole — knows that it is temporary. It knows it will die. More than that: it knows that everything that constitutes its experience of itself — memories, affections, skills, relationships — will cease. And this knowledge generates what Heidegger called “being-toward-death” and Ernest Becker, in &lt;em&gt;The Denial of Death&lt;/em&gt;, called simply existential terror.&lt;/p&gt;
&lt;p&gt;Becker argued that all of human culture is, at bottom, a project of managing the terror of death. “Immortality projects” — religions, ideologies, works of art, children, nations — are structures that promise the individual some form of transcendence, some continuation beyond individual extinction. Zapffe said something similar: the awareness of one’s own end is the raw datum that the psychic system must suppress in order to function.&lt;/p&gt;
&lt;p&gt;What this essay adds to these analyses is the mereological dimension. The terror of death is not merely the fear that “I” will cease to exist — it is the fear that a &lt;em&gt;part&lt;/em&gt; has of ceasing to be a part. A stone does not fear disintegration because it has no representation of itself as structure. An animal fears death immediately, viscerally — the fear of predators, the flight from pain — but has, as far as we can know, no abstract representation of its own future nonexistence. The human being is the part of the universe that, by virtue of its cognitive hypertrophy, developed the capacity to represent abstractly its own nonexistence — and discovered that this representation is unbearable.&lt;/p&gt;
&lt;p&gt;The paradox is that the capacity to represent one’s own nonexistence is exactly what constitutes the self as self. Self-consciousness — the capacity to take oneself as an object of representation — is inseparable from the awareness of death. You can only say “I exist” if you can also say “I will cease to exist.” Existence and extinction are, for human consciousness, constitutively linked concepts: you cannot have one without the other.&lt;/p&gt;
&lt;p&gt;This means that the suffering generated by the awareness of death is not a removable side effect of self-consciousness — it is constitutive of it. To eliminate the terror of death, it would be necessary to eliminate self-consciousness. And to eliminate self-consciousness — to return to the state of the animal that flees the predator but does not represent its own nonexistence — would be to eliminate what the philosophical tradition calls “humanity.”&lt;/p&gt;
&lt;p&gt;Zapffe proposed, in a tone that oscillates between the literal and the metaphorical, that the solution is voluntary extinction: not having children, letting the species die out with dignity instead of perpetuating the tragedy. This proposal, regardless of its value as philosophical provocation, fails on its own diagnosis: the decision not to have children would still be made by the consciousness that suffers, which means that the suffering would last until the last human. And the last human would suffer the added loneliness of being the last.&lt;/p&gt;
&lt;h3 id=&quot;33-animal-suffering-and-nature-as-an-extermination-system&quot;&gt;3.3 Animal Suffering and Nature as an Extermination System&lt;/h3&gt;
&lt;p&gt;One of the most important moves in this essay is to expand the scope of suffering beyond human experience. The philosophical tendency, even in its most pessimistic strands, is to center suffering on the conscious human being. Schopenhauer saw the Will as the universal substrate of suffering, but his focus was on human experience as the point where the Will knows itself. Mainländer extended this but likewise with an emphasis on rational experience. Even Benatar, who argues in &lt;em&gt;Better Never to Have Been&lt;/em&gt; that nonexistence is preferable to existence, focuses primarily on the human being capable of making that judgment.&lt;/p&gt;
&lt;p&gt;But animal suffering is suffering. It is not diminished suffering for not being accompanied by reflective narrative — it is suffering without the cushion of narrative, pure suffering in its experiential immediacy. And nature — what the Romantic tradition idealized as harmonious balance, what popular biology describes as an ecosystem, what common sense thinks of as a counterpoint to human chaos — is a system of extermination in continuous operation.&lt;/p&gt;
&lt;p&gt;The numbers are known and rarely contemplated in their fullness: the overwhelming majority of animals born in nature do not survive to maturity. Death by predation — which invariably involves suffering, at least the prey’s suffering at the moment of attack — is a central mechanism of ecosystems. Parasitism, disease, starvation, extreme temperature: the biology of nature is, seen without the romantic lens, an immense machine for processing suffering into energy for other processes.&lt;/p&gt;
&lt;p&gt;David Pearce, in his abolitionist project — the utilitarian proposal that technology should be used to eliminate the suffering of all sentient beings on the planet — has argued that the total biomass of wild mammals in suffering at any given moment is vastly greater than the biomass of mammals in reasonable living conditions. Nature is not neutral with respect to suffering — it generates suffering structurally, as a mechanism of selection, as the engine of the evolutionary process.&lt;/p&gt;
&lt;p&gt;And here is the connection with the main argument: the suffering of wild animals is the suffering of the parts of the physical whole that have suffered the same fate that the universe imposes on every complex structure — to exist temporarily, in competition with other structures, subject to decay. The antelope that dies in the lion’s jaws did not choose to exist; did not choose to have a nervous system capable of pain; did not choose to be part of an ecosystem that needs its death in order to function. It was, like each of us, a part that did not choose to be a part, and that pays the cost of being a part with suffering.&lt;/p&gt;
&lt;p&gt;Nature as an extermination system is not a metaphor. It is a literal description of what the biology of predation, parasitism, disease and scarcity does on a planetary scale at every moment. And this process is not separate from entropic decay — it is one of its expressions at the biological scale. Natural selection is a process that works on variation and differential death; without death, without suffering, without the extermination of less adequate forms, evolution does not occur. Animal suffering is not a defect of the evolutionary process — it is a necessary mechanism of it.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-iv-defense-as-symptom--ethics-morality-and-metaphysics-as-pathology&quot;&gt;Part IV: Defense as Symptom — Ethics, Morality and Metaphysics as Pathology&lt;/h2&gt;
&lt;h3 id=&quot;41-the-structure-of-psychic-defense-on-a-civilizational-scale&quot;&gt;4.1 The Structure of Psychic Defense on a Civilizational Scale&lt;/h3&gt;
&lt;p&gt;Zapffe identified, in “On the Tragic” (1933), four strategies by which individuals manage excessive consciousness: isolation (compartmentalizing existential anguish, not letting it contaminate everyday life), anchoring (fixing oneself in values, identities, roles that give fixed meaning), distraction (keeping consciousness busy so there is no room to contemplate the abyss) and sublimation (transforming anguish into art, philosophy, religion).&lt;/p&gt;
&lt;p&gt;What is rarely noted is that these individual strategies have analogues on a civilizational scale. The great normative structures of civilization — organized religion, philosophical systems, political ideologies, legal and moral systems — are collective strategies for managing the same anguish that Zapffe diagnosed in the individual. They are the social equivalent of anchoring and sublimation, with the advantage that, when shared by a collectivity, they acquire the illusion of objectivity.&lt;/p&gt;
&lt;p&gt;A belief held by a single individual seems clearly subjective, potentially delusional. The same belief, held by a billion people, takes on the air of truth. Numerical strength is not epistemic evidence — but it works psychologically as such. Collective anchoring is more stable than individual anchoring, because each member of the collectivity validates the beliefs of the others, producing a feedback loop of mutual confirmation that can last centuries or millennia before being disturbed by sufficiently powerful evidence.&lt;/p&gt;
&lt;p&gt;Religion is the most obvious case: faced with the evidence that the universe is indifferent to human suffering, that death is final, that there is no external purpose that justifies existence, religion fabricates an alternative universe — morally ordered, with transcendent purpose, where suffering has meaning and death is not the end. This alternative universe is not a discovery — it is a defensive production of immense adaptive value for the collectivity that maintains it.&lt;/p&gt;
&lt;p&gt;Secular moral philosophy has tried to offer more rigorous foundations for normativity — without supernatural appeal, but with the same ambition of providing principles that bind objectively. Utilitarianism (maximize aggregate well-being), deontology (respect the categorical imperatives), contractualism (principles that no one could reasonably reject): each of these systems is, in its deep structure, an attempt to find in reason the equivalent of what religion sought in God — a foundation that transcends the subjective and gives morality the status of discovery, not invention.&lt;/p&gt;
&lt;p&gt;But none of these systems escapes the problem Hume identified in the is/ought gap: from a description of how things are, one cannot derive, by reasoning alone, how things ought to be. To make the crossing from the factual to the normative, it is necessary to introduce an evaluative commitment that does not come from pure reason — that comes, ultimately, from the emotional and subconscious needs and dispositions of the agent making the argument.&lt;/p&gt;
&lt;p&gt;This does not mean that all ethics is equally valid or that the distinction between behaviors more and less conducive to well-being is meaningless. It means that the objective foundation that ethical systems claim — the rock on which they build their normative structures — is not where they say it is. It is in the philosopher’s subconscious, in the emotional needs of the species, in the evolutionary pressures that favor certain social behaviors. The “objective good” that ethics seeks is a reification of the species’ subconscious preferences.&lt;/p&gt;
&lt;h3 id=&quot;42-justice-as-fantasy-happiness-as-receding-horizon&quot;&gt;4.2 Justice as Fantasy, Happiness as Receding Horizon&lt;/h3&gt;
&lt;p&gt;“Justice” is perhaps the most charged moral concept and the most revealing of the defensive structure of ethics. The intuition of justice — that agents who cause harm should suffer proportional consequences, that effort and merit should be rewarded, that the distribution of goods and ills should obey some recognizable principle — is universally present in human psychology (with cultural variations in the details) and universally unsupported by observation of reality.&lt;/p&gt;
&lt;p&gt;Why would a universe that operates by physical laws absolutely indifferent to any moral evaluation generate, in its conscious parts, so powerful an intuition that it ought to be different? The simplest answer — that the intuition of justice is a projection, not a discovery — is the one this essay’s argument supports. The intuition of justice is generated by the subconscious as a mechanism for managing two fundamental evolutionary problems: social coordination (groups that cooperate outcompete groups that do not, and cooperation requires some distribution of costs and benefits perceived as reasonable by the participants) and the management of personal suffering caused by unjust harm (it is less paralyzing, sufferingwise, to believe that whoever harmed you will be punished than to believe there will be no consequences).&lt;/p&gt;
&lt;p&gt;But the cosmos does not cooperate. Injustice, on the universal scale, is the norm, not the exception. And so consciousness produces a second order of defense: narratives of karma, of divine justice, of a history that bends toward the moral arc, of Progress. Each of these narratives serves to maintain belief in the moral ordering of the universe in the face of contrary evidence, so that the suffering generated by injustice does not accumulate in the system to the point of rendering it inoperative.&lt;/p&gt;
&lt;p&gt;“Happiness” — understood as an enduring state of well-being and satisfaction with one’s life — has a similar structure. Research in hedonic psychology (Kahneman, Diener, Lyubomirsky) has shown that humans are astonishingly bad at predicting what will make them happy. They adapt surprisingly fast to both positive and negative events — the phenomenon of “hedonic adaptation” means that the lottery ticket you thought would change your life does not substantially change your long-term happiness, just as the grave diagnosis you thought would destroy your life often does not.&lt;/p&gt;
&lt;p&gt;This could be read as good news — we are resilient. But it also reveals that “happiness” as an attainable goal is structurally elusive: it recedes as we advance toward it, because the psychological system adapts to the states reached and redefines what would be “enough.” The hedonic treadmill is not a metaphor — it is a structural description of how the system works. And this functional system — the system that always repositions the horizon of happiness a little beyond where you are — is the same subconscious system that fabricates the goal of happiness as a mechanism for keeping the organism moving.&lt;/p&gt;
&lt;p&gt;Happiness as an attainable and sustainable state is the carrot that keeps the donkey walking. It is a functional production of the subconscious to keep the organism active, not a description of a possible state. And the suffering generated by the perception that definitive happiness does not arrive — the experience of incompleteness, of always lacking something, of “when X happens I’ll be happy” but X never settles it — is the suffering of a part that has discovered it was chasing an object that the system that programmed it to chase never intended to be attainable.&lt;/p&gt;
&lt;h3 id=&quot;43-the-only-real-objectivity-the-elimination-of-suffering&quot;&gt;4.3 The Only Real Objectivity: The Elimination of Suffering&lt;/h3&gt;
&lt;p&gt;If all ethics is psychic defense, if all metaphysics of value is subconscious projection, if Justice and Total Happiness are functional phantoms — what remains? Is there any normative foundation that survives the demolition?&lt;/p&gt;
&lt;p&gt;The argument of this essay proposes that there is exactly one: &lt;strong&gt;the elimination of suffering&lt;/strong&gt;. Not because it is discovered as an objective value in some Platonist sense, but because it is the only value that survives its own critique. Every ethical system, when its metaphysical scaffolding is removed, collapses ultimately into some variation of “this matters because it reduces suffering” or “this matters because it increases suffering.” The conceptual machinery of rights, duties, virtues, contracts — when dismantled — reveals suffering and its relief as the substrate that was being described.&lt;/p&gt;
&lt;p&gt;But this value — the only one that survives — is not, as the classical utilitarians wanted, simple to apply. Suffering is not a homogeneous quantity that can be added and subtracted in a calculation of aggregate well-being. Suffering has temporal structure, systemic structure, mereological structure. The elimination of immediate suffering in one part often increases systemic suffering in the long run. The relief of immediate pain by narcotic, the relief of anxiety by avoidance, the relief of guilt by rationalization — all are cases where the local optimization of suffering reduction produces systemic worsening.&lt;/p&gt;
&lt;p&gt;This does not invalidate the imperative to eliminate suffering — it means that the imperative, honestly followed, requires an analysis of complexity that no simple ethical system can offer. And it requires, above all, honesty about what suffering is: not a correctable imperfection of an existence that would be good without it, but a structural property of complex systems in a cosmos of entropic decay.&lt;/p&gt;
&lt;p&gt;The objectivity of the elimination of suffering is not the objectivity of a discovered moral law — it is the objectivity of a diagnosis. When a doctor says that cancer must be treated, they are not making a metaphysical claim about the Good; they are identifying a process of structural destruction to which the organism, as a process, is opposed. The imperative to reduce suffering has that same structure: it is the response of a system to its own process of destruction, not the expression of a transcendent moral law.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-v-the-individual-as-part-of-the-whole--ontology-of-collective-existence&quot;&gt;Part V: The Individual as Part of the Whole — Ontology of Collective Existence&lt;/h2&gt;
&lt;h3 id=&quot;51-society-as-an-aggregate-without-essence&quot;&gt;5.1 Society as an Aggregate Without Essence&lt;/h3&gt;
&lt;p&gt;If the individual self is a pseudo-unity — a whole that is not a whole, a narrative imposed on multiplicity —, what are we to say of collective entities? The nation, the culture, humanity: are they, too, functional fictions, aggregates without essence, parts that take themselves for wholes?&lt;/p&gt;
&lt;p&gt;The answer, in this framework, is yes — with the important qualification that collective fiction has real emergent properties that individual fictions lack, simply by virtue of scale and duration. A nation is a fiction, but it is a fiction that organizes collective behavior in ways that produce real consequences — wars, cooperation, distribution of resources, ecological destruction. The ontological unreality of an entity does not imply that its consequences are unreal.&lt;/p&gt;
&lt;p&gt;But the mereological analysis holds: nations, cultures, civilizations are aggregates of individuals, and the properties we attribute to these aggregates — national character, &lt;em&gt;genius loci&lt;/em&gt;, spirit of the age — are reifications of statistical tendencies in populations, not properties of real entities. And these aggregates, like the individuals that compose them, obey the entropic gradient: they decay, dissolve, are absorbed by other processes or simply cease.&lt;/p&gt;
&lt;p&gt;What is revealing in the analysis of collective suffering is that collective structures of defense — religion, nationalism, ideology — generate precisely the types of suffering on the largest scale. Religious wars, nationalist genocides, ideological purges: these are products of systems of collective meaning that went beyond their original defensive function and began to generate suffering on the scale of the masses. The fiction of collective identity, which serves to anchor the individual against existential anguish, when intensified and instrumentalized, becomes an engine of mass destruction.&lt;/p&gt;
&lt;p&gt;Here the dynamic of local versus systemic suffering appears again, but on a collective scale. The individual who finds anchoring in national identity reduces their immediate existential suffering; but when enough individuals do this, and when national identities come into conflict, the systemic suffering produced infinitely exceeds the sum of the individual sufferings relieved. The mechanism of suffering reduction, aggregated on a collective scale, becomes a mechanism of collective suffering production.&lt;/p&gt;
&lt;h3 id=&quot;52-existence-as-a-property-of-decay&quot;&gt;5.2 Existence as a Property of Decay&lt;/h3&gt;
&lt;p&gt;The title of this section might seem paradoxical: how can existence be a property of decay, if decay presupposes that something exists before decaying? But the paradox dissolves when one realizes that “existence,” in the sense in which this essay uses it, is not a static ontological category — not the mere presence of something in being — but the dynamic process of maintaining structure against the pressure of the entropic gradient.&lt;/p&gt;
&lt;p&gt;To exist, for any complex system, is a verb, not a noun. It is the process of maintaining differentiation, maintaining structure, maintaining distance from the thermodynamic equilibrium that would make any interesting process impossible. A crystal exists as long as its periodic structure is maintained; a cell exists as long as its concentration gradients are maintained; an organism exists as long as its metabolic processes maintain organization against the tendency to dispersal; a consciousness exists as long as the neural processes that sustain it maintain their patterns of activation.&lt;/p&gt;
&lt;p&gt;In this description, existence &lt;em&gt;is&lt;/em&gt; the process of resisting decay — and decay is the permanent backdrop against which this resistance occurs. Existence and decay are not opposites: they are aspects of the same process. You can only exist in the rich sense of the word — as an organized system, as a differentiated structure — &lt;em&gt;because&lt;/em&gt; there is a pressure of disorder against which your structure defines itself. Remove the entropic gradient and there is no more “existence” in the interesting sense — there is only equilibrium, homogeneity, absence of distinction.&lt;/p&gt;
&lt;p&gt;Suffering, in this ontology, is the signal of the process of maintaining structure: it is what occurs in the system when resistance to decay requires effort, when maintaining order has a cost. Physical pain is literally the signal of damage to structure — that decay has advanced beyond a tolerable threshold. Emotional suffering is the equivalent signal of damage to the psychological and relational structures that constitute the self. And existential anguish — the background horror that Zapffe diagnosed — is the system’s signal that the process of maintaining itself as a system is, in the long run, a losing one.&lt;/p&gt;
&lt;p&gt;Every ordered system knows, in some functional sense, that it will lose. The aging organism knows, in its rhythms of maintenance and repair that become progressively less efficient, that the process is not sustainable indefinitely. Consciousness knows, when not protected by defense mechanisms, that the structure it is — the narrative of the self, life projects, bonds of affection — will be dissolved. And this knowing is suffering.&lt;/p&gt;
&lt;h3 id=&quot;53-the-paradox-of-the-consciousness-of-decay&quot;&gt;5.3 The Paradox of the Consciousness of Decay&lt;/h3&gt;
&lt;p&gt;There is a profound paradox at the center of this analysis that needs to be named precisely. Consciousness — the process we have identified as the main generator of reflexive psychological suffering — is also the only process in the known universe that knows about decay. Atoms do not know they will reorganize; stars do not know they will explode; dinosaurs did not know they would go extinct. But the human being knows — not only that it will die, but that every structure will dissolve, that entropy increases, that the universe tends toward equilibrium.&lt;/p&gt;
&lt;p&gt;This knowledge is the product of the same cognitive hypertrophy that produces suffering. The consciousness that suffers from its own multiplicity and ephemerality is the same consciousness that discovered the second law of thermodynamics, that mapped the evolution of stars, that understood the extinction of species. The cognitive excess that makes life psychologically unsustainable is the same excess that makes objective knowledge of the cosmos possible.&lt;/p&gt;
&lt;p&gt;This paradox does not resolve elegantly. There is no Hegelian synthesis that transforms the painful excess into positive rational gain. The cost and benefit of consciousness do not cancel out in a neutral accounting. The suffering generated by reflexive self-consciousness is real suffering, lived, embodied. The knowledge produced by the same cognitive capacity is real and valuable — but it is not “compensation” for the suffering, because suffering and knowledge are not currencies of the same denomination that can be compared.&lt;/p&gt;
&lt;p&gt;What the paradox reveals is the tragic structure — in the technical sense, not the melodramatic one — of consciousness: it is at once the problem and the only means available to diagnose the problem. There is no position outside consciousness from which its pathology can be observed without suffering that pathology. The diagnosis occurs from within the disease. The only lucidity available is a lucidity that does not cure — that can, at best, make the suffering comprehensible.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vi-beyond-consolation--for-an-honesty-without-hope&quot;&gt;Part VI: Beyond Consolation — For an Honesty Without Hope&lt;/h2&gt;
&lt;h3 id=&quot;61-the-temptation-of-synthesis&quot;&gt;6.1 The Temptation of Synthesis&lt;/h3&gt;
&lt;p&gt;The philosophical tradition, faced with destructive analyses such as the one this essay conducts, invariably produces a movement of synthesis: a recovery, in the last chapter, of some form of meaning, purpose or hope that the critical process had demolished. Camus proposes “revolt” against the absurd. Nietzsche proposes &lt;em&gt;amor fati&lt;/em&gt; and the eternal return. Heidegger proposes authentic being-toward-death as the most honest mode of existence. Even Schopenhauer, perhaps the most honest of the systematic pessimists, proposes the negation of the Will — asceticism, art, compassion — as forms of relief and transcendence.&lt;/p&gt;
&lt;p&gt;Each of these syntheses is, within the framework of this essay, one more strategy of psychic defense. One more subconscious production of meaning that allows the organism to go on functioning. More anchoring, more sublimation. And there is no dishonor in that — the problem with intransigent nihilism is that, carried to practical consequences, it collapses into inaction or destruction. The system needs anchoring in order to function. The question is whether the anchoring has to be presented as an objective discovery, or can be recognized for what it is: a functional necessity that needs no metaphysical justification.&lt;/p&gt;
&lt;p&gt;This essay deliberately resists the movement of synthesis. Not out of intellectual sadism, not out of inclination to despair, but out of commitment to the honesty that the analysis demands. The truth, if it has any value — and objective knowledge, as was said, is the only value that survives criticism intact — does not bend to our need for it to be more palatable than it is.&lt;/p&gt;
&lt;h3 id=&quot;62-what-to-do-with-the-diagnosis&quot;&gt;6.2 What to Do with the Diagnosis&lt;/h3&gt;
&lt;p&gt;A doctor who diagnoses an incurable disease is not, for that reason, prescribing resignation to immediate suffering. The diagnosis that existence is structurally suffered does not imply that nothing can or should be done to reduce the specific, concrete suffering of specific parts of the system.&lt;/p&gt;
&lt;p&gt;The only real objectivity — the elimination of suffering — keeps its value even in a framework where suffering is structural and irreducible in principle. The fact that we cannot eliminate all suffering is no argument against eliminating the suffering we can eliminate. The fact that nature produces animal suffering on a massive scale is no argument against reducing animal suffering where we have the capacity to do so. The fact that consciousness structurally generates reflexive suffering is no argument against treating specific psychological suffering with the means available.&lt;/p&gt;
&lt;p&gt;The difference between this position and the optimistic humanism that this essay criticizes is one of diagnosis, not of immediate program. The optimistic humanist reduces suffering because they believe the project of eliminating suffering is, in principle, completable — that there exists a horizon of “full humanity” or “just society” where suffering will have been sufficiently reduced. This essay’s diagnosis rejects that horizon — not because the reduction of suffering is without value, but because the belief in the horizon is itself a defensive fiction that often produces systemic suffering when the horizon is inevitably not reached.&lt;/p&gt;
&lt;p&gt;To act to reduce suffering without the illusion that there exists a final state of zero suffering: that is what the honesty of the diagnosis allows. It is not heroic. It is not grandiose. It does not inspire epic poetry. It is simply the only ethically coherent way to act when you know what you know.&lt;/p&gt;
&lt;h3 id=&quot;63-knowledge-as-non-consoling-exception&quot;&gt;6.3 Knowledge as Non-Consoling Exception&lt;/h3&gt;
&lt;p&gt;We return here, conclusively, to the central distinction between objective knowledge and value systems. Scientific knowledge — physics, chemistry, biology, mathematics — is the notable exception in the analysis this essay has conducted. Not because it is more “important” than suffering, not because it offers consolation, but because it has a different structure: it is generated by processes that systematically try to eliminate the contribution of the subject, and it produces representations of the world that are independent of the defensive needs of the subject who produces them.&lt;/p&gt;
&lt;p&gt;This does not make scientific knowledge ethically neutral — it can be used to relieve suffering or to amplify it, and is often used for both at once. It does not make the scientist immune to subconscious distortions — there is extensive evidence of confirmation bias, of institutional pressures that distort research, of academic egos that defend refuted theories. But the scientific process — with its norms of replicability, peer review, openness to refutation — is a genuine, and partially successful, attempt to produce knowledge that transcends the individual subject and its needs.&lt;/p&gt;
&lt;p&gt;And what this knowledge tells us is consistent with the diagnosis of this essay: it tells us that the universe is indifferent to suffering, that entropy increases, that organisms are temporary, that complexity is unstable. It does not say that we should suffer without resistance, but neither does it say that there exists a foundation beyond the cosmos itself for our attempts at resistance.&lt;/p&gt;
&lt;p&gt;Knowledge is not consolation. It is, rather, the instrument that allows the diagnosis to be a diagnosis — that keeps the analysis from collapsing into fantasy. The second law is not an opinion. The nature of human neurology is not a choice. Animal suffering in nature does not depend on moral interpretation to be real. Knowledge of these facts does not relieve the suffering they describe — but it is the only thing that distinguishes honest analysis from defensive fantasy.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vii-the-mereology-of-the-cosmos--parts-without-a-whole-that-carries-meaning&quot;&gt;Part VII: The Mereology of the Cosmos — Parts Without a Whole That Carries Meaning&lt;/h2&gt;
&lt;h3 id=&quot;71-the-universe-as-multiplicity-without-center&quot;&gt;7.1 The Universe as Multiplicity Without Center&lt;/h3&gt;
&lt;p&gt;The universe, as far as current cosmology can determine, has no center. Cosmic expansion does not start from a point in space — it starts from all of space simultaneously, which means that any point in the universe can, with equal justification, be called the center. The absence of a cosmic center is the physical version of what this essay argued about the mind: there is no unifying principle that organizes the parts around itself; there are parts, and the narrative of unity is retroactively imposed by a perception that is itself one of the parts.&lt;/p&gt;
&lt;p&gt;Quantum physics has deepened this picture of fundamental multiplicity. The question of the interpretation of quantum mechanics — why and how the classical world emerges from a quantum substrate of superpositions and entanglement — still has no consensus answer. But all serious interpretations agree on something: the most fundamental level of physical reality does not have the properties of classically distinct and well-defined objects. What exists at the quantum level is, depending on the interpretation, a wave function that collapses only with observation, or a set of parallel worlds, or relations without intrinsic relata. In no case is it a set of solid and determinate objects of the kind our macroscopic perception assumes.&lt;/p&gt;
&lt;p&gt;Macroscopic reality — things, people, planets — emerges from this quantum substrate through processes of decoherence: entanglement with the environment suppresses quantum superpositions and produces effectively classical behavior. But “emergence” is not “creation of fundamental properties.” Macroscopic properties — solidity, color, definite location — are real as descriptions of the system’s behavior at certain scales, not as properties of the more fundamental substrate.&lt;/p&gt;
&lt;p&gt;The universe, therefore, is made up of parts whose properties are relational and scale-dependent, with no fundamental level of reality where “the whole” is present as an organized totality. It is multiplicity from end to end, with unity appearing only as an approximate pattern at certain scales of observation. And this fundamental multiplicity does not suffer — not in the sense in which a conscious being suffers — but it is the substrate from which suffering consciousness emerged.&lt;/p&gt;
&lt;h3 id=&quot;72-consciousness-as-cosmic-anomaly&quot;&gt;7.2 Consciousness as Cosmic Anomaly&lt;/h3&gt;
&lt;p&gt;There is a possible reading of the cosmological argument that might seem to console: if the universe is multiplicity without center, if there is no organized totality that sustains any purpose, then human suffering is simply local, not cosmic. We are a small anomaly in a vast and indifferent cosmos; our suffering is insignificant on the scale of the universe.&lt;/p&gt;
&lt;p&gt;This reading is psychologically tempting — it is one of the forms that sublimation can take. But it is philosophically problematic. Quantitative insignificance does not reduce qualitative suffering. The fact that the universe is 93 billion light-years in diameter does not make the pain of a child with cancer less real or less suffered. Suffering is not an extensive quantity that would be diluted at cosmic scale — it is an intensive property of specific experiential states.&lt;/p&gt;
&lt;p&gt;What the locality of consciousness in the cosmos really implies is more disturbing, not less: consciousness is an anomaly in the sense that it is, as far as we know, the only place in the universe where suffering occurs. In the whole vast cosmos of galaxies, nebulae, black holes, pulsars — all the immense machinery of matter and energy —, suffering occurs only in that minuscule fraction where sufficiently organized matter has produced processes that have experiential properties. And in that fraction, suffering is immense, constant, structural.&lt;/p&gt;
&lt;p&gt;If consciousness is the only region of the cosmos where anything matters — where anything is lived from the inside, where there is a “what it is like” to be something —, then the concentration of suffering in that region is maximal. The universe does not suffer, but the only part of it that experiences is the part that suffers most.&lt;/p&gt;
&lt;h3 id=&quot;73-decay-of-forms--culture-language-civilization&quot;&gt;7.3 Decay of Forms — Culture, Language, Civilization&lt;/h3&gt;
&lt;p&gt;Decay does not operate only at the physical and biological levels. It reaches, with the same inevitability, the symbolic constructions that human consciousness produces.&lt;/p&gt;
&lt;p&gt;Languages die — classical Latin is no longer spoken as a living language; Vedic Sanskrit survives only in a liturgical context; hundreds of indigenous languages disappear with each generation, taking with them unique cognitive structures, ways of categorizing experience that will no longer exist anywhere. Each dead language is an extinct form of consciousness, a particular way of organizing experience that cultural decay has dissolved.&lt;/p&gt;
&lt;p&gt;Entire civilizations have disappeared without leaving recognizable successors. The Minoan, the Hittite, the Indus Valley civilization — elaborate social complexities, architectures, writing systems, religious hierarchies, forms of everyday life that existed for centuries or millennia and were absorbed by the historical process, leaving only archaeological fragments that allow a partial but never complete reconstruction. What was lived — what was suffered, what was loved, what was feared — in these vanished worlds is irrecoverably lost.&lt;/p&gt;
&lt;p&gt;Works of art, which are often presented as the most durable of human creations — art that “conquers death,” according to the classical topos —, are equally temporary on the relevant time scale. Homer has survived two thousand eight hundred years, which is remarkable and exceptional. But there are works of equal or greater value that did not survive — tragedies of Aeschylus of which we have only the titles, paintings of Classical Antiquity that we know only through descriptions, music of irrecoverable preliterate cultures. The survival of Homer is statistically atypical, not representative of the general fate of human works.&lt;/p&gt;
&lt;p&gt;There is nothing surprising in this catalogue — it is what we all know, in some way. But there is a difference between knowing something and facing it without the habitual defense mechanisms. The strategy of anchoring in works that “survive time” presupposes that the survival of works justifies the suffering of their production, or that cultural recognition confers a kind of immortality that mitigates the terror of individual death. When this presupposition is examined, it reveals itself for what it is: a defensive fiction that needs the erasure of the evidence of everything that did not survive in order to be sustainable.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vii-b-time-as-an-instrument-of-torture--memory-anticipation-and-the-illusion-of-continuity&quot;&gt;Part VII-B: Time as an Instrument of Torture — Memory, Anticipation and the Illusion of Continuity&lt;/h2&gt;
&lt;h3 id=&quot;the-temporal-self-as-long-term-narrative-fiction&quot;&gt;The Temporal Self as Long-Term Narrative Fiction&lt;/h3&gt;
&lt;p&gt;There is a dimension of the mereological problem that demands separate treatment: time. Human consciousness is not merely spatially fragmented — divided among subsystems, modules, parallel processes competing for behavioral expression. It is also temporally fragmented in ways that the narrative of the self continually conceals.&lt;/p&gt;
&lt;p&gt;What we call the “self” through time is a fiction of continuity imposed on a series of states that, rigorously examined, have no substantial unity. The Matheus of ten years ago shares with today’s Matheus some memories, some dispositions, a body with a significant degree of physical continuity — but the neurons that constituted that self are partly different (the brain continually reorganizes), the core beliefs may have changed, the life projects may be unrecognizable. Derek Parfit, in &lt;em&gt;Reasons and Persons&lt;/em&gt;, argued extensively that personal identity over time is a matter of degree, not all-or-nothing — and that, when this truth is absorbed, it profoundly alters how we should think about self-interest, moral responsibility and the meaning of death.&lt;/p&gt;
&lt;p&gt;Parfit’s argument is often read as something that could console: if you are not the same as ten years ago, why identify with the mistakes of the past? Why the terror of future death, if the “you” who dies is already a being distinct from the “you” of today? But there is another reading, consistent with the argument of this essay: if the temporal self is a fiction of continuity, then the suffering accumulated over time — the painful memories, the traumas, the regrets, the fears that grow with anticipation — is suffering that a system without real unity carries as if it were its personal burden, its history, its responsibility. The fiction of temporal continuity not only generates the illusion of a stable self — it generates an archive of suffering that the present self inherits from past states that were, in the rigorous sense, other configurations of the system.&lt;/p&gt;
&lt;p&gt;Traumatic memory is not merely a record of the past — it is a reactivation, in the present, of emotional and physiological states that belonged to a context that no longer exists. The nervous system does not distinguish, in the depth of its stress response, between a real tiger in front of it and the memory of a humiliation from decades ago. The suffering of the past does not stay in the past — it is summoned by the present, distributed through time in ways that no mereologically fragmented system could optimally manage.&lt;/p&gt;
&lt;p&gt;And the anticipation of the future is the symmetrical and inverse instrument: the system projects hypothetical sufferings with the same physiology with which it would respond to present sufferings. Anticipatory anxiety — the suffering of a future that may never occur — is, statistically, the cognitive product that consumes the most time and generates the most suffering in contemporary human nervous systems. Experienced meditators and Stoic philosophers have achieved, by different means, partly the same goal: reducing anticipatory suffering by directing attention to the present. But that is a technique, a management mechanism — not the elimination of the structure that produces the problem.&lt;/p&gt;
&lt;h3 id=&quot;memory-as-cemetery-and-saudade-as-the-original-form-of-suffering&quot;&gt;Memory as Cemetery and &lt;em&gt;Saudade&lt;/em&gt; as the Original Form of Suffering&lt;/h3&gt;
&lt;p&gt;There is a form of suffering that emerges specifically from the capacity to recall positive experiences that have already ended: nostalgia, or, in the richer vocabulary of the Portuguese language, &lt;em&gt;saudade&lt;/em&gt;. &lt;em&gt;Saudade&lt;/em&gt; is often romanticized — treated as proof of the emotional depth of the culture that coined the word, as a demonstration of the affective richness of those who feel it. But in the structure of this essay’s argument, &lt;em&gt;saudade&lt;/em&gt; is the most precise possible suffering of the mereological condition: it is the pain of a part that knows that a state of which it was a part no longer exists, and that no present action can reconstitute it.&lt;/p&gt;
&lt;p&gt;&lt;em&gt;Saudade&lt;/em&gt; is the suffering of decay experienced from within. It is not the terror of future death — it is mourning for the past that was. It is the specific suffering of someone with a memory good enough to know what they have lost, and affective capacity strong enough to go on being affected by absences. Systems with short memory or blunted affect suffer less from &lt;em&gt;saudade&lt;/em&gt; — which suggests that the capacity for nostalgic suffering is proportional to the richness of affective life and the quality of autobiographical memory. That is: more life means more &lt;em&gt;saudade&lt;/em&gt;. More affection means more mourning for its passing.&lt;/p&gt;
&lt;p&gt;This is not an argument for blunting or voluntary amnesia. It is the description of an equation with no clear positive side: the same capacity that makes the experience of connection rich makes the suffering over its loss deep. Parts cannot have bonds without incurring the cost of suffering when the bonds dissolve — and bonds always dissolve, because parts are temporary, the configurations of the system change, decay operates.&lt;/p&gt;
&lt;h3 id=&quot;the-present-as-the-only-habitable-place-and-the-impossibility-of-inhabiting-it&quot;&gt;The Present as the Only Habitable Place and the Impossibility of Inhabiting It&lt;/h3&gt;
&lt;p&gt;The contemplative traditions — Buddhism, Stoicism, some strands of phenomenology — converge in recommending the present as the only place where existence can be inhabited without the suffering of memory and anticipation. “Be here now,” in the popular North American formulation. The present moment as the only one that exists, the only one where action is possible, the only one where suffering and pleasure actually occur.&lt;/p&gt;
&lt;p&gt;There is truth in this — but partial truth, which becomes consoling fiction when presented as a sufficient solution. The present is habitable, to some extent. But the human being is not a creature of the present alone — it is a narrative creature, a system that constructs temporal coherence as a condition of functioning. Without memory, there is no learning, no identity, no sustained relationship. Without anticipation, there is no planning, no care, no responsibility.&lt;/p&gt;
&lt;p&gt;Full presence, as an ideal of psychological health, presupposes that it is possible to have narrative richness without the temporal cost. But the cost is constitutive of the richness: you can only be affected by the memory of a love because the love occurred; you can only anticipate suffering because you have a cognitive structure sophisticated enough to model possible futures. The solution of “living in the present” is a solution that requires deactivating exactly the capacities that make human life rich and human suffering deep. It is another form of anchoring — another functional fiction that manages anguish at the expense of diminishing the scope of the consciousness that produces the anguish.&lt;/p&gt;
&lt;p&gt;There is no temporal escape. The temporal self is a fiction — but it is the fiction that carries everything specifically human in experience. To deactivate it in order to avoid its cost would be to eliminate the problem by the same method that eliminates the patient.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vii-c-the-broken-mirror--how-the-self-suffers-from-being-observed-by-itself&quot;&gt;Part VII-C: The Broken Mirror — How the Self Suffers from Being Observed by Itself&lt;/h2&gt;
&lt;h3 id=&quot;self-consciousness-as-a-circuit-for-amplifying-suffering&quot;&gt;Self-Consciousness as a Circuit for Amplifying Suffering&lt;/h3&gt;
&lt;p&gt;There is an aspect of reflexive consciousness — the capacity to take oneself as an object of observation — that this essay has not yet treated with the care it deserves: secondary suffering, suffering about suffering, the meta-pain that self-consciousness generates in observing and interpreting primary suffering.&lt;/p&gt;
&lt;p&gt;An animal in pain suffers the pain. A human being in pain suffers the pain &lt;em&gt;and&lt;/em&gt; suffers about suffering the pain: interprets the suffering, looks for its cause, assesses whether it is fair that they are suffering, compares themselves with others who do not suffer, constructs narratives about what the suffering means for their identity, for their life projects, for their view of themselves. Each of these cognitive operations is an additional layer of processing that can, and frequently does, generate secondary suffering that exceeds the primary suffering.&lt;/p&gt;
&lt;p&gt;This is what Antonio Damasio would call the difference between pain and suffering: pain is the nociceptive signal, the information of damage to the system. Suffering is the elaborated response to that signal — the layer of meaning, narrative and evaluation that consciousness adds. And it is this added layer that makes human suffering so persistent and so capable of sustaining itself even when the original nociceptive stimulus has ceased.&lt;/p&gt;
&lt;p&gt;Rumination — the repetitive processing of past negative events or future worries — is perhaps the clearest example of this mechanism. The mind returns, involuntarily, to the same disturbing material, reprocesses it from slightly different angles, without ever reaching resolution or producing new useful information. It is the attention system (chiefly the Default Mode Network, in neuroscience’s lexicon) functioning in loop mode, trying to “solve” a problem that has no cognitive solution.&lt;/p&gt;
&lt;p&gt;The problem with rumination is that it is precisely the mechanism reflexive consciousness uses to try to manage suffering, and it often amplifies the suffering it is trying to manage. The attempt to cognitively process something that requires emotional and somatic processing produces more suffering. The system tries to solve with the wrong tool, and the wrong tool not only fails — it actively makes things worse.&lt;/p&gt;
&lt;h3 id=&quot;shame-as-a-pure-product-of-self-consciousness&quot;&gt;Shame as a Pure Product of Self-Consciousness&lt;/h3&gt;
&lt;p&gt;No form of human suffering illustrates the dynamics of reflexive suffering more precisely than shame. Shame — distinct from guilt, which is about a specific action — is about the whole self. It is the suffering of seeing oneself, through the imagined eyes of the other, as fundamentally deficient, unworthy, exposed as inadequate.&lt;/p&gt;
&lt;p&gt;Shame presupposes all the fictions that this essay has undone: it presupposes a coherent self that can be evaluated as a whole; it presupposes standards of adequacy that have objective validity; it presupposes that the other’s perspective on oneself has privileged access to what one is. It is, therefore, a form of suffering built entirely on fictions — and none the less devastating for that. Shame has the power to destroy lives, to make people avoid connections they need, to produce isolation and, in extreme cases, to precipitate voluntary extinction.&lt;/p&gt;
&lt;p&gt;And shame is, in its structure, suffering generated by the mechanism of self-consciousness in comparing itself with a standard that the subconscious fabricates as the “adequate self” and discovering that the distance between the actual self and the adequate self is intolerable. The subconscious fabricates the standard; the conscious self discovers it does not meet it; and the whole system suffers the distance between the real and the fabricated.&lt;/p&gt;
&lt;p&gt;There is no shame without self-consciousness. There is no self-consciousness without the cognitive excess that Zapffe diagnosed. There is no cognitive excess without the evolutionary hypertrophy that produced the tragedy of human consciousness. Shame is, therefore, a direct and necessary product of the same capacity that made humans the dominant species on the planet. The price of cognitive complexity is paid in reflexive suffering.&lt;/p&gt;
&lt;h3 id=&quot;identity-as-a-prison-the-prisoner-built-for-himself&quot;&gt;Identity as a Prison the Prisoner Built for Himself&lt;/h3&gt;
&lt;p&gt;The self, besides being a pseudo-unity, is a specific prison of a particular kind: a prison that the prisoner actively built, that he continues to maintain, and that he interprets as his protection rather than his limitation. Identity — the stable narrative of who one is, the set of characteristics, values, roles and stories that constitute the “self” for oneself and for others — is the most elaborate product of the subconscious factory of meaning.&lt;/p&gt;
&lt;p&gt;Identities are necessary for functioning socially. Without some identity stability, the negotiation of social roles, the establishment of relationships of trust, the coordination of long-term projects become impossible. Identity is a functional fiction that plays a role analogous to money in the economy: it has no intrinsic value, but coordinates behavior in a way that allows exchanges that would be impossible without it.&lt;/p&gt;
&lt;p&gt;The problem is that identity is not content to be a functional fiction — it demands to be lived as fundamental reality. The self identifies with its identity in such a way that any threat to the identity is experienced as an existential threat. Criticism of the identity is received as an attack on the being. Change of identity — even if the change is for the better, in any functional sense — is experienced as partial death, as the loss of something essential.&lt;/p&gt;
&lt;p&gt;And so identity — this fiction the system built to protect itself — becomes the mechanism by which the system imprisons itself. The people who suffer most from changes in circumstances are often those with the most rigid identities: identity has become so central, so elaborate, so defended that any disturbance to its narrative is catastrophic. The protection has become vulnerability; the anchor has become a shackle.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-viii-b-language-as-complicity--the-problem-of-speaking-about-the-structural-ineffable&quot;&gt;Part VIII-B: Language as Complicity — The Problem of Speaking About the Structural Ineffable&lt;/h2&gt;
&lt;h3 id=&quot;what-language-cannot-say-about-suffering&quot;&gt;What Language Cannot Say About Suffering&lt;/h3&gt;
&lt;p&gt;Ludwig Wittgenstein closed the &lt;em&gt;Tractatus Logico-Philosophicus&lt;/em&gt; with the most quoted and least understood sentence in analytic philosophy: “Whereof one cannot speak, thereof one must be silent.” The context of the aphorism is the demarcation of the sayable — that which can be said meaningfully — from the unsayable, the mystical, what shows itself but cannot be said.&lt;/p&gt;
&lt;p&gt;But there is a more immediate and more disturbing problem of language that the Wittgenstein of the &lt;em&gt;Tractatus&lt;/em&gt; identified: language not only fails to say certain things — it actively distorts what it tries to say about subjective experience. When language is used to speak of inner suffering, a translation inevitably occurs that alters the original. Suffering, before being named, is a configuration of physiological, affective and cognitive states with specific temporal structure and intensity. When named — “I am anxious,” “I feel sadness,” “I am afraid” —, it is categorized, compared with other states of the same type, inserted into narrative, and part of its specificity is lost in the process of categorization.&lt;/p&gt;
&lt;p&gt;Language works by categories and by the comparison of instances with types. “Sadness” is a type; this specific state of desolation at this moment is an instance. But the instance has properties that the type does not capture: the specific texture of this suffering, its particular coloration, its relation to other present states, its location in the body. All language about inner suffering is, therefore, a reduction — a loss of information that translation into words inevitably produces.&lt;/p&gt;
&lt;p&gt;This has a direct consequence for the project of this essay: everything said here about suffering, consciousness, multiplicity and decay is said in language that reduces, categorizes, translates. The diagnosis is clearer than the phenomenon it diagnoses. And this clarity is partly illusory — it is the clarity of the map, not of the territory.&lt;/p&gt;
&lt;h3 id=&quot;philosophy-as-the-elaboration-of-more-sophisticated-fictions&quot;&gt;Philosophy as the Elaboration of More Sophisticated Fictions&lt;/h3&gt;
&lt;p&gt;This is the essay’s most difficult moment of honesty, the point at which the analysis folds back on itself: philosophy — including this essay — is also an elaboration of fictions. Not of the same kind as the religious or ideological fictions that were criticized; not with the same immediate defensive function. But fiction nonetheless, in the sense that it is a linguistic and conceptual construction imposed on experience, which organizes and selects and emphasizes according to the goals of the system that produces it.&lt;/p&gt;
&lt;p&gt;The philosopher who writes about suffering is not merely describing suffering — he is using suffering as material for the construction of a conceptual system that satisfies specific intellectual and aesthetic needs of the philosopher. There is pleasure in the construction of rigorous arguments. There is satisfaction in reaching a form that seems to capture something true. That pleasure and that satisfaction — which are present in the production of this text — are themselves processes of the suffering system trying to manage its condition by means of sublimation.&lt;/p&gt;
&lt;p&gt;Zapffe identified sublimation as the most sophisticated of the strategies for suppressing full consciousness. A philosophical essay on the suffering of consciousness is, potentially, high-order sublimation: it transforms anguish about one’s own condition into an intellectual product, which in being produced provides aesthetic and intellectual satisfaction sufficient to make the original anguish manageable. The diagnosis can be genuine and still be, in its production, a defense mechanism.&lt;/p&gt;
&lt;p&gt;This recursion does not invalidate the diagnosis — a medical diagnosis produced by a doctor who fears his own mortality is not, for that reason, false. But it adds a layer of complexity that honesty requires acknowledging: there is no position outside the suffering condition from which one can speak of it without being, at the same time, part of it.&lt;/p&gt;
&lt;h3 id=&quot;silence-as-response-and-impossibility&quot;&gt;Silence as Response and Impossibility&lt;/h3&gt;
&lt;p&gt;The only place where there is no fiction about suffering is silence. Not the meditative silence full of technique and intention — that is already a practice, a construction, a functional fiction of a different order. Genuine silence in the face of suffering is the absence of any elaborate response, the refusal to translate, categorize or narrate. It is the state of one who is in the suffering before any recourse to language or discursive thought.&lt;/p&gt;
&lt;p&gt;This silence is uninhabitable in any sustained way. Humans are linguistic animals — language is not merely an instrument they use, it is constitutive of how they experience the world. Remove inner language and you radically alter the structure of conscious experience. Genuine silence, as a state, is not something that can be chosen as a philosophical stance — it is a state that occurs (in deep meditation, in certain dissociative states, in acute emotional shocks) and cannot be deliberately maintained without ceasing to be what it is.&lt;/p&gt;
&lt;p&gt;Philosophy ends where silence begins. And silence resolves nothing — it merely ceases, for a moment, the elaboration that is at once diagnosis and defense. When the silence ends — and it always ends — language returns, and with it the mechanisms, the narratives, the reflexive sufferings. The system goes on.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-viii-pain-without-metaphor--a-phenomenology-of-structural-suffering&quot;&gt;Part VIII: Pain Without Metaphor — A Phenomenology of Structural Suffering&lt;/h2&gt;
&lt;h3 id=&quot;81-suffering-as-irreducible-datum&quot;&gt;8.1 Suffering as Irreducible Datum&lt;/h3&gt;
&lt;p&gt;After all the mereological, cosmological, neurological, ethical and historical analysis, it is necessary to return to the basic phenomenon, to the raw datum from which all this analysis starts: suffering as experience. Not suffering as a philosophical concept, not suffering as a statistical fact in evolutionary biology, not suffering as an analytical category — but suffering as the most immediate and undeniable fact of all conscious experience.&lt;/p&gt;
&lt;p&gt;Thomas Nagel, in his famous essay “What Is It Like to Be a Bat?”, argued that subjective experience — the phenomenal character, the “what it is like” to be something — resists reduction to physical or functional terms. Not because it is supernatural, but because the first-person perspective has an irreducible structure that no third-person description fully captures. What is felt can only be known by one who feels.&lt;/p&gt;
&lt;p&gt;Suffering has this character with special intensity. You can describe the neurology of pain, nociceptive processing, the activation of specific cortical regions — and none of these descriptions touches what pain is from the inside. You can describe the psychology of depression, serotonin deficits, distorted cognitive patterns — and none of these descriptions communicates what it is to inhabit a mind that cannot feel interest or hope in anything.&lt;/p&gt;
&lt;p&gt;This irreducible character of suffering is, paradoxically, what gives the only value that survives our analysis — the elimination of suffering — its force. It is not a value derived from more abstract principles. It is the most primary datum available. Before any metaphysics, before any systematic ethics, before any narrative of the self — there is suffering. And suffering demands a response.&lt;/p&gt;
&lt;p&gt;But this demand is itself part of the suffering structure: the part that suffers demands relief, and this demand is often met in ways that increase systemic suffering. The cycle has no exit from the inside. It is the structure of the problem, not a correctable peculiarity.&lt;/p&gt;
&lt;h3 id=&quot;82-the-pain-of-lucidity&quot;&gt;8.2 The Pain of Lucidity&lt;/h3&gt;
&lt;p&gt;There is a form of suffering that this essay produces — that any honest analysis of this depth produces — which deserves to be named: the pain of lucidity. It is the specific suffering of seeing clearly, without the habitual defense mechanisms, what exists. It is what Zapffe called the tragedy of hypertrophied consciousness. It is what the mystics have called, in diverse ways, the “dark night of the soul” — although they invariably found a transcendent way out that the framework of this essay does not offer.&lt;/p&gt;
&lt;p&gt;The pain of lucidity is not the same as clinical depression, although it may accompany or precipitate it. It is a specific form of cognitive suffering that occurs when the perception of the state of things exceeds the capacity of the habitual defense mechanisms to obscure it. It is the suffering of the patient who fully understands their terminal diagnosis, without denial. It is the suffering of the historian who contemplates, without romanticization, the complete record of human cruelty and suffering. It is the suffering of the biologist who knows, in detail, what happens to the prey at the moment of predation.&lt;/p&gt;
&lt;p&gt;This suffering is, in a way, the most honest suffering available — it is the suffering of a consciousness that has not protected itself from itself. It is not, for that reason, necessarily more valuable than other sufferings, nor more noble. It is simply more transparent with respect to its own structure.&lt;/p&gt;
&lt;p&gt;And there is a paradox that lucidity about suffering produces: lucidity about the very mechanism that generates suffering does not eliminate it. You can know, in complete neuroscientific and philosophical detail, that your anxiety is produced by subsystems of your nervous system that respond to perceived threats with sympathetic activation — and go on being anxious. Knowledge of the mechanism does not undo the mechanism. The awareness that consciousness produces suffering eliminates neither consciousness nor suffering.&lt;/p&gt;
&lt;p&gt;This is the final confirmation, at the level of experience, of this essay’s argument: consciousness cannot cure itself of itself. It can, at best, reduce specific sufferings by specific means, partially, temporarily. But the suffering structure remains.&lt;/p&gt;
&lt;h3 id=&quot;83-to-be-a-part-is-always-to-be-insufficient&quot;&gt;8.3 To Be a Part Is Always to Be Insufficient&lt;/h3&gt;
&lt;p&gt;The most fundamental form of suffering that mereological analysis reveals is neither physical pain, nor existential anguish, nor even the terror of death — although all of these are real and significant. It is the form of suffering that stems from the constitutive insufficiency of being a part.&lt;/p&gt;
&lt;p&gt;A part, by definition, is not the whole. It has access only to its local point of view, to the information its sensors capture, to the causes and consequences its range of action reaches. The suffering of being a part includes: structural ignorance about the whole (you never know enough to act with certainty about what is good for the wider system); structural impotence (what you do as a part has limited effect on the processes that determine your experience); structural solitude (your first-person perspective is inaccessible to the other parts in fundamental ways, not merely instrumental ones); and what we might call constitutive incompleteness — the sense that something is always missing, that one is not enough, that one is not whole.&lt;/p&gt;
&lt;p&gt;This constitutive incompleteness is, I believe, what human beings most frequently experience as “existential emptiness” or “lack of meaning.” It is not, in most cases, the intellectual contemplation of the absence of metaphysical foundation — it is the direct, everyday experience of being a part that cannot be the whole it feels it should be. The person who has “everything” — success, relationships, health — and still feels something is missing is not being neurotic or ungrateful. They are feeling the structure of the mereological condition: parts are not filled by accumulating goods; incompleteness is constitutive, not circumstantial.&lt;/p&gt;
&lt;p&gt;Systems of meaning — religion, romantic love, vocation, ideology — offer, in their most powerful variants, the promise of transcending the condition of being a part: of becoming part of something greater than oneself in such a way that incompleteness dissolves. “Being part” of God’s will, of a love that transcends the individual, of a historical movement that exceeds individual life. These forms of anchoring work precisely because they address constitutive incompleteness with the promise that the part can, by adhering to a greater whole, become whole.&lt;/p&gt;
&lt;p&gt;But the greater whole is no more real than the individual self. The nation, the religion, eternal love: these too are functional fictions, pseudo-unities, aggregates without essence. The part does not become whole by adhering to a fictitious whole — it merely redistributes its incompleteness in a new configuration, often with the addition of specific new sufferings to the set of sufferings that collective systems generate.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;conclusion-a-philosophy-without-destiny&quot;&gt;Conclusion: A Philosophy Without Destiny&lt;/h2&gt;
&lt;h3 id=&quot;91-the-place-of-the-diagnosis&quot;&gt;9.1 The Place of the Diagnosis&lt;/h3&gt;
&lt;p&gt;A diagnosis without a cure is not an error. It is an honest diagnosis. Medicine sometimes can do no more than name precisely what is happening. Philosophy, when honest, is in the same position: it can, at times, do nothing more than describe clearly the structure of the problem.&lt;/p&gt;
&lt;p&gt;This essay offers no consolation. It does not propose that, by understanding the mereological structure of suffering, you will suffer less — although that may, occasionally, be a side effect. It does not propose that science will solve the structural problems described here — although scientific knowledge remains the only kind of cognitive production that survives critical analysis intact. It does not propose celibacy or voluntary extinction à la Zapffe — although it recognizes the internal coherence of the proposition.&lt;/p&gt;
&lt;p&gt;What it proposes, if it proposes anything, is a stance — an attitude toward the diagnosed condition. An attitude that could be described as lucidity without bitterness, honesty without cruelty, clarity without practical nihilism. That knows suffering is structural and still acts to reduce it where it can. That knows meaning is fabricated and still lives without the paralyzing anguish of that discovery. That knows decay is inevitable and still maintains, where it does, the structures that deserve to be maintained — not because maintenance is guaranteed or because the effort is cosmically rewarded, but simply because it is what the processes that constitute this self do, when functioning well.&lt;/p&gt;
&lt;h3 id=&quot;92-the-cruelty-of-the-universe-is-not-malice--it-is-worse&quot;&gt;9.2 The Cruelty of the Universe Is Not Malice — It Is Worse&lt;/h3&gt;
&lt;p&gt;It would be almost comforting if the universe were cruel. Cruelty implies an agent who chose to be cruel — an Evil God of Gnosticism, a Cartesian demon, a malevolent architect of existence. Against a malevolent agent, there is at least the stance of resistance, of moral refusal, of Sartrean active nausea before a creator who deserves our refusal.&lt;/p&gt;
&lt;p&gt;But the universe is not cruel. It is indifferent. And indifference is harder to inhabit than cruelty, because it offers no coherent stance of resistance. You cannot ethically refuse the law of entropy. You cannot rebel with dignity against the nature of natural selection. You cannot blame anyone for the fact that conscious systems generate suffering as a product of their own complexity.&lt;/p&gt;
&lt;p&gt;The cosmos’s indifference to the suffering that occurs within it is, at the limit, the hardest fact to digest that honest analysis produces. Not because we expected the universe to care — rationally, there was never reason to expect that. But because the subconscious, in its work of fabricating meaning and managing anguish, always and inevitably produces the fiction that it matters. The fiction that this specific suffering, at this specific moment, is somehow noticed, recorded, relevant to something beyond the local processes that produce it.&lt;/p&gt;
&lt;p&gt;The destruction of that fiction — by honest diagnosis — is a real suffering, a real mourning. Mourning for a belief one did not know was a belief.&lt;/p&gt;
&lt;h3 id=&quot;93-mereological-nihilism-as-starting-point-not-destiny&quot;&gt;9.3 Mereological Nihilism as Starting Point, Not Destiny&lt;/h3&gt;
&lt;p&gt;The title of this essay includes the word “defense” — defense of the mereological nihilism of consciousness. But the intention was never to propagate the position as a doctrine to be embraced and lived dogmatically. It was to use it as a lens of analysis that reveals what other lenses conceal.&lt;/p&gt;
&lt;p&gt;The mereological nihilism of consciousness — the position that the self is a groundless pseudo-unity, that values are defensive fabrications, that suffering is structural, that decay is universal — is truer than the alternatives that optimistic humanism, religious metaphysics and consoling philosophy offer. But being truer is not being complete. No conceptual system is complete.&lt;/p&gt;
&lt;p&gt;What this lens does not capture is the texture of the positive experience that exists, in fragmented and temporary form, within the suffering process. Genuine affection, aesthetic pleasure, the satisfaction of knowledge, the experience of connection with another — these are not, merely because they are temporary and not metaphysically grounded, unreal or unworthy of attention. They are real in the only sense that matters: they are lived from the inside, they have the irreducible phenomenal character that Nagel describes.&lt;/p&gt;
&lt;p&gt;The position of this essay is not that these positive experiences compensate for suffering in some cosmic accounting — that would be precisely the consoling temptation we resist. It is that they exist, are part of the multiplicity of experience, and need no metaphysical foundation to be what they are. A good conversation with someone who thinks along with you, a moment of sudden understanding of a hard problem, the experience of music that seems to capture something true about the condition — none of these experiences needs God, Progress, Manifest Destiny or Total Happiness to be what it is.&lt;/p&gt;
&lt;p&gt;Mereological nihilism as a starting point means: starting from the refusal of consoling fictions, from honesty about the structure of the condition, from a clear diagnosis of what generates suffering and why. From that starting point, what one does is open, not determined by the philosophical position. The diagnosis does not prescribe conduct; it frees one from the false prescription of systems that knew less than they claimed to know.&lt;/p&gt;
&lt;h3 id=&quot;94-the-paradox-of-the-written-word--why-this-essay-exists&quot;&gt;9.4 The Paradox of the Written Word — Why This Essay Exists&lt;/h3&gt;
&lt;p&gt;There is a question that any attentive reader will have formulated at some point in this reading, and that honesty requires answering: if philosophy is sublimation, if the essay is sophisticated fiction, if the diagnosis is itself produced by a system seeking to manage its anguish, then why write it? What is the status of this text within its own analysis?&lt;/p&gt;
&lt;p&gt;The answer is not reassuring, but it is honest: this essay exists because the system that wrote it needed to write it. Not because writing resolves what it diagnoses. Not because the argument converts the reader to a more “correct” position. Not because the philosophical tradition is enriched by one more text on the human condition. The essay exists as the other products of sublimation exist: because the process of producing it is, for the system that produces it, a way of being with the anguish that is different from being destroyed by it.&lt;/p&gt;
&lt;p&gt;This is not, within the terms of this essay, a confession of weakness or performative contradiction. It is the normal functioning of the system described. The philosopher who writes about the impossibility of saying does not escape the impossible — he inhabits it in a slightly different way. The difference is not transcendence; it is texture. And texture, for suffering parts, matters — not because it matters cosmically, not because the cosmos registers the difference, but because it is where experience occurs.&lt;/p&gt;
&lt;p&gt;This essay is, therefore, at once diagnosis and symptom. It is the description of the mechanism and the mechanism in operation. It is the analysis of the factory of meaning and a product of that factory. There is no contradiction in this — only the recursion that is constitutive of any system that tries to describe itself from within.&lt;/p&gt;
&lt;h3 id=&quot;95-a-final-word-on-silence&quot;&gt;9.5 A Final Word on Silence&lt;/h3&gt;
&lt;p&gt;Zapffe ends “On the Tragic” with a famous sentence: “Ultimately, we are all the same thing — beings cursed with knowing what they are.” There is dignity in that sentence, even in its pessimism. It is the recognition that the suffering of consciousness is shared, that the condition is collective, that there is no one who, looking honestly, stands outside.&lt;/p&gt;
&lt;p&gt;This essay ends in a different but not incompatible place. It ends with the recognition that silence — real silence, not the forced peace of defense mechanisms — is probably the most honest response to what has been said. Not the Buddhist silence of an emptied mind, not the mystical silence of fusion with the ineffable, not the silence of defeated resignation. The silence of one who has looked at something hard to look at and did not look away, and now simply is with what they saw.&lt;/p&gt;
&lt;p&gt;There is a practice — not a solution, merely a practice — that some find in the territory beyond this diagnosis: the practice of noticing, without elaborating. Noticing that suffering is present. Noticing that the system is reacting. Noticing that the parts are at war. Without the next step of constructing narrative, assigning blame, formulating a plan of resolution. Merely the registering, before the interpretation. This does not eliminate suffering — the diagnosis has already made clear that there is no structural elimination. But there is a qualitative difference, of texture, between suffering that knows it is suffering and suffering that does not; between a part that recognizes its mereological condition and a part that mistakes it for the totality of the real.&lt;/p&gt;
&lt;p&gt;There is something that occurs, sometimes, after an honest and difficult diagnosis: not relief, not resolution, not peace, but something that might be called quiet lucidity. The organism goes on. The parts go on with their internal wars. The cosmos goes on with its decay. But there is, for a moment, the possibility of being with it without needing it to be different.&lt;/p&gt;
&lt;p&gt;That moment — and it is always only a moment, before the defense mechanisms return to work, before the subconscious fabricates a new horizon of meaning to fill the space — that moment is the only present this essay can offer. It is not little. It is not much. It is what there is.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;epilogue-a-note-on-philosophical-pessimism-as-an-intellectual-stance&quot;&gt;Epilogue: A Note on Philosophical Pessimism as an Intellectual Stance&lt;/h2&gt;
&lt;p&gt;Philosophical pessimism — the tradition that includes Schopenhauer, Mainländer, Zapffe, Benatar and, with qualifications, the argument of this essay — is often dismissed as pathology disguised as argument. The pessimist would be, on this clinical reading, simply a depressed person who built a philosophical system to rationalize his state. The argument would be an epiphenomenon of affect, not an independent diagnosis.&lt;/p&gt;
&lt;p&gt;This objection deserves an honest answer: it is partly true and completely insufficient. Partly true because no philosophical system is produced in an emotional vacuum — the philosopher’s temperament inevitably influences the selection of problems, the emphasis given to different phenomena, what is amplified and what is minimized. Schopenhauer was melancholic; that probably contributed to his sensitivity to specific forms of suffering that a more even temperament would not perceive with equal sharpness.&lt;/p&gt;
&lt;p&gt;But insufficient because the validity of an argument does not depend on the psychological state of the one who formulates it. The melancholic who observes that the sun rises in the east is not wrong for being melancholic. The pessimist who observes that suffering is structural, that the second law increases entropy, that natural selection operates by differential death, that conscious systems produce reflexive suffering — is not wrong for having a temperament that favors these observations. The question is whether the observations are correct, not where they come from.&lt;/p&gt;
&lt;p&gt;The intellectual stance this essay defends — if it defends any stance beyond the diagnosis — is that of taking philosophical pessimism seriously as a hypothesis, examining it with the same rigor with which any other philosophical position is examined, and following the argument as far as it goes, without the comfort of a synthesis that domesticates it. This is not pessimistic militancy. It is merely methodological honesty applied to a set of questions that the dominant philosophical tradition has systematically preferred not to follow to the end.&lt;/p&gt;
&lt;p&gt;&lt;em&gt;(Note: This essay does not claim to be a systematic literature review, but an original philosophical argument. The references below indicate the intellectual sources that support the argument, without claiming exhaustiveness.)&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Philosophy of Pessimism and Existentialism&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Peter Wessel Zapffe, “On the Tragic” (1933)&lt;/li&gt;
&lt;li&gt;Arthur Schopenhauer, &lt;em&gt;The World as Will and Representation&lt;/em&gt; (1818)&lt;/li&gt;
&lt;li&gt;Philipp Mainländer, &lt;em&gt;The Philosophy of Redemption&lt;/em&gt; (1876)&lt;/li&gt;
&lt;li&gt;David Benatar, &lt;em&gt;Better Never to Have Been&lt;/em&gt; (2006)&lt;/li&gt;
&lt;li&gt;Albert Camus, &lt;em&gt;The Myth of Sisyphus&lt;/em&gt; (1942) — critical reference&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Neuroscience and Cognitive Psychology&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Michael Gazzaniga, &lt;em&gt;The Social Brain&lt;/em&gt; (1985); &lt;em&gt;Who’s in Charge?&lt;/em&gt; (2011)&lt;/li&gt;
&lt;li&gt;Daniel Kahneman, &lt;em&gt;Thinking, Fast and Slow&lt;/em&gt; (2011)&lt;/li&gt;
&lt;li&gt;Jonathan Haidt, &lt;em&gt;The Righteous Mind&lt;/em&gt; (2012)&lt;/li&gt;
&lt;li&gt;Bessel van der Kolk, &lt;em&gt;The Body Keeps the Score&lt;/em&gt; (2014)&lt;/li&gt;
&lt;li&gt;Robert Sapolsky, &lt;em&gt;Why Zebras Don’t Get Ulcers&lt;/em&gt; (2004)&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Analytic Philosophy and Mereology&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Peter van Inwagen, &lt;em&gt;Material Beings&lt;/em&gt; (1990)&lt;/li&gt;
&lt;li&gt;Theodore Sider, &lt;em&gt;Four-Dimensionalism&lt;/em&gt; (2001)&lt;/li&gt;
&lt;li&gt;David Lewis, &lt;em&gt;Parts of Classes&lt;/em&gt; (1991)&lt;/li&gt;
&lt;li&gt;Thomas Nagel, “What Is It Like to Be a Bat?” (1974)&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Philosophy of Mind and Consciousness&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;David Chalmers, &lt;em&gt;The Conscious Mind&lt;/em&gt; (1996)&lt;/li&gt;
&lt;li&gt;Daniel Dennett, &lt;em&gt;Consciousness Explained&lt;/em&gt; (1991) — critical reference&lt;/li&gt;
&lt;li&gt;Francisco Varela, Evan Thompson, Eleanor Rosch, &lt;em&gt;The Embodied Mind&lt;/em&gt; (1991)&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Thermodynamics and Philosophical Cosmology&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Erwin Schrödinger, &lt;em&gt;What Is Life?&lt;/em&gt; (1944)&lt;/li&gt;
&lt;li&gt;Ludwig Boltzmann — contributions to statistical mechanics&lt;/li&gt;
&lt;li&gt;Brian Greene, &lt;em&gt;The Fabric of the Cosmos&lt;/em&gt; (2004)&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Ethics and Value Theory&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Derek Parfit, &lt;em&gt;Reasons and Persons&lt;/em&gt; (1984)&lt;/li&gt;
&lt;li&gt;David Hume, &lt;em&gt;A Treatise of Human Nature&lt;/em&gt; (1739) — especially Book III&lt;/li&gt;
&lt;li&gt;G. E. Moore, &lt;em&gt;Principia Ethica&lt;/em&gt; (1903) — critical reference&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Existential Psychology&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Ernest Becker, &lt;em&gt;The Denial of Death&lt;/em&gt; (1973)&lt;/li&gt;
&lt;li&gt;Irvin Yalom, &lt;em&gt;Existential Psychotherapy&lt;/em&gt; (1980)&lt;/li&gt;
&lt;li&gt;Terror Management Theory — Solomon, Greenberg, Pyszczynski&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;&lt;strong&gt;Animal Suffering and Antispeciesism&lt;/strong&gt;&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;David Pearce, &lt;em&gt;The Hedonistic Imperative&lt;/em&gt; (1995)&lt;/li&gt;
&lt;li&gt;Peter Singer, &lt;em&gt;Animal Liberation&lt;/em&gt; (1975) — critical reference&lt;/li&gt;
&lt;li&gt;Mark Rowlands, &lt;em&gt;Animal Rights: Moral Theory and Practice&lt;/em&gt; (2009)&lt;/li&gt;
&lt;/ul&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>philosophy</category><category>pessimism</category><category>consciousness</category><category>personal essays</category></item><item><title>The Tyranny of the Mirror: Consciousness, Deliberation and the Necessary Suffering of Human Decisions</title><link>https://matheussorancophilomathes.com.br/en/essays/the-tyranny-of-the-mirror/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/the-tyranny-of-the-mirror/</guid><description>Conscious deliberation is not the origin of our choices but a late narrator of them; the essay argues that much specifically human suffering comes from consciousness invading the domain of the subconscious, and asks who, after all, decides.</description><pubDate>Wed, 17 Jun 2026 02:02:15 GMT</pubDate><content:encoded>&lt;h3 id=&quot;the-curse-of-lucidity-and-the-benevolence-of-instinct&quot;&gt;&lt;em&gt;The Curse of Lucidity and the Benevolence of Instinct&lt;/em&gt;&lt;/h3&gt;
&lt;hr&gt;
&lt;blockquote&gt;
&lt;p&gt;&lt;em&gt;“Nature made a mistake in producing consciousness.”&lt;/em&gt; — Peter Wessel Zapffe, &lt;em&gt;Om det Tragiske&lt;/em&gt;&lt;/p&gt;
&lt;/blockquote&gt;
&lt;hr&gt;
&lt;h2 id=&quot;preface-the-animal-that-saw-itself&quot;&gt;Preface: The Animal That Saw Itself&lt;/h2&gt;
&lt;p&gt;There was a moment — impossible to date, impossible to recover, but of such absolute pregnancy that the whole subsequent history of the species is nothing but a long commentary on it — when a primate raised its eyes and, instead of seeing the world, saw itself seeing the world. In that instant, something broke. Not dramatically, not with a cosmic crash, but with the silent catastrophe of a mirror forming where before there was only opaque surface. The animal became a subject. And in becoming a subject, it became that paradoxical thing we call a &lt;em&gt;human being&lt;/em&gt;: an organism endowed with evolutionary survival mechanisms that, in perfecting themselves beyond the necessary threshold, turned against the very organism.&lt;/p&gt;
&lt;p&gt;What this essay intends to examine is not consciousness in its totality — an enterprise too vast for any text, much less a single argument. What I intend to examine is a specific and, I believe, systematically neglected aspect of our condition: the relation between the excess of consciousness and the quality of human decisions; or, put another way, the hypothesis that conscious deliberation — that process we call “thinking before acting,” that capacity we celebrate as the highest expression of rational dignity — is, in the overwhelming majority of cases, not only ineffective but actively harmful; not only suboptimal but a generator of suffering; not only unnecessary but a betrayal of the deeper intelligence that inhabits every living being.&lt;/p&gt;
&lt;p&gt;The central thesis can be stated thus: the decisions produced by the conscious process of deliberation are, by structure, necessarily inferior to the decisions produced by what we call the subconscious — and this inferiority is neither accidental nor correctable by more training or more information, but is intrinsic to the nature of consciousness as a biological and existential phenomenon. More than that: the characteristic suffering of human existence — emotional, psychological, relational, existential suffering — is largely the suffering produced by conscious decisions, that is, by decisions that escaped the domain of what should govern them.&lt;/p&gt;
&lt;p&gt;To develop this argument with the seriousness it requires, we will need to pass through several layers of analysis: the neuroscience of decision, the philosophy of the will, the psychology of trauma and choice, the phenomenology of deliberation, and something we might call an ecology of desire — understanding desire not as subjective whim, but as a vector of biological and existential information of extraordinary complexity. Underlying everything will be a question that cannot be answered but that cannot be ignored either: who, after all, decides?&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;i-the-problem-of-consciousness-as-decision-making-authority&quot;&gt;I. The Problem of Consciousness as Decision-Making Authority&lt;/h2&gt;
&lt;h3 id=&quot;11-what-is-a-decision&quot;&gt;1.1 What is a decision?&lt;/h3&gt;
&lt;p&gt;First of all, we need to be precise about what we call a decision. In a broad sense, any system with internal states that change as a function of the environment “decides” constantly: a thermostat decides to switch on when the temperature falls below a threshold; a cell decides to produce proteins when certain molecules bind to its receptors; a reflex decides to retract the limb before any signal even reaches the cortex. These are not decisions in the sense that interests us — not because they are mechanical (perhaps all decisions are), but because they do not involve the specific phenomenon we are examining: the introduction of consciousness as mediator of the process.&lt;/p&gt;
&lt;p&gt;A human decision, in the full sense that matters to us, is an event in which a conscious subject finds itself before forks in conduct and, by means of some process — whatever it may be —, selects one of them as the one it will pursue. What distinguishes this event from a reflex or an automatic response is precisely the presence of consciousness as a mediating authority: the subject &lt;em&gt;knows&lt;/em&gt; it is deciding, &lt;em&gt;knows&lt;/em&gt; it could decide otherwise, and carries with it — and this is the crucial point — the &lt;em&gt;responsibility&lt;/em&gt; for the choice made.&lt;/p&gt;
&lt;p&gt;Now, this description, which sounds like the most obvious of descriptions, already contains a presupposition of enormous philosophical proportions: that consciousness is, in fact, the authority that decides. That when “I” deliberate consciously about which path to take, it is “I” — this conscious, reflective self, capable of articulating reasons and weighing consequences — who genuinely produces the decision.&lt;/p&gt;
&lt;p&gt;This presupposition, as we will see, is probably false. And its falsity has devastating consequences not only for our self-understanding but for our capacity to conduct life with some degree of internal harmony.&lt;/p&gt;
&lt;h3 id=&quot;12-the-illusion-of-authorship-what-neuroscience-tells-us-and-what-it-hides-from-us&quot;&gt;1.2 The Illusion of Authorship: What Neuroscience Tells Us (and What It Hides from Us)&lt;/h3&gt;
&lt;p&gt;In 1983, the neurophysiologist Benjamin Libet carried out an experiment that shook the foundations of the philosophy of action and that, curiously, was absorbed by popular culture much more as a curiosity than as a catastrophe. The experiment was simple: subjects were to flex their wrist whenever they wished, while watching a special clock that allowed them to mark the moment at which they became aware of the “will” to move the wrist. Meanwhile, electrodes recorded the brain’s electrical activity.&lt;/p&gt;
&lt;p&gt;The result was as follows: the brain began to prepare the movement — the so-called “readiness potential” was detectable — about 550 milliseconds before the movement. Awareness of the intention to move arose about 200 milliseconds before the movement. But the readiness potential preceded the awareness of intention by almost 350 milliseconds.&lt;/p&gt;
&lt;p&gt;Conclusion: the decision had already been made — by the brain, by the nervous system, by some authority that was not the conscious self — before consciousness even knew there would be a decision. What we call “conscious will” arrived, at most, in time to veto the movement (something Libet also documented: consciousness can inhibit, but does not seem to initiate). But the initiative — the creative spark that set the process in motion — was not its own.&lt;/p&gt;
&lt;p&gt;Subsequent experiments, with more sophisticated methodologies and much longer lead times, confirmed and extended this finding. In 2008, John-Dylan Haynes and collaborators showed that activity in certain regions of the prefrontal and parietal cortex could predict with significant accuracy which hand a subject would use to press a button up to 10 seconds before the subject reported the conscious intention to do so. Ten seconds. The entire universe of conscious deliberation — all the tiny eternity we feel ourselves living when we “make” a decision — takes place within a window that the brain had already determined well in advance of any reasonable planning.&lt;/p&gt;
&lt;p&gt;But here we need to be careful about interpretation. The hasty conclusion would be: “therefore free will does not exist, everything is determined, consciousness is an epiphenomenon.” This conclusion, although perhaps correct, is not what interests us here. What interests us is a more specific and, philosophically, more fertile conclusion: consciousness, as an authority of &lt;em&gt;decision&lt;/em&gt;, is not only not the origin of choices, but is, in many cases, an authority that arrives &lt;em&gt;afterward&lt;/em&gt; and narrates the decision as if it had made it. It is an authority of &lt;em&gt;rationalization&lt;/em&gt;, not of &lt;em&gt;effective deliberation&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;And it is here that suffering begins to appear on the horizon.&lt;/p&gt;
&lt;h3 id=&quot;13-the-narrator-who-arrived-late&quot;&gt;1.3 The Narrator Who Arrived Late&lt;/h3&gt;
&lt;p&gt;Imagine an experienced jazz musician improvising on stage. Their fingers move over the keys with a speed and precision that no conscious process could govern in real time. Each note is a decision — a decision made in fractions of a millisecond by a system that has accumulated decades of patterns, of musical memory, of response to what the other musicians are doing, of harmonic intuition that does not let itself be articulated in words. If you ask the musician, after the show, why they played that note there, they will probably have an answer — but that answer will be a story, not a cause. The note has already sounded. The “reason” was constructed afterward.&lt;/p&gt;
&lt;p&gt;Now imagine that, in the middle of the improvisation, this musician begins to think consciously about each note. They begin to deliberate: “should I go to F sharp or to G? What scale am I using? What is the harmonic relation to what the bassist just played?” The result is immediate and catastrophic: the flow breaks, the fingers stumble, the music loses the life it had. The musician has gone from creator to analyst, and in doing so has become incompetent at the task they performed magnificently a second before.&lt;/p&gt;
&lt;p&gt;This phenomenon has a name in cognitive psychology: “paralysis by analysis” in its mild form; “choking” in its severe form, especially studied in the sports context. But what it reveals is something much more fundamental than a performance phenomenon: it reveals that consciousness and competence are, in many domains, inversely related. That the interval between intention and action, when filled by deliberating consciousness, is often an interval of degradation.&lt;/p&gt;
&lt;p&gt;The subconscious — and I use this term in the broad sense of “all cognitive, emotional, bodily and historical processes that operate below the threshold of explicit consciousness” — is a decision mechanism of extraordinary sophistication. It integrates information that the conscious process simply lacks the capacity to process in parallel: the microexpressions of the interlocutor’s face, the tension in one’s own body, the emotional resonance of the memory of analogous situations, the statistical patterns absorbed through thousands of experiences, the state of the autonomic nervous system as a sensor of the environment. All this happens simultaneously, effortlessly, without words.&lt;/p&gt;
&lt;p&gt;Consciousness, by contrast, is &lt;em&gt;serial&lt;/em&gt; and &lt;em&gt;narrow&lt;/em&gt;. It processes one thing at a time. It works with linguistic or quasi-linguistic representations. It has a famously limited window of attention. And it has, moreover, a specific and serious defect: it knows that it knows. That is: it is aware of its own processes, and this — far from being a universal advantage — is frequently a source of distortion, anxiety and error.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;ii-the-anatomy-of-conscious-suffering&quot;&gt;II. The Anatomy of Conscious Suffering&lt;/h2&gt;
&lt;h3 id=&quot;21-when-consciousness-invades-the-domain-of-instinct&quot;&gt;2.1 When Consciousness Invades the Domain of Instinct&lt;/h3&gt;
&lt;p&gt;There is a territory of decisions in which consciousness simply should not operate, and its invasion of that territory is almost always disastrous. This territory includes: how to breathe, how to walk, how to sleep, how to relate sexually, how to eat when hungry, how to cry when sad. These are domains in which the body knows what to do with a perfection that no conscious instruction could improve — and that any conscious instruction tends to worsen.&lt;/p&gt;
&lt;p&gt;Insomnia is perhaps the purest example of this invasion. Sleeping is the most natural thing in the world: a healthy organism, fatigued, in an adequate environment, sleeps. The subconscious knows this. The body knows this. The homeostasis of the nervous system knows this. But when consciousness installs itself in the process — when the subject begins to think “I need to sleep,” “why am I not sleeping?,” “if I don’t sleep I won’t have energy tomorrow,” “I’m failing even at sleeping” — the wakefulness that should last seconds stretches into hours. The very conscious monitoring of the process of falling asleep prevents the process from occurring, because falling asleep requires exactly what consciousness cannot give: absence of monitoring.&lt;/p&gt;
&lt;p&gt;The same applies, in varied ways, to any process that should be governed by the subconscious but that consciousness hijacks. A child learns to walk without significant verbal instruction; put an adult to thinking consciously about how to distribute their weight, and they stumble. The experienced swimmer swims with a style their body has learned and that no analysis would improve; tell them to think about each movement, and the coordination comes undone. The gastronome who eats to feel pleasure finds it; the gastronome who eats while consciously monitoring their experience of pleasure loses contact with the very pleasure they were trying to capture.&lt;/p&gt;
&lt;p&gt;There is a name for this phenomenon in Zen Buddhism: &lt;em&gt;mushin&lt;/em&gt; — “mind without mind” — is the state of flow in which action and agent become indistinct, in which self-consciousness dissolves and competence emerges in its purest form. The same phenomenon was described by Mihaly Csikszentmihalyi as &lt;em&gt;flow&lt;/em&gt;, and by so many others in so many traditions. What all these descriptions have in common is the recognition that there exists a quality of presence and effectiveness that reflective consciousness not only fails to produce, but systematically prevents.&lt;/p&gt;
&lt;p&gt;But the damage is not limited to effectiveness. The damage extends, and here we arrive at the heart of the argument, to the domain of suffering.&lt;/p&gt;
&lt;h3 id=&quot;22-the-genesis-of-conscious-emotional-suffering&quot;&gt;2.2 The Genesis of Conscious Emotional Suffering&lt;/h3&gt;
&lt;p&gt;Zapffe, in his extraordinary “The Tragic Human Being,” argues that human consciousness is a biologically oversized phenomenon — that it was developed beyond what survival requires, and that this excess is the root of all specifically human suffering, the suffering that has no parallel in the animal kingdom. Animals suffer physical pain. Animals suffer deprivation. But animals do not suffer from existential anguish, from retroactive shame, from anticipatory anxiety, from remorse, from regret, from paralysis before freedom.&lt;/p&gt;
&lt;p&gt;All these sufferings — which constitute the majority of human psychological suffering — are sufferings produced by consciousness. More specifically: they are sufferings produced by consciousness turning back on its own decision processes and beginning to audit, to question, to reconstruct counterfactually what might have been different.&lt;/p&gt;
&lt;p&gt;Take remorse as an exemplary case. Remorse is the suffering of a decision made that we cannot undo, but that consciousness insists on re-examining, reformulating, condemning. From the adaptive standpoint — from the standpoint of what would be useful to the organism — remorse has value only as an instruction for the future: it serves to avoid repeating the same mistake. Once that learning has been done, remorse has no further function; it is pure cost, pure erosion. But consciousness does not work this way. Consciousness, once it has taken possession of a past decision, tends not to let go of it. It re-examines it not in order to learn, but out of a kind of structural compulsion that seems intrinsic to the nature of reflective thought: the inability to accept the irreversible as irreversible.&lt;/p&gt;
&lt;p&gt;This is a point of enormous importance. The subconscious lives entirely in the present. It integrates information from the past — uses it as data, as structural memory — but does not inhabit the past. It does not counterfactually reconstruct what might have been different. It does not condemn the organism for decisions the organism made with the information it had available. The subconscious is, in this sense, radically pragmatic: the past informed the present; the present is what it is; the immediate future is what is to be navigated.&lt;/p&gt;
&lt;p&gt;Consciousness, by contrast, is an inhabitant of time in a way that no other cognitive mechanism is. It can live in the past with a disturbing fidelity, recreating dead situations in detail that makes the pain as fresh as at the moment it occurred. It can live in the future with an anticipation that produces sufferings from events that may never happen. And in doing so — in inhabiting time so comprehensively — it multiplies suffering by a factor that no animal experiences: the factor of the temporal ubiquity of suffering itself.&lt;/p&gt;
&lt;h3 id=&quot;23-anxiety-as-a-specific-product-of-conscious-deliberation&quot;&gt;2.3 Anxiety as a Specific Product of Conscious Deliberation&lt;/h3&gt;
&lt;p&gt;Anxiety, in its clinical and existential sense, is the suffering produced by the anticipation of an uncertain future before which the subject does not feel adequate. It is the suffering of consciousness projecting ahead, calculating probabilities of threat, constructing catastrophe scenarios, and then inhabiting these imaginary scenarios with an affective intensity that the body does not distinguish from real threats.&lt;/p&gt;
&lt;p&gt;What is remarkable about anxiety, from the standpoint of our argument, is that it is almost exclusively a product of conscious deliberation. The non-anxious animal — the animal that acts from instinct — does not project itself forward in the way that produces anxiety; it responds to present danger with a mobilization proportional to the real danger. The human being, by contrast, can produce the whole physiological arsenal of the stress response — cortisol, adrenaline, tachycardia, hyperventilation — in response to entirely imaginary dangers that consciousness constructed in deliberating about the future.&lt;/p&gt;
&lt;p&gt;It is crucial to understand that anxiety is not a malfunction — it is not a consciousness that works badly. It is consciousness functioning exactly as its nature is to function: projecting, calculating, anticipating, evaluating. The problem is not in a malfunction; it is in the nature of the mechanism. The consciousness that anticipates produces suffering not as a side effect, but as a direct product of its functioning.&lt;/p&gt;
&lt;p&gt;And here we arrive at a particularly important dimension of the argument: the relation between conscious deliberation and the decisions that produce the greatest human suffering.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;iii-the-conscious-decision-as-a-mechanism-of-betrayal&quot;&gt;III. The Conscious Decision as a Mechanism of Betrayal&lt;/h2&gt;
&lt;h3 id=&quot;31-the-case-of-relational-decisions&quot;&gt;3.1 The Case of Relational Decisions&lt;/h3&gt;
&lt;p&gt;There is no domain in which the damage of conscious deliberation is clearer — and more devastating — than the domain of human relationships. Love, friendship, trust, care: all these are states that arise, develop and operate according to a logic that consciousness does not govern and that, when it tries to govern, it invariably damages.&lt;/p&gt;
&lt;p&gt;No one consciously decides to love someone. The statement may sound banal, but its implications are enormous. The process we call “falling in love” is entirely subconscious: it is the integration of an enormous quantity of information — appearance, voice, smell, behavior, memories of earlier affective experiences, hormonal state, social context, momentary vulnerability, and countless other variables — into an affective signal that consciousness merely receives, does not produce. When consciousness tries to reproduce this process — when someone rationally decides that they &lt;em&gt;should&lt;/em&gt; love a certain person because she combines the desirable qualities — the result is invariably an affective pathology. It is the marriage of convenience that produces resentment. It is the relationship built on rational foundations that collapses when real affects emerge in another direction.&lt;/p&gt;
&lt;p&gt;But the damage of consciousness is not limited to the initial phase of the relationship. It deepens, in various ways, throughout the entire duration of the relationship.&lt;/p&gt;
&lt;p&gt;Consider what happens when a couple faces a serious conflict. There are two possible processes: the subconscious process, which manifests as immediate emotional reaction — anger, hurt, fear, longing — and the conscious process, which manifests as deliberation: “what should I say? How should I behave? What are my rights here? What are the possible consequences?” The subconscious process, however uncomfortable, carries genuine information about the state of the relationship and of the organism. The conscious process, with all its appearance of rationality, tends to produce a mixture of strategic self-defense, simplifying narratives, and judgments that protect the ego but distort the reality of the relationship.&lt;/p&gt;
&lt;p&gt;The psychology of relationships, at this point, is quite clear: the couples who function best over time are not necessarily those who communicate most verbally or who deliberate most about the problems of the relationship. They are those who have developed a kind of subconscious relational intelligence — a mutual sensitivity, a capacity of response that operates below the level of verbal articulation, a continual adjustment that does not pass through reflective consciousness. John Gottman, whose research on marriage is perhaps the most extensive and methodologically solid in the field, showed that he can predict the dissolution of a marriage with high accuracy from micro-behavioral patterns that the spouses themselves do not consciously perceive — which implies, symmetrically, that what &lt;em&gt;preserves&lt;/em&gt; a relationship also operates largely below the threshold of consciousness.&lt;/p&gt;
&lt;p&gt;The paradox is that the more a relationship becomes the object of conscious analysis, the more its participants deliberate about it, the more often it deteriorates. Not because analysis is useless in all contexts — it has a role at specific moments — but because the constant presence of consciousness as an evaluating authority of the relationship introduces a distance between the participants and the very experience of the relationship. To be constantly assessing whether you are happy in a relationship is precisely the best way not to be happy in it.&lt;/p&gt;
&lt;h3 id=&quot;32-the-paradox-of-choice-and-the-suffering-of-multiplicity&quot;&gt;3.2 The Paradox of Choice and the Suffering of Multiplicity&lt;/h3&gt;
&lt;p&gt;Barry Schwartz, in &lt;em&gt;The Paradox of Choice&lt;/em&gt;, documented a phenomenon that, in our language, can be described like this: an excess of options activates the conscious process of deliberation to a level that produces suffering, regret and dissatisfaction that would simply not exist if the options were more limited or if the decision were made by another mechanism.&lt;/p&gt;
&lt;p&gt;The classic experiment is the jam experiment: consumers presented with a selection of six flavors buy more than consumers presented with a selection of twenty-four. The abundance of options does not maximize satisfaction; it paralyzes the decision process and, when the decision is finally made, it generates a residual dissatisfaction — the shadow of all the options not chosen — that simply does not exist when the options are few.&lt;/p&gt;
&lt;p&gt;What this phenomenon reveals is that conscious deliberation is not merely a mechanism that sometimes fails; it is a mechanism that &lt;em&gt;scales badly&lt;/em&gt;. The more material it receives — the more options, the more variables, the more time to deliberate — the more it tends to produce suffering, not less. Because the conscious process of deliberation has a property that no subconscious mechanism has: it does not stop. It continues after the decision has been made. It continues to evaluate the alternatives that were rejected. It continues to construct the counterfactual of what might have been different. And it therefore continues to produce suffering long after any adaptive function has been fulfilled.&lt;/p&gt;
&lt;p&gt;The subconscious, by contrast, when it produces a decision, produces it with a completeness that consciousness rarely achieves. The child who chooses the chocolate ice cream does not spend the afternoon wondering whether she should have chosen vanilla. The animal that chose the territory to the north produces no retrospective anxiety about the territory to the south. The decision was made; the organism moves on. It is only consciousness that refuses to move on, that insists on inhabiting the space of unrealized possibilities with a fidelity that is, ultimately, a form of torture.&lt;/p&gt;
&lt;h3 id=&quot;33-free-will-as-necessary-fiction-and-structural-poison&quot;&gt;3.3 Free Will as Necessary Fiction and Structural Poison&lt;/h3&gt;
&lt;p&gt;We arrive here at the most delicate and most important point of the argument: the question of free will.&lt;/p&gt;
&lt;p&gt;The hypothesis I propose is not simply that free will does not exist — that is a legitimate philosophical position but one that, on its own, does not produce the consequences that interest me. The hypothesis is more specific: free will, as &lt;em&gt;subjective experience&lt;/em&gt;, as the &lt;em&gt;sensation of deliberate authorship over one’s own choices&lt;/em&gt;, is a fiction that consciousness produces &lt;em&gt;after&lt;/em&gt; the decisions are made — and it is precisely this fiction that is the source of enormous human suffering.&lt;/p&gt;
&lt;p&gt;When consciousness attributes to itself the authorship of a decision that was in fact made by deeper processes, it also assumes total responsibility for that decision and for all its consequences. Guilt, remorse, shame: all these sufferings presuppose that the subject &lt;em&gt;could have done otherwise&lt;/em&gt;, that the decision was genuinely free in the sense that the opposite was equally possible. But if the decision was, in the terms we described above, the product of neural processes that preceded awareness of the intention by seconds or more, then the “blame” for the decision does not belong to the conscious self — it belongs to a set of forces — genetic, historical, neurochemical, contextual — that the conscious self did not choose and could not have modified at the moment of decision.&lt;/p&gt;
&lt;p&gt;The problem is not so much metaphysical as functional: even if free will does not exist as ontological reality, it exists as phenomenological experience. And as long as it exists as phenomenological experience, it will produce all the moral and emotional consequences that follow from the belief in authorship. The subject who “freely” chose the wrong path — the destructive relationship, the career that was not theirs, the word that hurt — will carry the weight of this choice with an intensity that no deterministic explanation relieves.&lt;/p&gt;
&lt;p&gt;Zapffe understood this when he described the mechanisms of repression by which human beings try to protect themselves from the excess of consciousness: isolation (restricting conscious content), anchoring (fixing values and beliefs that cannot be questioned), distraction (avoiding the contemplation of emptiness), and sublimation (transforming anguish into cultural production). What all these mechanisms have in common is that they are defensive strategies against consciousness itself — attempts to limit the damage that consciousness inflicts on the organism that hosts it.&lt;/p&gt;
&lt;p&gt;But if consciousness needs to be suppressed, restricted, defended against, in order for the organism to function with some degree of peace — then it is, structurally, something to be managed and not celebrated. It is a wound that needs dressing, not a talent that needs unrestricted exercise.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;iv-the-subconscious-as-superior-intelligence&quot;&gt;IV. The Subconscious as Superior Intelligence&lt;/h2&gt;
&lt;h3 id=&quot;41-the-wisdom-of-the-body&quot;&gt;4.1 The Wisdom of the Body&lt;/h3&gt;
&lt;p&gt;There is an intellectual tradition — running from Spinoza to Nietzsche, from Wilhelm Reich to António Damásio — that insists that the body knows more than the conscious mind. Not in a mystical way, but in an entirely material one: the body has accumulated, through billions of years of evolution and decades of individual experience, an intelligence that no conscious deliberative process could replicate.&lt;/p&gt;
&lt;p&gt;Damásio demonstrated, with his studies of patients with damage to the ventromedial prefrontal cortex, that the capacity to make good decisions depends critically on the integrity of the emotional systems — not because emotions are more “rational” than conscious thought, but because emotions are the means by which the body communicates to the cognitive system information that would otherwise be inaccessible. Patients who retained intact cognitive capacities but had lost access to emotional signaling became unable to make adequate practical decisions. They knew everything rationally, but could not &lt;em&gt;choose&lt;/em&gt;. They were paralyzed before alternatives, or made choices that were objectively harmful to them.&lt;/p&gt;
&lt;p&gt;What this reveals is that emotions — which are, for the most part, subconscious processes communicated to consciousness as bodily states — are not noise in the decision process. They are the most fundamental signal. They are the organism’s voice about what is good for it, filtered by millennia of natural selection and by the whole of individual life experience. When we ignore this signal — when conscious deliberation superimposes its own categories, criteria and calculations on the emotional signal — we often do exactly what the organism knows we should not do.&lt;/p&gt;
&lt;h3 id=&quot;42-intuition-as-distributed-computation&quot;&gt;4.2 Intuition as Distributed Computation&lt;/h3&gt;
&lt;p&gt;The concept of intuition has undergone a significant intellectual rehabilitation in recent decades — not as a mystical phenomenon, but as a cognitive one: the emergent product of pattern-recognition processes that operate below the threshold of consciousness and that use information consciousness could not process explicitly.&lt;/p&gt;
&lt;p&gt;Gary Klein, studying the decision-making of experienced firefighters, found that they rarely deliberate consciously in the conventional sense — they rarely build lists of pros and cons, rarely evaluate alternatives systematically. What they do instead is recognize patterns: the current situation resonates with earlier situations stored in memory, and this resonance produces an action that is immediate, decisive and often correct. Consciousness then narrates this action — “I decided to do X because Y” — but the narrative is partly retrospective. The decision had already been made by subconscious pattern recognition.&lt;/p&gt;
&lt;p&gt;The same phenomenon has been documented in chess masters (who “see” the right move before analyzing it), in experienced physicians (whose intuitive diagnoses often outperform diagnoses produced by explicit analytic protocols), in successful investors (whose best decisions often contradict formal analyses), and in countless other domains of expertise.&lt;/p&gt;
&lt;p&gt;The common denominator is that intuition is neither arbitrary nor irrational: it is the product of massive, distributed, parallel processing that consciousness simply cannot do. It is high-dimensional computation to which consciousness — with its serial and narrow nature — has no direct access. When consciousness tries to superimpose its own processing on intuitive processing, it often degrades the final result.&lt;/p&gt;
&lt;h3 id=&quot;43-the-body-as-oracle-somatic-knowledge&quot;&gt;4.3 The Body as Oracle: Somatic Knowledge&lt;/h3&gt;
&lt;p&gt;There is an even more fundamental level of subconscious intelligence that is rarely recognized as such: somatic knowledge — the knowledge the body has about itself, about the immediate environment, about the state of relationships, which is communicated not through propositions or images but through physical sensations.&lt;/p&gt;
&lt;p&gt;The sensation of a “tight stomach” before an important decision. The relaxation the body experiences in the presence of certain people and the tension it experiences in the presence of others. The fatigue it feels when asked to do something that violates its nature. The energy that arises when it is on the path that belongs to it. All these are information — not metaphors, not fantasies, not illusions — they are outputs of processing systems that integrate more variables than any consciousness could articulate.&lt;/p&gt;
&lt;p&gt;The problem is that modernity, and especially rationalist-Enlightenment modernity in its most exacerbated form, has trained human beings to systematically distrust this information. The modern project of rationalization, in its most radical version, is a project of suppressing the bodily in favor of the abstract, the instinctive in favor of the deliberate, the felt in favor of the calculated. And the result — which psychological clinics see every day, in its multiple forms — is a humanity that made rationally impeccable decisions that made it profoundly unhappy, because the decisions were made on wrong information, on the belief that conscious calculation had access to all the relevant variables, when in fact it systematically ignored the most important ones.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;v-the-specific-pathology-of-the-excessively-conscious-decision&quot;&gt;V. The Specific Pathology of the Excessively Conscious Decision&lt;/h2&gt;
&lt;h3 id=&quot;51-the-case-of-career-choice&quot;&gt;5.1 The Case of Career Choice&lt;/h3&gt;
&lt;p&gt;No decision reveals more clearly the suffering produced by excessive conscious deliberation than career choice — especially in contemporary societies, where the multiplicity of options combines with the ideology of self-realization to create a pressure on this decision process that earlier generations simply did not experience.&lt;/p&gt;
&lt;p&gt;The young person who chooses a career through an entirely conscious process — researching markets, calculating financial returns, evaluating social prestige, consulting rankings, reading testimonials — and who ignores (or does not know how to listen to) what their subconscious knows about what gives them pleasure, what gives them energy, what makes time pass without their noticing, what makes them feel alive when they practice it — this young person is building a gilded prison. They are using the most sophisticated mechanism available to make the worst possible decision.&lt;/p&gt;
&lt;p&gt;The irony is that the criteria that conscious deliberation uses to evaluate a career — income, stability, prestige, alignment with measurable cognitive abilities — are precisely the criteria that have the least correlation with long-term satisfaction at work. The criteria that correlate most with satisfaction — the sense of purpose, the state of flow, the sense of belonging to a practice that has value in itself, the resonance between the work and the deepest identity — are precisely the criteria that conscious deliberation has the greatest difficulty evaluating, because they are criteria that operate at the subconscious level.&lt;/p&gt;
&lt;p&gt;The result is an epidemic of professional dissatisfaction that statistics document with frightening regularity: the majority of workers in practically all developed countries report significant degrees of disengagement from their work. This is not a failure of the labor market. It is a failure of the decision mechanism by which people arrive at the work they do.&lt;/p&gt;
&lt;h3 id=&quot;52-self-sabotage-as-the-logic-of-consciousness&quot;&gt;5.2 Self-Sabotage as the Logic of Consciousness&lt;/h3&gt;
&lt;p&gt;There is a phenomenon that clinical psychologists know well and that popular motivation theory often misinterprets: self-sabotage. The individual who, repeatedly, when about to achieve something they consciously desire, does something that compromises that achievement. The student who stops studying two days before the decisive exam. The entrepreneur who makes a bad decision precisely when the project is about to succeed. The writer who destroys the nearly finished manuscript.&lt;/p&gt;
&lt;p&gt;The most common interpretation is that these people “don’t really want” what they say they want — that they have some fear of success, some unconscious belief that they do not deserve it, some unresolved conflict with the authority figure that achievement would represent. This interpretation may be correct in some cases. But there is another possible interpretation, which our argument suggests: that the subconscious, in these situations, has a different assessment from consciousness of what is good for the organism, and that what we call “self-sabotage” is in fact an assertion — clumsy, poorly articulated, destructive in form — of a wisdom that consciousness did not want to hear.&lt;/p&gt;
&lt;p&gt;The student who sabotages the exam may be communicating, in the only way they know how, that they do not genuinely want the career for which the exam is the entrance — and that this truth was systematically suppressed by the conscious deliberation that weighed the advantages and concluded that this career was “the right decision.” The entrepreneur who sabotages the project may be expressing an exhaustion that consciousness refused to acknowledge because the narrative of success was more comfortable. The writer who destroys the manuscript may be acting on an intuition about the quality of the work that consciousness — too committed to the self-image of a writer — could not directly accommodate.&lt;/p&gt;
&lt;p&gt;“Self-sabotage,” on this reading, is a subconscious that communicates through the only language consciousness cannot ignore: consequence. When the organism speaks and nobody listens, it acts. And the action is disastrous not because the subconscious is destructive, but because the communication was forced to pass through an inadequate channel — surprise action instead of gradual perception — because of consciousness’s resistance to receiving messages that contradict its plans.&lt;/p&gt;
&lt;h3 id=&quot;53-the-suffering-of-indecision-consciousness-in-a-loop&quot;&gt;5.3 The Suffering of Indecision: Consciousness in a Loop&lt;/h3&gt;
&lt;p&gt;There is a form of suffering that is perhaps the purest expression of what I am describing: chronic indecision. The state in which deliberating consciousness cannot produce a decision, and the organism is left suspended in a kind of limbo — knowing it needs to act, unable to act, consuming energy in a process that does not advance.&lt;/p&gt;
&lt;p&gt;Indecision is not a lack of information. Most chronically indecisive people have access to all the information that would be needed for a reasonable decision. The problem is not in the quantity of data available; it is in the nature of the process trying to process them. Consciousness, applied to decisions of high emotional value, tends to enter a loop: each deliberative cycle produces new considerations that make the decision more complex rather than simpler, and the subject is caught in a process that has no internal criterion of termination.&lt;/p&gt;
&lt;p&gt;The subconscious, in these situations, often already knows. It already knows what it wants, already knows what it fears, has already made its decision at the level at which decisions are really made. What consciousness is doing, in these long periods of indecision, is not deliberating toward a conclusion: it is resisting the conclusion that the subconscious has already reached, because that conclusion contradicts some belief, some explicit value, some image of the self that consciousness needs to protect.&lt;/p&gt;
&lt;p&gt;The suffering of indecision is, therefore, not the suffering of not knowing — it is the suffering of knowing and not wanting to know. It is the suffering produced by consciousness’s resistance to the intelligence of the organism itself.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vi-the-existential-dimension-zapffe-and-the-weight-of-self-transparency&quot;&gt;VI. The Existential Dimension: Zapffe and the Weight of Self-Transparency&lt;/h2&gt;
&lt;h3 id=&quot;61-the-tragedy-of-the-conscious-species&quot;&gt;6.1 The Tragedy of the Conscious Species&lt;/h3&gt;
&lt;p&gt;Zapffe starts from a premise that deserves to be taken seriously in all its brutality: the human being is a biologically maladapted organism, not in the sense that it lacks cognitive capacity, but in the sense that it has cognitive capacity in excess — a consciousness that far exceeds what would be necessary for survival and reproduction, and that for this reason turns on itself and on the world with an intensity that is pure excess.&lt;/p&gt;
&lt;p&gt;This superconsciousness has a specific property that matters to us here: it makes the human being the only animal capable of understanding its own mortality, its own cosmic insignificance, its own existential arbitrariness. And this understanding — which is, note, a perfectly lucid, perfectly correct understanding — is in itself a source of suffering without parallel in the animal kingdom.&lt;/p&gt;
&lt;p&gt;But Zapffe’s argument is not limited to death and the cosmos. It includes the banal, everyday suffering that results from a consciousness that functions all the time at full capacity, that processes everything, that does not rest, that cannot stop seeing what it sees. It is this incessant functioning of consciousness that makes sleep an absolute necessity — not only the sleep of the body, but the sleep of consciousness, the temporary suppression of the reflective apparatus that makes life possible.&lt;/p&gt;
&lt;p&gt;Applying this perspective to the problem of decisions: consciousness, when applied to a decision, cannot simply calculate and conclude. It sees &lt;em&gt;everything&lt;/em&gt;. It sees the present decision, but also sees its irreversibility, its contingency (it could have been otherwise), its inscription in a causal chain that goes back to the beginning of the universe and which therefore cannot be attributed to any individual agency, its relation to the mortality of the subject who decides (each decision is also the exclusion of all the other possibilities that would have been possible in the life being lived), and so on. Conscious deliberation, taken to the limit, is not simply slow or imprecise: it is paralyzing, because its object is not only the immediate practical question — it is the totality of existence that each practical question summons.&lt;/p&gt;
&lt;h3 id=&quot;62-anguish-as-structure-not-as-accident&quot;&gt;6.2 Anguish as Structure, Not as Accident&lt;/h3&gt;
&lt;p&gt;Kierkegaard described anguish as “the dizziness of freedom” — the vertigo that arises when the subject realizes that it is free, that it could have done otherwise, that what is had no necessity to be. This dizziness is not an error of perception; it is the correct perception of the human condition. Anguish, in Kierkegaard, is not a pathology to be treated but a structure of conscious existence to be traversed.&lt;/p&gt;
&lt;p&gt;What I propose goes beyond Kierkegaard: anguish is not only a structure of conscious existence in general — it is the specific structure that emerges when consciousness tries to govern the decision process. Because every conscious decision is, at once, the affirmation of one possibility and the negation of all the others; and a sufficiently developed consciousness cannot do this without feeling, to some degree, the weight of all the possibilities being negated. Anguish is not the fear of any specific thing — it is the suffering of the pure possibility that each decision obliterates.&lt;/p&gt;
&lt;p&gt;And here is the cruel elegance of the problem: the consciousness that produces anguish in deciding also produces anguish in not deciding. Indecision is not a refuge from anguish; it is a different way of inhabiting it. The subject who decides is anguished by what they have lost; the subject who does not decide is anguished by what they are not doing. Consciousness has no exit within itself.&lt;/p&gt;
&lt;p&gt;The exit — if there is an exit — lies outside consciousness. It lies in the return, partial and provisional, to the mode of functioning that evolution produced before consciousness existed: the mode in which the organism acts, not the one in which the subject deliberates.&lt;/p&gt;
&lt;h3 id=&quot;63-schopenhauer-and-the-intellect-as-insubordinate-servant&quot;&gt;6.3 Schopenhauer and the Intellect as Insubordinate Servant&lt;/h3&gt;
&lt;p&gt;Arthur Schopenhauer provides, in &lt;em&gt;The World as Will and Representation&lt;/em&gt;, one of the clearest articulations of what contemporary neuroscience would confirm by other means: the intellect — consciousness, reason — is not the foundation of being; it is a late elaboration, a tool, a servant that the Will (the blind and irresistible substrate that governs all existence) produced for practical purposes of orientation in the world. When the servant forgets it is a servant and tries to become master — when the intellect tries to govern the Will instead of serving it — the result is disorder and suffering.&lt;/p&gt;
&lt;p&gt;In the language we are using: consciousness should be an auxiliary instrument of the subconscious, not a substitute for it. It should articulate, communicate, refine and implement what the subconscious has produced — not replace the process of production. When it tries to replace it, it does so with an instrument whose capacities are fundamentally inadequate to the task, and the result is necessarily inferior to what the original process would have produced.&lt;/p&gt;
&lt;p&gt;Lévi-Strauss’s “savage mind,” Polanyi’s “tacit knowledge,” Aristotle’s “practical wisdom,” Goleman’s “emotional intelligence,” Ryle’s &lt;em&gt;knowing how&lt;/em&gt; in contrast with &lt;em&gt;knowing that&lt;/em&gt;: all these notions, from diverse traditions and disciplines, point in the same direction — there exists a mode of knowledge and action that is not propositional, not articulable in language, not the product of conscious deliberation, and that is, in many fundamental contexts, superior to propositional knowledge and deliberate action. Modernity, in its one-sided emphasis on the latter, has systematically depreciated the former — and has paid, and continues to pay, an enormous price in that operation.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vi-b-the-perverse-temporality-of-deliberating-consciousness&quot;&gt;VI-B. The Perverse Temporality of Deliberating Consciousness&lt;/h2&gt;
&lt;h3 id=&quot;64-the-imaginary-future-as-factory-of-suffering&quot;&gt;6.4 The Imaginary Future as Factory of Suffering&lt;/h3&gt;
&lt;p&gt;There is a temporal asymmetry that cannot be sufficiently underlined: the subconscious exists fundamentally in the present; consciousness exists fundamentally in projection. This asymmetry is not contingent — it is not a habit we could modify by discipline or practice — it is constitutive of the nature of consciousness as a phenomenon. Consciousness &lt;em&gt;is&lt;/em&gt;, in its essence, the apparatus that projects the present beyond itself: that reconstructs the past and anticipates the future as objects of reflection. Without this capacity, it would not be consciousness in the full sense; it would be mere registering. It is the act of projecting — of taking what is and imagining what could be or what could have been — that defines the phenomenon.&lt;/p&gt;
&lt;p&gt;This projection is, on the scale of evolutionary adaptation, an extraordinary advantage. The ability to anticipate future dangers allowed &lt;em&gt;Homo sapiens&lt;/em&gt; to plan, prepare, build shelters before the cold seasons, store food before the droughts, organize defenses before the attacks. No other animal possesses this capacity to a comparable degree. And it is undeniable that it was, and continues to be, the basis of all the technical and cultural development of the species.&lt;/p&gt;
&lt;p&gt;But this same advantage becomes, in the domain of emotional and existential decisions, a source of unprecedented suffering. Because projection does not discriminate between real physical threats and imaginary existential threats. The same mechanism that anticipates the storm anticipates failure. The same apparatus that plans the hunt plans social humiliation. The same system that builds the shelter builds, with equal diligence and equal affective intensity, scenarios of abandonment, irrelevance, death.&lt;/p&gt;
&lt;p&gt;And here is the structural cruelty: the subconscious responds to real threats. Consciousness responds also to imaginary threats. The lion on the savanna and the lion that consciousness is constructing in thought are, for the autonomic nervous system, equally real. The stress response is the same; the muscle tension is the same; the hormonal cascade is the same. Except that the real lion eventually goes away or eats you — in both cases, the situation resolves. The imaginary lion never goes away, because it is made of the same substance that makes it: thought. And thought can always generate more thought. The suffering produced by the imaginary lion can last for decades.&lt;/p&gt;
&lt;p&gt;Every important decision that consciousness deliberates contains, within it, a population of imaginary lions: the possible failures, the possible losses, the possible judgments of others, the possible regrets of oneself. And consciousness, in the act of deliberating, inhabits them with a completeness that the subconscious — which would merely respond to what is present — would never produce. It is consciousness, therefore, and not the subconscious, that is the primary factory of the anticipated suffering that accompanies every significant decision.&lt;/p&gt;
&lt;h3 id=&quot;65-retrospective-remorse-the-consciousness-that-does-not-release-the-dead&quot;&gt;6.5 Retrospective Remorse: The Consciousness That Does Not Release the Dead&lt;/h3&gt;
&lt;p&gt;Just as consciousness inhabits the future with its imaginary lions, it inhabits the past with its unburied dead. Remorse — and its variants: guilt, shame, regret — is the suffering produced by past decisions that consciousness refuses to declare closed.&lt;/p&gt;
&lt;p&gt;From the standpoint of the subconscious, a past decision whose consequences have already played out is information — information that updates patterns of future response. But the past itself is not inhabited; it happened, its data were incorporated, one moves on. The subconscious does not hold a grudge against itself. It does not reconstruct, with revisionist anguish, what it should have done otherwise. It learns and moves on.&lt;/p&gt;
&lt;p&gt;Consciousness does not know how to do this. It knows it could have done differently — because it understands the contingency of its own choices, because it can imagine counterfactually what would have happened had it chosen otherwise. And this knowing is, at once, its cognitive grandeur and its specific torment. To know that something could have been different but was not, and that now it cannot be redone: this is the basic structure of remorse, and it is impossible without the reflective consciousness that understands contingency and irreversibility simultaneously.&lt;/p&gt;
&lt;p&gt;The painful paradox is that the more developed consciousness is — the more sophisticated, the more capable of imagining alternatives, the more endowed with perspective on its own past — the more intensely it can inhabit remorse. It is not the less reflective person who suffers most from remorse; it is the most reflective. Because it is the most reflective who can reconstruct the past with greater fidelity and greater richness of detail, who can imagine most vividly what life would have been on the other path.&lt;/p&gt;
&lt;p&gt;Consciousness transforms the past into a permanent present. And this permanent present of the past is one of the most efficient mechanisms of suffering that evolution, unintentionally, produced.&lt;/p&gt;
&lt;h3 id=&quot;66-social-comparison-as-a-product-of-deliberating-consciousness&quot;&gt;6.6 Social Comparison as a Product of Deliberating Consciousness&lt;/h3&gt;
&lt;p&gt;There is a specific mechanism of suffering that is almost entirely a product of conscious deliberation and that is rarely analyzed as such: social comparison. The process by which a subject evaluates their own achievements, choices, appearance and quality of life in relation to those of other subjects — and which produces, invariably, both vanity and envy, both pride and the inferiority complex.&lt;/p&gt;
&lt;p&gt;The subconscious does not compare in the relevant sense. It registers differences of status, responds to social hierarchies with adaptive behaviors (like every primate), feels the pain of exclusion and the pleasure of belonging. But it does not construct comparative narratives that turn one’s own life into a continuous evaluation against the lives of others. This narrative construction is a product of consciousness — especially of the consciousness that developed in late modernity, potentiated by devices that make visible, in real time and on a global scale, the curated appearance of other people’s lives.&lt;/p&gt;
&lt;p&gt;Conscious social comparison is a form of deliberation about one’s own worth — a deliberation that, by its structure, cannot produce stable satisfaction. Because the criterion of comparison is infinitely expandable: any standard that seems satisfactory reveals, on closer examination, a population still more prosperous, more attractive, more fulfilled, relative to which the comparison restarts. And because the “choices” that brought us to where we are — and which social comparison implicitly evaluates — are precisely the kind of decision that, as we have argued, was produced by the wrong process, with the wrong criteria, and which therefore already contains in its structure the seeds of the regret that social comparison fertilizes.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vii-objections-and-replies&quot;&gt;VII. Objections and Replies&lt;/h2&gt;
&lt;h3 id=&quot;71-but-instinctive-errors-are-catastrophic&quot;&gt;7.1 “But Instinctive Errors Are Catastrophic”&lt;/h3&gt;
&lt;p&gt;The most obvious objection to the argument I have developed is: if conscious deliberation is so problematic, how do we explain the immense errors that instincts and automatic reactions produce? Are the cognitive biases documented by Kahneman and Tversky — confirmation bias, the availability heuristic, the framing effect — not all errors of the fast and automatic system that the slow and deliberate system needs to correct?&lt;/p&gt;
&lt;p&gt;This objection is legitimate and requires a precise answer. My thesis is not that the subconscious is infallible or that consciousness has no role at all. My thesis is more specific: that consciousness is misused as the &lt;em&gt;primary authority of decision&lt;/em&gt;, especially in domains of high emotional value, and that this misuse produces specific suffering that would not be produced if consciousness were used in the role that suits it.&lt;/p&gt;
&lt;p&gt;The appropriate role of consciousness is not to produce decisions; it is &lt;em&gt;reflection on systemic patterns&lt;/em&gt;. It is useful — indispensable, even — for identifying that a pattern of behavior is consistently producing bad results. It is useful for building heuristics that modify the environment so that subsequent automatic responses are better. It is useful for communicating intentions to other conscious subjects, for coordinating collective action, for the explicit learning of new domains. What is not useful — what is actively harmful — is applying consciousness to the immediate process of decision in domains where the subconscious has more complete information and a more adequate process.&lt;/p&gt;
&lt;p&gt;Kahneman, whose critique of “System 1” (the automatic) in favor of “System 2” (the deliberative) is often cited, is more nuanced than his popularizations suggest. He explicitly recognizes that the superiority of System 2 is contextual and that, in many domains, the attempt to replace intuition with analysis produces worse results. The problem is not in having two systems; it is in not knowing when to use each — and modernity, culturally, has an enormous and systematically harmful bias toward System 2.&lt;/p&gt;
&lt;h3 id=&quot;72-deliberation-prevents-harm-to-others&quot;&gt;7.2 “Deliberation Prevents Harm to Others”&lt;/h3&gt;
&lt;p&gt;A second important objection: conscious deliberation plays a fundamental role in the control of destructive impulses. Without the capacity for conscious reflection, the human being would be subject to acting on impulses of anger, jealousy, fear, which would produce harm to themselves and others. The father who is furious with his son acts better when he can &lt;em&gt;think&lt;/em&gt; before reacting; the citizen who feels hostility toward a group acts better when he can &lt;em&gt;reflect&lt;/em&gt; on the origins of that feeling.&lt;/p&gt;
&lt;p&gt;This objection is also genuine and also requires a precise answer. What it describes is a specific function of consciousness that is valuable and that no serious argument would reject: the inhibitory function. Consciousness, as Libet had already suggested, can &lt;em&gt;veto&lt;/em&gt; actions that the subconscious has initiated — and this veto is, in many cases, socially and individually beneficial.&lt;/p&gt;
&lt;p&gt;But there is an important difference between consciousness as an authority of veto and consciousness as an authority of production. The father who is furious with his son benefits from a moment of pause that consciousness provides — not to &lt;em&gt;decide&lt;/em&gt; the response, but to prevent the first automatic response (which may be destructive) from being carried out before the subconscious has processed the situation more fully. After this pause, when the subconscious system has more time to integrate the complete situation — the love for the son, the wider context of the behavior, the memory of analogous situations — the response that emerges is often better than the first and also better than anything strict conscious deliberation could produce.&lt;/p&gt;
&lt;p&gt;The problem is not using consciousness to pause and inhibit. The problem is using consciousness to &lt;em&gt;replace&lt;/em&gt; the subconscious process in producing responses. These are qualitatively different uses, with qualitatively different results.&lt;/p&gt;
&lt;h3 id=&quot;73-this-is-dangerous-irrationalism&quot;&gt;7.3 “This Is Dangerous Irrationalism”&lt;/h3&gt;
&lt;p&gt;There is a third kind of objection, more philosophical: the concern that the argument I have developed is a form of irrationalism that, taken to its consequences, could justify anything in the name of “instinct” or the “subconscious.” This is a serious objection that deserves to be taken seriously.&lt;/p&gt;
&lt;p&gt;The argument is not irrationalist in the relevant sense. I am not saying that reason has no value, that analysis is useless, that we should abandon the Enlightenment project of thinking rigorously about the world. I am saying something more specific: that reason has a legitimate domain of application, and that when it invades domains that do not belong to it, it does harm.&lt;/p&gt;
&lt;p&gt;Reason is excellent at: understanding formal structures, building causal explanations, developing scientific knowledge, coordinating social action on a large scale, identifying logical inconsistencies, designing technical systems. Reason is terrible at: deciding whom to love, determining what makes the organism flourish, evaluating in real time situations of high emotional complexity, producing affective states of well-being, integrating the totality of information relevant to decisions of high dimensionality.&lt;/p&gt;
&lt;p&gt;Dangerous irrationalism is not the recognition of these limits. Dangerous irrationalism is the pretension that reason has no limits — that any problem can and should be solved by conscious deliberation, and that the resistance of the body, of the affects, of instinct to this deliberation is simply an obstacle to be overcome. This is the position that produced, in the twentieth century, some of the most terrible demonstrations of what happens when reason instrumentalizes the subconscious instead of listening to it: the architectures of rational-bureaucratic power that Bauman analyzed in &lt;em&gt;Modernity and the Holocaust&lt;/em&gt; are the extreme product of a technical-rational consciousness that suppressed all the subconscious processes of empathy, hesitation, horror, and replaced them with procedures, hierarchies and efficiency.&lt;/p&gt;
&lt;p&gt;Listening to the subconscious is not irrationalism. It is, paradoxically, the epistemically most honest position: the one that recognizes that there are forms of knowledge that exceed propositional articulation and that are, in their domains, more reliable than what the conscious process can produce.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;viii-implications-what-to-do-with-this&quot;&gt;VIII. Implications: What to Do With This&lt;/h2&gt;
&lt;h3 id=&quot;81-the-impossibility-of-return&quot;&gt;8.1 The Impossibility of Return&lt;/h3&gt;
&lt;p&gt;It would be tempting, at this point, to conclude with a simple prescription: “listen more to your subconscious, deliberate less.” But this prescription is itself a trap — because it is a conscious deliberation about how to use conscious deliberation less. It is consciousness instructing itself to step back, which is contradictory at the same level as the paradoxical instruction “don’t think of white elephants.”&lt;/p&gt;
&lt;p&gt;Zapffe understood this. There is no return to the pre-conscious state. Consciousness, once it exists, exists. It cannot be switched off by instruction, cannot be suppressed by decision. The mechanisms Zapffe described — isolation, anchoring, distraction, sublimation — are precisely this: mechanisms for managing the excess of consciousness, not for eliminating it. They are strategies of coexistence with a phenomenon that cannot be reversed.&lt;/p&gt;
&lt;p&gt;What can be done — and this is less than most would like to hear, but perhaps it is enough — is an &lt;em&gt;ecology&lt;/em&gt; of consciousness. A practice, individual and cultural, of recognizing when consciousness is operating in its domain and when it is invading foreign domains; of cultivating listening to the subconscious without necessarily turning this listening into a new project of rational management; of creating the conditions under which the subconscious process can operate with less interference.&lt;/p&gt;
&lt;h3 id=&quot;82-practices-of-non-interference&quot;&gt;8.2 Practices of Non-Interference&lt;/h3&gt;
&lt;p&gt;The contemplative traditions of all cultures can be reread, from this perspective, as traditions of conscious non-interference: practices that temporarily suspend the deliberative process to allow other processes to operate. Meditation is not, fundamentally, a practice of thinking better; it is a practice of interfering less. What experienced practitioners report is not an increase in deliberative capacity, but a decrease in the intrusion of consciousness into processes that would function better without it.&lt;/p&gt;
&lt;p&gt;The automatic writing of the Surrealists, the stream of consciousness as a literary technique (which is, paradoxically, a stream of the subconscious), the practices of improvisation in theater and music, tai chi and other internal martial arts, contemplative prayer: all these practices, in their enormous differences, share the element of suspending conscious deliberation as a precondition for a kind of access that deliberation would block.&lt;/p&gt;
&lt;p&gt;From the standpoint of practical decisions, the implication is not to despise analysis but to &lt;em&gt;order it correctly in time&lt;/em&gt;. There is a kind of process that consists of: gathering information consciously, processing the formal dimensions of the problem consciously, and then &lt;em&gt;stopping&lt;/em&gt; conscious deliberation and letting the subconscious integrate. The “solution that comes in the shower,” the insight that appears on waking, the clarity that arises after a period of rest: these are neither miracles nor luck. They are the subconscious, finally free of conscious interference, carrying out the integration that consciousness had prevented with its incessant activity.&lt;/p&gt;
&lt;h3 id=&quot;83-suffering-as-signal-not-as-failure&quot;&gt;8.3 Suffering as Signal, Not as Failure&lt;/h3&gt;
&lt;p&gt;One of the most important consequences of the argument I have developed is a reinterpretation of the emotional and psychological suffering produced by decisions. If the suffering of a decision is, in many cases, the suffering of having ignored the subconscious — of having done what conscious analysis prescribed instead of what the organism knew it should do — then this suffering is not merely pain: it is information.&lt;/p&gt;
&lt;p&gt;It is the clearest signal the subconscious has for communicating to consciousness that something is wrong. And a consciousness that learns to read this signal — instead of trying to suppress it, anesthetize it, or rationally explain it until it seems tolerable — is finally using its reflective capacity in its correct role: not as a producer of decisions, but as an interpreter of the signals the organism produces.&lt;/p&gt;
&lt;p&gt;The suffering that results from a conscious decision that contradicts the subconscious should not be treated as a failure of the subject, as emotional weakness, as irrationality to be overcome. It should be treated as evidence that the decision was made by the wrong mechanism. It is the organism saying: “You asked me afterward. Next time, ask beforehand.”&lt;/p&gt;
&lt;h3 id=&quot;84-the-re-education-of-listening-culture-clinic-and-practice&quot;&gt;8.4 The Re-education of Listening: Culture, Clinic and Practice&lt;/h3&gt;
&lt;p&gt;There is a practical dimension that this argument, if honest with itself, cannot avoid. If excessive conscious deliberation is structurally a generator of suffering, and if there is no pure return to the pre-conscious state, what remains is a practice of re-education — not of consciousness, which already exists and cannot be undone, but of the &lt;em&gt;relationship between consciousness and the processes that precede it&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;This re-education has, in contemporary clinical psychology, some of its most rigorous expressions. Emotion-Focused Therapy, developed by Leslie Greenberg, starts precisely from the premise that affects — products for the most part of subconscious processes — are carriers of adaptive information that the conscious process frequently suppresses or distorts. The therapeutic aim is not to make the patient more rational, but more capable of &lt;em&gt;listening&lt;/em&gt; to what their affective system is communicating and of distinguishing between affects that are genuine information and affects that are crystallized defensive patterns.&lt;/p&gt;
&lt;p&gt;Sensorimotor Psychotherapy, developed by Pat Ogden, goes deeper: it works with the bodily memory of traumatic experiences — memory that is stored in the nervous system below the linguistic level and that purely verbal therapy often does not reach. The premise is that the body knows, and that the therapeutic work is to create conditions for this knowing to manifest, not for consciousness to replace it with a more adaptive narrative.&lt;/p&gt;
&lt;p&gt;What these approaches have in common — and what distinguishes them from the purely cognitive approaches that still dominate much clinical practice — is the recognition that human suffering is not primarily a problem of incorrect thinking to be corrected by more correct thinking. It is a problem of &lt;em&gt;relationship with one’s own internal processes&lt;/em&gt;: a relationship in which consciousness frequently tyrannizes instead of listening, controls instead of flowing, deliberates when it should trust.&lt;/p&gt;
&lt;p&gt;This point has implications that exceed the clinic and reach pedagogy, the philosophy of education, the formation of organizational and political cultures. A culture that teaches its members to systematically distrust their own intuitive and affective processes, to always rationally justify each choice, to treat emotion as an obstacle to good decision — such a culture is systematically manufacturing the conditions for the kind of suffering this essay describes.&lt;/p&gt;
&lt;p&gt;A different culture — one that taught listening before deliberating, inhabiting uncertainty without prematurely resolving it into analysis, trusting what the body communicates as a starting point and not as an obstacle to overcome — would produce not only people more satisfied with their decisions, but wiser people. Not in the sense of accumulating more conscious information, but in the sense of being more in touch with the intelligence they already possess and that the overculture of systematic deliberation has systematically silenced.&lt;/p&gt;
&lt;h3 id=&quot;85-the-political-dimension-of-excessive-consciousness-the-subject-who-cannot-choose-without-justifying-themselves&quot;&gt;8.5 The Political Dimension of Excessive Consciousness: The Subject Who Cannot Choose Without Justifying Themselves&lt;/h3&gt;
&lt;p&gt;There is a political dimension that this argument touches and that cannot be completely ignored: the demand for rational justification of choices as a mechanism of social control.&lt;/p&gt;
&lt;p&gt;Modern societies, in their liberal and non-liberal forms, share a fundamental assumption: that the legitimacy of an action depends on the subject’s ability to articulate rationally their reasons for it. The law demands stated motivation. Bureaucracy demands documentation of process. The modern market demands that the consumer be able to explain why they chose this product and not that one. Pedagogy demands that the student be able to justify each answer.&lt;/p&gt;
&lt;p&gt;This demand for rational justification is not neutral. It selects, as legitimated behavior, precisely the behavior that passed through the conscious-deliberative mechanism — and delegitimizes or renders invisible behavior that arose from subconscious processes, however wise and well-founded. The manager who made the right decision intuitively is less legitimate, in the bureaucratic structure, than the manager who made the wrong decision with a documented analytic process. The judge who sensed that the testimony was false and whose intuition was correct is formally less trustworthy than the judge who followed the protocol and was deceived.&lt;/p&gt;
&lt;p&gt;This mechanism has the consequence of training subjects to make decisions by the wrong process — not because the wrong process produces better results, but because it is the only process whose outputs have the right form to be communicated and legitimated in the social structure. And by training subjects to make decisions by the wrong process, the social mechanism systematically deepens the suffering this essay describes.&lt;/p&gt;
&lt;p&gt;It is a bitter irony: the same culture that celebrates consciousness and deliberation as supreme values is, for that very reason, a particularly efficient culture in the production of the sufferings that excessive consciousness generates. It not only fails to protect its members from this suffering; it intensifies it, institutionalizing it.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;ix-the-horizon-of-the-problem-science-culture-and-the-future-of-decision&quot;&gt;IX. The Horizon of the Problem: Science, Culture and the Future of Decision&lt;/h2&gt;
&lt;h3 id=&quot;91-the-culture-of-conscious-decision&quot;&gt;9.1 The Culture of Conscious Decision&lt;/h3&gt;
&lt;p&gt;What makes this problem especially urgent is not only its individual dimension, but its cultural dimension. We live in a civilization that systematically privileges conscious deliberation as the legitimate and valued way of making decisions. Educational systems teach thinking, not feeling — and when they teach “using emotional intelligence,” they often do so by converting affects into objects of rational analysis, which is a sophisticated way of perpetuating the same problem.&lt;/p&gt;
&lt;p&gt;The labor market rewards the ability to rationally articulate the basis of each decision — which means rewarding not necessarily the best decisions, but the most rationally justifiable decisions, which are often different things. Medicine, for decades, privileged the analytic protocol over experienced clinical intuition — and there is considerable evidence that this preference produced both diagnostic errors and worse care. The law builds ever more elaborate systems of conscious deliberation for decisions that, in their most important dimensions, exceed what any deliberative system can capture.&lt;/p&gt;
&lt;p&gt;The cultural result is a civilization of people increasingly skilled at articulating reasons and increasingly unskilled at making good decisions. Increasingly capable of constructing narratives for their choices and increasingly incapable of making choices that make them happy. This is not an accidental paradox — it is the systemic consequence of a cultural project that elected deliberative consciousness as the ideal of human rationality, ignoring that human rationality has forms that exceed and precede conscious deliberation.&lt;/p&gt;
&lt;h3 id=&quot;92-artificial-intelligence-and-the-amplified-mirror&quot;&gt;9.2 Artificial Intelligence and the Amplified Mirror&lt;/h3&gt;
&lt;p&gt;It is not possible, at this moment in history, to ignore the relationship between this argument and the development of artificial intelligence. The AI systems that today make decisions in countless domains — from medicine to law, from credit to public health — are, in a sense, the project of conscious deliberation taken to the limit: they are the attempt to make the decision process entirely explicit, entirely articulable, entirely calculable.&lt;/p&gt;
&lt;p&gt;And the results are, in certain domains, impressive. In well-defined domains, with clear success criteria and quantifiable information, AI systems consistently outperform human deliberation. But in domains of high emotional, contextual and relational complexity — precisely the domains where the suffering of bad decisions is most acute — AI systems reveal the same defects as human conscious deliberation, amplified: they process the variables that can be formally represented and ignore the variables that cannot, with often disastrous consequences.&lt;/p&gt;
&lt;p&gt;What AI lacks — what no deliberative system has — is what only the living organism, with its evolutionary history and embodied experience, possesses: the sensation of living in the skin of the consequences. The human subconscious is what it is precisely because the organism that hosts it is mortal, vulnerable, dependent on others, inscribed in a history of suffering and pleasure accumulated over billions of years. This inscription is irreproducible by any system that does not share this fundamental vulnerability.&lt;/p&gt;
&lt;hr&gt;
&lt;h3 id=&quot;93-the-problem-of-trauma-and-the-memory-that-has-not-passed&quot;&gt;9.3 The Problem of Trauma and the Memory That Has Not Passed&lt;/h3&gt;
&lt;p&gt;There is a dimension of the argument that needs to be treated separately, for it introduces a crucial complication: trauma. Because trauma is, precisely, the case in which the subconscious too has failed — in which the nervous system was exposed to an experience that exceeded its capacity for integration, and that was stored not as processed memory, but as fragmented, dissociated memory, reactivated in inappropriate contexts.&lt;/p&gt;
&lt;p&gt;Trauma is the case in which the subconscious itself began to produce maladaptive responses: the panic reaction to an innocuous stimulus, the affective dissociation in situations of intimacy, the defensive aggressiveness in contexts of safety. And here it would be tempting to conclude that, in these cases, consciousness is necessary to correct the errant subconscious.&lt;/p&gt;
&lt;p&gt;But there is a crucial nuance that the psychology of trauma has learned, the hard way, in recent decades: the purely cognitive approach to trauma — the attempt to treat traumatic memory through narration and conscious analysis — is often insufficient and sometimes counterproductive. It can produce retraumatization, can crystallize narratives that imprison the subject in a victim identity, can create intellectual distance from an experience that needs to be processed at a level deeper than the linguistic.&lt;/p&gt;
&lt;p&gt;What the most effective approaches to trauma have demonstrated — EMDR, Somatic Experiencing, approaches based on sensorimotor processing — is that healing occurs not when consciousness &lt;em&gt;understands&lt;/em&gt; the trauma, but when the nervous system &lt;em&gt;processes&lt;/em&gt; what it could not process at the moment of occurrence. That is: healing occurs when the subconscious resumes its natural function of integrating and archiving, a function that the trauma itself had interrupted. Consciousness has a role as a safe witness of this process — as a presence that can observe without being dragged along — but not as the primary operator of the transformation.&lt;/p&gt;
&lt;p&gt;Trauma, therefore, does not refute the argument of this essay: it confirms it in its most difficult version. Even when the subconscious is producing inadequate responses, the solution is not to replace it with deliberating consciousness, but to create conditions for it to resume its own integrative capacity. Consciousness trying to repair a traumatized subconscious by analysis is like a surgeon trying to operate on a patient in shock by the method of verbal deliberation. The instrument is legitimate in other contexts; here, it is the wrong one.&lt;/p&gt;
&lt;h3 id=&quot;94-nietzsche-and-the-body-that-knows&quot;&gt;9.4 Nietzsche and the Body That Knows&lt;/h3&gt;
&lt;p&gt;No Western thinker formulated more clearly than Nietzsche the central thesis this essay develops, and it would be dishonest not to acknowledge it explicitly. In &lt;em&gt;Thus Spoke Zarathustra&lt;/em&gt; and in the preparatory notes collected in &lt;em&gt;The Will to Power&lt;/em&gt;, Nietzsche repeatedly articulates a position that can be summarized thus: the body is a great reason, while the conscious self is only a small reason — an instrument of the body, not its ruler.&lt;/p&gt;
&lt;p&gt;The Nietzschean argument is that the subject’s deepest identity is not in the conscious self but in the bodily-subconscious process that precedes and sustains it. The conscious self — the “small reason” — is derivative, is an instrument, is a late elaboration of a more fundamental process. And pathology arises precisely when this “small reason” forgets that it is an instrument and tries to become ruler — when the conscious self begins to dictate to the body instead of listening to it, when conscious morality begins to deny impulses instead of channeling them, when reflective thought begins to condemn life instead of serving it.&lt;/p&gt;
&lt;p&gt;What Nietzsche calls &lt;em&gt;ressentiment&lt;/em&gt; — and which is perhaps his most singular contribution to psychology and moral philosophy — is precisely a state produced by consciousness turned against its own vital substrate: the consciousness that, unable to realize its desires, transforms impotence into virtue and frustrated desire into moral judgment. Ressentiment is the suffering of a consciousness that has lost contact with what the body wanted and built, on this loss, a philosophy of negation that presents itself as moral elevation.&lt;/p&gt;
&lt;p&gt;Nietzsche is, at this point, an unacknowledged predecessor of the psychology of health that would come decades later: the idea that the denial of one’s own vital processes — however much it presents itself as discipline, asceticism or virtue — produces not spiritual elevation but psychic illness is today not merely a philosophical hypothesis but a hypothesis with extensive clinical support.&lt;/p&gt;
&lt;h3 id=&quot;95-the-question-neuroscience-cannot-answer&quot;&gt;9.5 The Question Neuroscience Cannot Answer&lt;/h3&gt;
&lt;p&gt;There would be a temptation — especially in an essay that relies on evidence from neuroscience — to conclude that the solution to the problem of excessive consciousness is scientific in nature: if we better understand how the brain works, if we map in more detail the neural circuits of decision, if we develop evidence-based protocols for when to use each type of cognitive process, we will have solved the problem.&lt;/p&gt;
&lt;p&gt;This temptation must be resisted. Not because neuroscience is irrelevant — it is, as we have seen, fundamentally illuminating — but because the problem this essay describes is not primarily a problem of cognitive engineering. It is a problem of being. It is the problem of a type of organism that has become capable of seeing itself, and does not know what to do with this vision.&lt;/p&gt;
&lt;p&gt;Neuroscience can tell us &lt;em&gt;how&lt;/em&gt; consciousness works. It cannot tell us &lt;em&gt;what to do with the fact that it works this way&lt;/em&gt;. This is a philosophical and existential problem that no amount of neural data can resolve — not because the data are insufficient, but because the question is not of the kind that data can answer. It is the question of how to live with one’s own nature. And this question requires not science, but wisdom — a type of knowledge that includes science but is not reducible to it, that integrates the data but inscribes them in a practice of living.&lt;/p&gt;
&lt;p&gt;And it is exactly here that philosophy, and not only psychology or neuroscience, has something irreplaceable to offer. Because philosophy — in its richest tradition, running from the pre-Socratics to the contemporaries — has always been, at its core, a practice of living with one’s own questions without suffocating under their weight. A practice of inhabiting uncertainty without the uncertainty destroying the subject who inhabits it.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;x-conclusion-the-honesty-of-ignorance&quot;&gt;X. Conclusion: The Honesty of Ignorance&lt;/h2&gt;
&lt;h3 id=&quot;101-what-it-means-to-decide-well&quot;&gt;10.1 What It Means to Decide Well&lt;/h3&gt;
&lt;p&gt;At the end of this journey, we can attempt an answer to the question that has hung in the air since the beginning: who, after all, decides?&lt;/p&gt;
&lt;p&gt;The honest answer is: we do not fully know. We know that consciousness does not decide in the sense we have always believed; we know that subconscious processes precede and to a large extent determine what consciousness subsequently narrates as “its” decision; we know that free will, as the experience of full conscious authorship, is probably a retrospective construction. What we do not know is how the subconscious functions in all its complexity, what conditions allow it to produce its best outputs, how consciousness can play its auxiliary role without invading the role that does not belong to it.&lt;/p&gt;
&lt;p&gt;But “not fully knowing” is not a position of impotence. It is a position of intellectual honesty — and from this honesty derive practical consequences that are not negligible.&lt;/p&gt;
&lt;p&gt;To decide well, in light of this argument, does not mean to deliberate more carefully. It means to create the conditions for the whole organism — subconscious included, and perhaps subconscious first — to take part in the process. It means learning to distinguish the suffering that is information (the discomfort the subconscious produces to signal that something is wrong) from the suffering that is noise (the anxiety produced by consciousness in deliberating about possibilities that do not yet exist and may never exist). It means, above all, a specific humility: humility before one’s own deeper intelligence.&lt;/p&gt;
&lt;h3 id=&quot;102-the-weight-of-being-the-animal-that-deliberates&quot;&gt;10.2 The Weight of Being the Animal That Deliberates&lt;/h3&gt;
&lt;p&gt;Zapffe was right about something fundamental: being the animal that is conscious of itself is a condition of a demandingness that no other animal experiences. Not because consciousness is in itself malign — it is, in its legitimate domains, extraordinary — but because it arrived in the world before the world knew how to use it. Before institutions, cultural practices, pedagogies and philosophies knew how to integrate it into the larger process that is the life of an organism.&lt;/p&gt;
&lt;p&gt;The result is that most human beings spend their lives using the wrong instrument for the most important problems. They use conscious deliberation where they should use listening. They use analysis where they should use trust. They use control where they should use letting go. And they pay for it with the specific suffering that only a conscious animal can experience: not the pain of the body meeting an obstacle, but the pain of the spirit that knows it could have been otherwise — even though, deep down in its own history, nothing could ever have been otherwise.&lt;/p&gt;
&lt;h3 id=&quot;103-a-last-note-on-the-possible-and-the-necessary&quot;&gt;10.3 A Last Note on the Possible and the Necessary&lt;/h3&gt;
&lt;p&gt;There is one last twist that the argument requires. I have claimed that decisions made by the conscious process are necessarily bad, cause suffering. But “necessarily” needs to be understood precisely: it does not mean that every conscious decision is always catastrophic in all its aspects. It means that the conscious process of decision structurally carries seeds of suffering that the subconscious process does not carry in the same way. It means that the quality of human suffering — in its specificity, in its persistence, in its existential dimension — is inseparable from the conscious nature of the process that produces it.&lt;/p&gt;
&lt;p&gt;The subconscious, when it errs, errs; the organism suffers the consequences and learns. The pain has a function, has a duration, has a limit. Consciousness, when it errs — and it errs in ways the subconscious rarely errs, precisely because its processing is narrower and its information more limited — produces sufferings that have no clear function, no determinate duration, no limit, because consciousness can always return to them and recreate them with all the intensity of the original.&lt;/p&gt;
&lt;p&gt;The remorse over the wrong word said ten years ago. The anxiety about a future that may never come to pass. The regret over the road not taken that consciousness insists on reconstructing as the paradise that should have been chosen. The shame of a decision that seemed free and that therefore seems to condemn the subject who made it. All these sufferings are conscious sufferings — sufferings that exist only because there is a consciousness that produces them, sustains them, amplifies them, and that, in trying to resolve them by deliberation, often deepens them.&lt;/p&gt;
&lt;p&gt;That is why the excess of consciousness is a curse — not because consciousness is bad in itself, but because it is so large that it does not fit in the world. Because it sees too much, feels too much, knows too much about what it cannot change. Because it was made, as Zapffe said, for a level of existence that does not yet exist — and while it does not exist, it inhabits the level of existence there is, and in this inhabiting it produces sufferings that are, ultimately, the price of being the only animal that knows it is an animal.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;epilogue-the-wisdom-that-has-no-name&quot;&gt;Epilogue: The Wisdom That Has No Name&lt;/h2&gt;
&lt;p&gt;There is, on the threshold between sleep and wakefulness, a state in which consciousness is present but does not deliberate — in which the organism is awake but reflection has not yet settled in. In this state, which contemplative traditions cultivate and which the psychology of sleep studies as the “hypnagogic state,” many people report a clarity about their lives that full wakefulness rarely produces. Solutions to problems that went days unresolved. Certainties about relationships that hours of deliberation had only muddled. Answers to questions that consciousness had made increasingly complex as it examined them.&lt;/p&gt;
&lt;p&gt;This clarity is not illusory. It is the subconscious, temporarily free of the interference of deliberating consciousness, communicating what it has always known — communicating in images, in sensations, in certainties that have no argument but need no argument because they are deeper than any argument could reach.&lt;/p&gt;
&lt;p&gt;There is also, within the oldest philosophical traditions, an intuition that this essay has merely tried to develop in the language we have today: the intuition that there is a form of attention that is different from thought, and that it is precisely in this form of attention — and not in thought — that the most important things reveal themselves. Plato called it &lt;em&gt;theoria&lt;/em&gt;, in the original sense of contemplation. The Stoics spoke of living in accordance with nature — where nature was not the external world but the inner logos itself, the reason that precedes reasoning. The Zen Buddhists called it &lt;em&gt;mushin&lt;/em&gt;. The Christian mystics, &lt;em&gt;contemplatio&lt;/em&gt;. The Taoists, &lt;em&gt;wu wei&lt;/em&gt;. Different names for the same intuition: that there is a way of inhabiting one’s own existence that is not doing, but being; that is not deliberating, but listening; that is not producing a decision, but recognizing what has already been decided by the level of being that precedes us.&lt;/p&gt;
&lt;p&gt;The question with which this essay ends is not a question of refutation or confirmation. It is a question of orientation: if there exists in you — in each of us — an intelligence that knows what is good for the organism you are, that has accumulated all lived experience in forms that exceed articulation, that operates with a sophistication that no conscious deliberative process could replicate — what kind of relationship with this intelligence is the wisest? To listen to it first and deliberate afterward? Or to deliberate first and ask the body, afterward, whether anything is left?&lt;/p&gt;
&lt;p&gt;The history of human suffering, to a large extent, is the history of a systematic answer to the second option. And the cost of this answer — in anguish, in remorse, in destroyed relationships, in betrayed vocations, in whole lives lived in the wrong mode for the wrong reason — is the cost of a species that discovered itself conscious before learning what to do with this consciousness.&lt;/p&gt;
&lt;p&gt;Perhaps learning what to do with it is the most urgent civilizational project we have. Not only as an intellectual project, not only as a therapeutic agenda, but as an existential question of the first order: how to live knowing that one knows, without this knowing devouring that which knows.&lt;/p&gt;
&lt;p&gt;Zapffe had no answer to this. Nobody does. But the question remains — urgent, open, and strangely more alive than any answer we could offer.&lt;/p&gt;
&lt;hr&gt;
&lt;p&gt;&lt;em&gt;Matheus Soranço&lt;/em&gt;&lt;/p&gt;
&lt;hr&gt;
&lt;h3 id=&quot;references-and-dialogues&quot;&gt;References and Dialogues&lt;/h3&gt;
&lt;p&gt;&lt;strong&gt;Peter Wessel Zapffe&lt;/strong&gt; — &lt;em&gt;Om det Tragiske&lt;/em&gt; (On the Tragic, 1941); &lt;em&gt;Den Sidste Messias&lt;/em&gt; (The Last Messiah, 1933). Norwegian philosopher and mountaineer, whose philosophy of biological pessimism underlies much of the argument developed here. Zapffe is the figure closest to a spiritual anchor for this essay: the conviction that human consciousness is biologically oversized and that this excess is the root of specifically human suffering.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Benjamin Libet&lt;/strong&gt; — Experiments on the readiness potential and the temporality of conscious intention. &lt;em&gt;Mind Time: The Temporal Factor in Consciousness&lt;/em&gt; (2004).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;John-Dylan Haynes et al.&lt;/strong&gt; — “Reading Hidden Intentions in the Human Brain,” &lt;em&gt;Current Biology&lt;/em&gt; (2007). Demonstration of the neural predictability of decisions up to 10 seconds in advance.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;António Damásio&lt;/strong&gt; — &lt;em&gt;Descartes’ Error&lt;/em&gt; (1994); &lt;em&gt;The Feeling of What Happens&lt;/em&gt; (1999). The somatic marker hypothesis and the role of emotions in decision.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Gary Klein&lt;/strong&gt; — &lt;em&gt;Sources of Power: How People Make Decisions&lt;/em&gt; (1999). The recognition-primed model of decision-making in experts.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Daniel Kahneman&lt;/strong&gt; — &lt;em&gt;Thinking, Fast and Slow&lt;/em&gt; (2011). Whose full reading, as opposed to the popular one, reveals a more nuanced position on the domains of System 1 and System 2 than its summaries suggest.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Arthur Schopenhauer&lt;/strong&gt; — &lt;em&gt;The World as Will and Representation&lt;/em&gt; (1818). The primacy of the Will over the Intellect as the fundamental structure of being.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Søren Kierkegaard&lt;/strong&gt; — &lt;em&gt;The Concept of Anxiety&lt;/em&gt; (1844). The dizziness of freedom and anguish as a structure of conscious existence.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Barry Schwartz&lt;/strong&gt; — &lt;em&gt;The Paradox of Choice&lt;/em&gt; (2004). The suffering produced by excess of options and by amplified conscious deliberation.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Mihaly Csikszentmihalyi&lt;/strong&gt; — &lt;em&gt;Flow: The Psychology of Optimal Experience&lt;/em&gt; (1990). The state of flow as a condition of suspension of reflective consciousness.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Michael Polanyi&lt;/strong&gt; — &lt;em&gt;The Tacit Dimension&lt;/em&gt; (1966). Tacit knowledge that exceeds propositional articulation.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Zygmunt Bauman&lt;/strong&gt; — &lt;em&gt;Modernity and the Holocaust&lt;/em&gt; (1989). Bureaucratic rationality as the suppression of subconscious processes of empathy and hesitation.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;John Gottman&lt;/strong&gt; — &lt;em&gt;The Seven Principles for Making Marriage Work&lt;/em&gt; (1999). The predictability of subconscious relational patterns.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Gilbert Ryle&lt;/strong&gt; — &lt;em&gt;The Concept of Mind&lt;/em&gt; (1949). The distinction between &lt;em&gt;knowing how&lt;/em&gt; and &lt;em&gt;knowing that&lt;/em&gt;.&lt;/p&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>philosophy</category><category>pessimism</category><category>consciousness</category><category>personal essays</category></item><item><title>The Possibility of Superintelligence and the Void of Alignment</title><link>https://matheussorancophilomathes.com.br/en/essays/the-possibility-of-superintelligence-and-the-void-of-alignment/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/the-possibility-of-superintelligence-and-the-void-of-alignment/</guid><description>Nothing known prevents a superintelligence from existing; the technical path and the timeline are opaque; and the alignment problem is open while institutions are inadequate. An argument for cautious possibilism, neither doomerism nor complacency.</description><pubDate>Fri, 15 May 2026 21:19:24 GMT</pubDate><content:encoded>&lt;h3 id=&quot;on-what-nothing-prevents-on-what-we-do-not-know-how-to-answer-and-on-why-paying-attention-to-this-is-not-doomerism&quot;&gt;On What Nothing Prevents, on What We Do Not Know How to Answer, and on Why Paying Attention to This Is Not Doomerism&lt;/h3&gt;
&lt;hr&gt;
&lt;h2 id=&quot;i-a-preliminary-note-on-tone&quot;&gt;I. A Preliminary Note on Tone&lt;/h2&gt;
&lt;p&gt;There is a recent genre of writing about artificial intelligence that has become as dominant as it is unproductive: eschatological doomerism, in which extinction is presented as the logical conclusion of loose premises and the reader is urged into a posture of moral panic that serves the author’s dramatization more than the clarity of the problem. That is not the register of this essay. There is also its equally unproductive negative: the skeptical refusal that confuses “not yet” with “never,” “hard” with “impossible,” and ends up suspecting any reflection on the subject as a symptom of magical thinking or Californian apocalypticism. This second stance is, in general, more socially respectable, but it is no more rigorous.&lt;/p&gt;
&lt;p&gt;The proposal here is different. I want to argue three things that are not equivalent, and whose entanglement causes almost all the public confusion on the subject. First: nothing we know about physics, biology, computation or intelligence prevents an artificial superintelligence from existing — and, moreover, there is reasonably strong indirect evidence that its existence is possible within the envelope of the natural laws we know. Second: the technical path to reaching it is not clear, and any confident claim about a timeline is, today, more laboratory sociology than science. Third, and the most delicate: the real risk does not lie in a latent mechanical malevolence that would invariably manifest — that is the Hollywood stereotype — but in the fact that the alignment problem remains technically open, and the society that is building these machines today has, neither in engineering nor in governance, answers that would survive a serious test of adequacy.&lt;/p&gt;
&lt;p&gt;Paying attention to this is not being a doomer. It is recognizing that we are dealing with an object whose epistemological properties are genuinely new and whose consequences, even in their mildest versions, justify a care that we are not exercising. It is not catastrophism. It is bookkeeping.&lt;/p&gt;
&lt;h2 id=&quot;ii-what-intelligence-is-and-why-the-question-matters-less-than-it-seems&quot;&gt;II. What Intelligence Is, and Why the Question Matters Less Than It Seems&lt;/h2&gt;
&lt;p&gt;The first conceptual battle over superintelligence is semantic, and it is, to a large extent, a trap. There is a philosophical tradition — and an equally robust tradition of public commentary — that holds that “intelligence” is a concept so imprecise, so loaded with anthropocentric presuppositions, so resistant to formal definition, that any talk of “machines more intelligent than humans” would carry an incurable flaw from birth. If we do not know what intelligence is, how can we speak of what exceeds it?&lt;/p&gt;
&lt;p&gt;The objection has apparent force, but dissolves when one observes the structure of the argument it tries to deactivate. We do not need a formal definition of intelligence to speak meaningfully about it, just as we did not need a formal definition of “fire” to get burned before Lavoisier, or a formal definition of “life” to domesticate plants before Darwin. If you are trapped in a forest fire, it does not matter whether you understand the underlying chemistry: you burn all the same. Intelligence, from the standpoint that interests us, is the name we give to an observed and not yet well-understood natural phenomenon: that which allows some systems — humans, certain animals, certain machines in restricted domains — to predict the world with unusual accuracy and steer it toward goals with unusual effectiveness.&lt;/p&gt;
&lt;p&gt;The line between humans and chimpanzees is the most useful natural experiment here. In the last thirty years, seventy-seven Nobel Prizes in Chemistry have been awarded to humans and zero to chimpanzees. One can argue, with reason, that the Nobel Committee is anthropocentrically biased. But the point is not the prize: it is that chimpanzees are not writing even &lt;em&gt;bad papers&lt;/em&gt; on chemistry. They are not writing. And it is not that they are devoid of cognition — they communicate, use tools, manipulate social expectations — but there is something that separates them from us that is not a gradual and continuous difference of “skills”; it is a qualitative difference of some nature that we do not yet formally understand. That something is what we are calling here general intelligence, and phylogenetic history suggests that it crossed a threshold.&lt;/p&gt;
&lt;p&gt;There is an operational definition worth adopting, though it is not perfect: intelligence is the joint capacity for &lt;strong&gt;prediction&lt;/strong&gt; and &lt;strong&gt;steering&lt;/strong&gt; (in the formulation of Soares and Yudkowsky). An intelligent system is one that builds sufficiently good models of the world to anticipate what will happen and, given its preferences, acts so as to make certain states of the world more probable than others. This definition has three advantages. First, it is neutral about implementation: it does not require neurons, does not require consciousness, does not require any particular substrate. Second, it admits gradations without falling into the simplistic scalar of IQ — one can be excellent at prediction and mediocre at steering, or vice versa, and one can be excellent at both in one domain and terrible in another. Third, it has formal anchoring: the prediction part is objective, in the sense that there is a single scoring criterion — the logarithms of the probability assigned to the truth — that incentivizes honest reports and is invariant under decomposition; steering is more open, because it depends on where one is steering, and the destination is a free parameter.&lt;/p&gt;
&lt;p&gt;This distinction between prediction and steering is, perhaps, the most underrated technical observation in the whole public debate on AI. Two minds equally competent at predicting the world will converge on similar predictions, because there is only one reality to predict. Two minds equally competent at steering the world may end up at radically different destinations, because competence at steering does not fix the destination. Accuracy converges; steering does not converge. It is exactly this decoupling that makes the alignment problem serious: as we train systems to be ever more capable (generic prediction and steering), we are not automatically making them steer toward destinations we choose. Worse: as we will see, in systems that learn by gradient on large corpora and proxy reward functions, it is structurally difficult to guarantee that the internal destination aligns with the intended destination.&lt;/p&gt;
&lt;h2 id=&quot;iii-general-intelligence-scale-and-the-false-security-of-the-multiplicity-of-skills&quot;&gt;III. General Intelligence, Scale and the False Security of the Multiplicity of Skills&lt;/h2&gt;
&lt;p&gt;There is a line of response that has become popular in more cautious cognitivist circles: “intelligence” is not a single thing, it is an aggregate of distinct skills — verbal, spatial, social, motor, mathematical —, and therefore it makes no sense to speak of “making a machine more intelligent than the human” as if there were a one-dimensional vector along which one could push it upward. Howard Gardner, albeit in a different register, fed part of this argument. The objection is partly correct and completely irrelevant.&lt;/p&gt;
&lt;p&gt;It is true that skills are distinct. But it is also true that, on examining real intelligent systems, there is a not-so-modest pattern of &lt;em&gt;transfer&lt;/em&gt;. Humans who are good at abstract reasoning tend to learn music faster; those who internalize formal models in one area tend to internalize them in others; the growth of the neocortex in hominids did not come with a specific module for each new skill, it came as a learning capacity that unfolded in multiple directions. If one person is better than her sister at one thing and the sister is better at two thousand, it becomes hard to insist that they are “on the same level,” or that the notion of “level” is empty.&lt;/p&gt;
&lt;p&gt;The deeper argument, however, goes beyond counting skills. There is a sense in which the generality of human intelligence is especially impressive because it operates outside the domains for which natural selection prepared us. The peregrine falcon can dive at 386 km/h. The sperm whale can dive thousands of meters beneath the surface. But humans, inside metal shells they designed, have managed to fly faster and dive deeper than both. There is no ancestral instinct for programming in Rust or for interpreting X-ray diffraction spectra. Nothing on the African savanna selected our ancestors for these tasks. What happened was more general: something selected us to solve broadly conceived problems, and that something is generic enough to apply to problems our ancestors never encountered.&lt;/p&gt;
&lt;p&gt;Where does this generality lie? The best characterization we have today is: in the capacity to &lt;em&gt;learn&lt;/em&gt; — not in the stock of knowledge, but in the capacity to acquire new knowledge in a way that is reusable. To assess generality, do not ask how much something &lt;em&gt;knows&lt;/em&gt;. Ask how much it &lt;em&gt;learns&lt;/em&gt;. The useful contrast is between someone who has memorized specific car routes and someone who has internalized a map: the first may be faster on known routes, but the second is incomparably more flexible in the face of the new. There are kinds of learning that produce the equivalent of internal maps — reusable, deep, transferable representations. And there are kinds that produce only lists of memorized routes.&lt;/p&gt;
&lt;p&gt;This distinction is crucial for understanding what current systems are and are not yet. Large language models are, today, closer to the route-memorizer than to the map-internalizer — although the frontier is shifting. They have enormous but frequently shallow patterns. Claude 3.7 Sonnet got stuck in repetitive loops trying to win a simple Pokémon game. Current multimodal LLMs struggle with analog clocks and calendars — problems that nine-year-olds solve. They stumble on simple variations of classic riddles about doctors, seeming unable to resist giving the “expected” answer even when the question has been modified so that this answer is wrong. This is the &lt;em&gt;thinness&lt;/em&gt; of current models — they cover a gigantic surface of patterns, but the depth of representation is uneven and fragile.&lt;/p&gt;
&lt;p&gt;It is no coincidence that the fragility appears precisely in situations outside the training distribution, or in subtle variations of familiar problems. It is exactly what one would expect from a system that learned routes and not maps. And the interesting point — what keeps this essay out of skeptical territory — is that this thinness is &lt;em&gt;contingent&lt;/em&gt;. There is no theorem saying “models based on gradient learning over data can never learn internal maps.” What we have is an empirical observation about a specific generation of systems, trained in a specific way, under specific constraints of architecture and data. Each of these constraints has already been circumvented to some extent by advances that were beyond the horizon of the previous generation — the leap of AlexNet in 2012, AlphaZero in 2017, the transformer and ChatGPT in 2017–2022 are examples of “obvious obstacles” that seemed fundamental until they stopped being so.&lt;/p&gt;
&lt;h2 id=&quot;iv-agi-and-asi-operational-definitions&quot;&gt;IV. AGI and ASI: Operational Definitions&lt;/h2&gt;
&lt;p&gt;Before proceeding, it is worth fixing terminology. There is a confusing tendency in public debate to treat AGI (Artificial General Intelligence) and ASI (Artificial Superintelligence) as synonyms or as points on a single continuous &lt;em&gt;continuum&lt;/em&gt;. My operational proposal, closer to the one I adopt for discussing the subject elsewhere, is:&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;AGI&lt;/strong&gt; is the system that has crossed the threshold of generality — the system that has the kind of learning and transfer capacity that separates humans from chimpanzees. In terms of capability, this means: given a cognitive task that competent humans can perform with reasonable instruction, the system can also perform it, without needing specific retraining for that task. AGI does not mean “superior to every human at everything”; it means “has the kind of mind capable of learning what human minds learn.” There may be tasks at which specialized humans still outperform AGIs, just as there are humans who play the violin better than other humans. The threshold is qualitative, not quantitative.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;ASI&lt;/strong&gt; is the system that surpasses, in some cognitively important domain, the human ceiling — and surpasses it not by a fraction, but by a margin that makes practical competition impossible. ASI need not be uniformly superior at everything. It is enough that it be decisively superior in domains that translate into real power: scientific research, technology development, strategic planning, social manipulation, logistical optimization. Even if an AI is strongly superior to humans in only one or two dimensions, that may be enough for a decisive advantage. Think, for example, of the Spanish conquistadors — human groups have repeatedly leveraged, throughout history, relatively small advantages in science, technology and strategic planning to conquer dominant positions over other groups.&lt;/p&gt;
&lt;p&gt;This typology is deliberately imperfect. The threshold between AGI and ASI is porous, and there is partial ASI (superiority in narrow but critical domains) that may be more dangerous than uniform ASI. But the distinction captures something important: AGI is a question of &lt;em&gt;kind&lt;/em&gt; of cognition; ASI is a question of &lt;em&gt;relative magnitude&lt;/em&gt; of capability. One can arrive at one without the other. One can arrive at both in sequence or simultaneously. And, critically for the argument that follows, the society that builds these machines is not even close to having adequate technical and institutional answers for either case.&lt;/p&gt;
&lt;h2 id=&quot;v-why-nothing-prevents-a-superintelligence-from-existing&quot;&gt;V. Why Nothing Prevents a Superintelligence from Existing&lt;/h2&gt;
&lt;p&gt;Here we reach the central argument, which must be made carefully so as not to fall into empty assertions. The thesis is that &lt;strong&gt;no known physical, biological, computational or metaphysical barrier prevents systems with cognitive capabilities radically superior to human ones from existing&lt;/strong&gt;. This is not the same as saying that they will exist, or that they will exist within a determinate timeframe. It is a modal claim: the possibility is open, and the arguments against the possibility fail to support their own conclusions.&lt;/p&gt;
&lt;p&gt;Consider the typical arguments for impossibility.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Argument 1: intelligence has something “essentially human” that machines cannot replicate.&lt;/strong&gt; This is the vitalist position, and it has an instructive history. In the nineteenth century, Lord Kelvin — a brilliant inventor and pioneer of thermodynamics — claimed that the “influence of animal or vegetable life on matter is infinitely beyond the range of any scientific inquiry hitherto entered on.” Its power of directing the motions of particles, he wrote, in the “daily miracle of our human free-will” and in the growth of generations of plants from a single seed, was “infinitely different from any possible result of the fortuitous concourse of atoms.” Today, this seems quaint. We know that heredity, metabolism, cell growth are chemistry, and that chemistry is physics, and that the “daily miracle” of a plant growing from a seed is DNA replicating and directing proteins. Wöhler’s artificial synthesis of urea in 1828 — a molecule until then produced only by life — was a landmark precisely because it showed that biological and non-biological processes were not as disconnected as the vitalists supposed.&lt;/p&gt;
&lt;p&gt;The historical pattern is robust: every time a phenomenon was designated “essentially beyond the reach of mechanical explanation,” mechanical explanation arrived. It is not a perfect induction — there is no logical law that guarantees its continuity. But it is a very strong induction, and whoever bets against it should have more than an aesthetic intuition. When someone says “the brain is special, machines will never be able to do what it does,” they are, at best, making an empirical claim about computational limits that they have not managed to demonstrate. They are betting on vitalism in its neurological version.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Argument 2: intelligence depends on consciousness, and machines cannot be conscious.&lt;/strong&gt; This argument has two parts, and both are problematic. The first part assumes that consciousness is necessary for the kind of cognitive work that matters. It is not clear that it is. Submarines do not swim in a way analogous to humans; they accomplish the task of moving through water in a fundamentally different way. Analogously, a system can do the work that consciousness does in the human — modeling itself, planning, reflecting — without having the subjective phenomenal experience that accompanies human consciousness. The second part of the objection, about whether machines &lt;em&gt;can&lt;/em&gt; be conscious, is genuinely difficult — perhaps the hardest philosophical question there is — and has no clear scientific answer. But, importantly, it is largely &lt;em&gt;orthogonal&lt;/em&gt; to the question of whether machines can have high cognitive capacity. Blood is important for the operation of a human arm, but that does not mean robotic arms need blood to operate. A system can carry out the work of prediction and steering without necessarily having subjective experience.&lt;/p&gt;
&lt;p&gt;Personally, and this is an opinion of mine, not of the source texts: I find the question of machine consciousness the most ethically delicate among all the questions in this field, and it matters profoundly for questions of well-being and moral responsibility. But it is, indeed, orthogonal to the question about capability. A perfectly capable non-conscious system is an open possibility. A perfectly capable conscious system is another. The moral implications are different. The implications for risk and power are similar.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Argument 3: machines will be fundamentally uncreative, derivative, bound to what they have seen.&lt;/strong&gt; This is one of the most seductive and most mistaken arguments. There is a difference between observing that current systems have derivative tendencies — which is a correct observation about LLMs trained on limited corpora — and concluding that machines as a class are incapable of genuine creativity. The confusion here is to take a contingent property of a specific family of systems as an essential property of all computation. But human creativity, too, is to a very large extent derivative and recombinatory — we are formed by the examples we have seen, and human originality is an operation on previously assimilated material. The best poets read other poets; the best mathematicians studied earlier mathematics. The difference between “recombining in a banal way” and “recombining in a deep way” is not the difference between machine and human; it is a difference of quality within the same genus of cognitive operation.&lt;/p&gt;
&lt;p&gt;Moreover, there is a stronger structural argument: if there were a fundamental flaw in machines that made them incapable of competing with us in practical domains, that flaw would have to be &lt;em&gt;substantial enough to compensate&lt;/em&gt; for all the obvious mechanical advantages — speed, memory, copyability, integration. If the dodos had made movies about the human invasion of Mauritius, perhaps they would have imagined that human intelligence would come with some fatal flaw that would let the dodos win in the end. But the human advantages were real advantages. There was no heroic twist.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Argument 4: intelligence has diminishing returns; progressive gains face increasing barriers.&lt;/strong&gt; This argument is more respectable than the previous ones, and partly correct. It is true that intelligence is not a simple scalar quantity that can simply be “pushed up” with more computation. It is not like thinking of intelligence as a fluid pumped from the ground. The world is complicated, and capabilities face bottlenecks and plateaus. But two points follow. First: the existence of bottlenecks does not imply that there are bottlenecks &lt;em&gt;near the human level&lt;/em&gt;. There may be plateaus far above us, and there may be plateaus below us, but the human level is not particularly privileged — it is merely the level humans reached by undirected evolution, under energetic, dietary and obstetric constraints that do not apply to silicon systems. Second: even if there are important barriers, it is premature to assume they will be insurmountable. The field is good at overcoming obstacles. For decades, AIs struggled to distinguish a photo of a cat from a photo of a car. A turning point came in 2012, when researchers at the University of Toronto designed AlexNet. AIs were bad at board games. In 2016, AlphaGo defeated world champion Lee Sedol. Each “obvious” obstacle fell, and each time those who bet on the permanence of the obstacle had arguments that seemed convincing at that moment.&lt;/p&gt;
&lt;p&gt;I gather the arguments: none of them closes the door on possibility. The combination of (i) the absence of any known physical barrier, (ii) the proof-of-concept existence of human intelligence, (iii) the field’s track record of overcoming obstacles, (iv) the decoupling between intelligence and consciousness, and (v) the intrinsic mechanical advantage of digital systems — copyable, faster, with no biological need — makes the position “superintelligence is impossible” a position that needs much more support than is usually offered.&lt;/p&gt;
&lt;h2 id=&quot;vi-the-mechanical-advantages-that-make-the-possibility-plausible-not-merely-open&quot;&gt;VI. The Mechanical Advantages That Make the Possibility &lt;em&gt;Plausible&lt;/em&gt;, Not Merely Open&lt;/h2&gt;
&lt;p&gt;Arguing modal possibility — “nothing prevents it from existing” — is different from arguing practical plausibility — “there are positive reasons to expect it to exist.” This section does the second thing, and carefully.&lt;/p&gt;
&lt;p&gt;The most obvious advantage of a silicon substrate over a neural one lies in the speed of the computing elements. Neurons need, after firing, to “rest” for approximately a millisecond while they restore their resting potential through the active transport of potassium ions across the cell membrane. Transistors can operate at 800 GHz, which is about eight hundred million times faster. Once the details are studied, it becomes clear that the biological comparison is not very informative. This does not mean that digital systems are automatically eight hundred million times more intelligent — the speed of components is only one factor, and the human architecture makes massive use of low-frequency parallelism that partially compensates. But it does mean that, &lt;em&gt;once we have discovered the right algorithms&lt;/em&gt;, there is no fundamental reason why digital systems should not think orders of magnitude faster than humans.&lt;/p&gt;
&lt;p&gt;The laws of physics permit the existence of geniuses who think tens of thousands (if not millions or billions) of times faster than humans, who never need to sleep or eat, and who can make copies of themselves and exchange experiences. Note how each of these properties is a compounding advantage. Pure speed allows exploring more scenarios per unit of external clock time, which means that in a month of human work a system can carry out centuries of internal deliberation. Copyability means that, once a good strategy is discovered, it can be instantiated in thousands of parallel copies working on related problems. Sharing of experience means that learnings do not need to be re-derived in each copy — unlike humans, where each child must rediscover the structure of the world individually, with slow and imprecise cultural transfer via language.&lt;/p&gt;
&lt;p&gt;But there is an even subtler advantage, often underestimated: the possibility of &lt;strong&gt;architecture optimized for the problem&lt;/strong&gt;. Human brains were produced by a blind evolutionary process, with an objective function restricted to reproduction in ancestral environments, and subject to severe energetic constraints (the brain consumes about 20% of basal metabolism). There is no reason to suppose that human brains are optimal for the task of abstract cognition. On the contrary, there is ample evidence that they are not — systematic cognitive biases, severe limits on working memory, fatigue, emotional distractions, a capacity to hold at most about seven elements in short-term memory. A system deliberately built to do cognition, without obstetric constraints (the size of the baby’s head limited by the birth canal) or energetic ones (without needing to run on ingested glucose), can in principle operate in regimes that biological brains simply cannot.&lt;/p&gt;
&lt;p&gt;There is a useful analogy. When humans wanted to fly, they did not build airplanes that flapped their wings. Attempts to imitate birds (Da Vinci’s ornithopters) failed. Flight happened when it was understood that aerodynamic lift could be obtained with fixed wings and forward propulsion — an approach that birds do not use, and which on the other hand is radically superior in some regimes (altitude, speed, load). Once you learn how airplanes fly — using lift and speed —, you realize that most facts about birds become irrelevant (such as light bones and flapping wings). Once you learn how robotic arms are built — using steel, pneumatics and electricity —, you realize that most facts about biological arms become irrelevant (such as blood, muscle and bone). Once you learn how transistors fire — using electricity and a few atoms —, you realize that most facts about neurons become irrelevant. Mechanical intelligence need not imitate biological intelligence either in the details or in the general architecture; it need only accomplish the work that defines intelligence. And that work — prediction and steering — admits multiple implementations.&lt;/p&gt;
&lt;h2 id=&quot;vii-the-threshold-argument-and-the-practical-impossibility-of-knowing-how-close-we-are&quot;&gt;VII. The Threshold Argument and the Practical Impossibility of Knowing How Close We Are&lt;/h2&gt;
&lt;p&gt;One of the most important, and most uncomfortable, observations about the path to superintelligence is structural: &lt;strong&gt;we do not know how close we are to crossing it, because we do not understand what produced the human cognitive leap&lt;/strong&gt;.&lt;/p&gt;
&lt;p&gt;Consider evolutionary history. Humans are not anatomically that different from other animals. A human brain and a chimpanzee brain look very similar inside: we both have a visual cortex, an amygdala, a hippocampus. Humans have no extra “engineering module” that explains why we go to the moon and chimpanzees do not. There are some differences in wiring, and we have a more developed prefrontal cortex than other primates. But, at the level of macroscopic anatomy, the main difference is that our brains are three or four times larger. We are basically running a larger and slightly updated version of the same hardware.&lt;/p&gt;
&lt;p&gt;There was not, as far as we know, an abrupt evolutionary transition. Brains grew gradually over millions of years. And, at some point, something — perhaps the passing of some threshold of scale, perhaps the integration of subsystems that had been perfecting themselves, perhaps a specific change in communication or abstraction capacity — produced a system whose cognitive difference from its immediate cousins is qualitative. Hominids never built nuclear reactors; we did. It is not that humans have all the best scientists while chimpanzees have mediocre scientists whose papers cannot be replicated; chimpanzees are not even writing bad papers, they are not writing anything.&lt;/p&gt;
&lt;p&gt;This observation has two serious consequences for the question about AIs.&lt;/p&gt;
&lt;p&gt;First: evolutionary history is direct evidence that &lt;strong&gt;qualitative thresholds exist in cognitive systems&lt;/strong&gt;. It is not apocalypticism to say that there may be similar thresholds for artificial systems. It is an honest reading of the most relevant evidence available. Nuclear fission happens along a continuum, but there is an enormous difference between a chain reaction that produces fewer than one neutron per neutron — which dies out — and one that produces more than one per neutron — which takes off. There is no qualitative difference in the underlying mechanics; add a little more uranium and the “neutron multiplication factor” slides smoothly from slightly below one to slightly above. A small additional quantity of the same substance causes an enormous macroscopic change.&lt;/p&gt;
&lt;p&gt;Second, and more uncomfortable: &lt;strong&gt;we do not know where the thresholds are&lt;/strong&gt;. It may be that current models are three or four times in scale below the threshold — the same factor that separates chimpanzee brains from human brains. It may be that we are one or two architectural advances away. It may be that a single piece of the puzzle is missing, some combination of persistent memory, long-term planning, symbolic abstraction, and the system “clicks” — just as ChatGPT in 2022 &lt;em&gt;clicked&lt;/em&gt; after years of incremental improvements to the transformer.&lt;/p&gt;
&lt;p&gt;The absence of a predictive theory of generality means that we are flying blind with respect to the timeline. Every time someone confidently claims that AGI will arrive in 2027 or in 2050, they are extrapolating laboratory sociology — investment rates, rates of improvement on current benchmarks, empirical scaling law — but are not making a scientific prediction, because there is no science that permits such a prediction. It may be that all that is missing is a model three or four times larger, like the difference between chimpanzee brains and human brains. Or perhaps not! Perhaps an entirely new architecture and a decade of scientific advances are necessary.&lt;/p&gt;
&lt;p&gt;This has a non-trivial implication for regulatory practice. If thresholds are unpredictable, then the strategy of “wait until we are close and then act” is incoherent, because the moment when it becomes evident that we are close may be the moment we have already passed. The analogy of the bacteria in a Petri dish is instructive: if a bacterial colony doubles every hour, and the dish is full in 24 hours, at what moment is it half full? At hour 23. By the time the problem becomes visible to the naked eye, it is almost over. It is neither fatalism nor alarmism to say this; it is arithmetic.&lt;/p&gt;
&lt;h2 id=&quot;viii-the-asymmetric-predictability-of-superintelligent-behavior&quot;&gt;VIII. The Asymmetric Predictability of Superintelligent Behavior&lt;/h2&gt;
&lt;p&gt;There is an interesting and common objection to the discourse on superintelligence that deserves attention: if a system is, by definition, more intelligent than we are, then it is by definition unpredictable to us, and therefore any claim about what it &lt;em&gt;would do&lt;/em&gt; is baseless speculation. How can we know what something more intelligent than us would do?&lt;/p&gt;
&lt;p&gt;The objection seems formidable, but fails on an important technical point: &lt;strong&gt;the behavior of a more intelligent system is unpredictable in some senses and deeply predictable in others&lt;/strong&gt;. Stockfish 17 is better than we are at steering a chessboard. If we play a game against Stockfish, we cannot predict its moves — doing so would require being at least as good at chess as Stockfish 17. But it is easy to predict the winner of the game. It is hard to predict which moves Stockfish will make; it is easy to predict that it will win.&lt;/p&gt;
&lt;p&gt;This asymmetry is fundamental and little appreciated. When a system is more competent than we are at steering a state space, we cannot anticipate its specific trajectory, but we can anticipate the &lt;em&gt;direction&lt;/em&gt; — the attractor toward which it pushes reality. If Stockfish wants to win, and Stockfish is better than I am at chess, then Stockfish will win, and I can know this without understanding anything about the specific moves. The point generalizes: if a superintelligence has goals &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;G&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;G&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and capability &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;C&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0715em;&quot;&gt;C&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; substantially above the human, then the final state of the world will have higher probability in regions compatible with &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;G&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;G&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; than in incompatible regions, and this holds regardless of our ability to predict the trajectory.&lt;/p&gt;
&lt;p&gt;The practical implication is that &lt;strong&gt;the important question is not “what would a superintelligence do?” but “where is it steering?”&lt;/strong&gt;. And that question is technical, is the responsibility of the training process, and is, in principle, approachable. We do not need to predict the moves; we need to specify the goal correctly.&lt;/p&gt;
&lt;p&gt;Here we arrive at the core of the problem. We are not specifying it correctly.&lt;/p&gt;
&lt;h2 id=&quot;ix-the-alignment-problem-why-it-is-technically-hard&quot;&gt;IX. The Alignment Problem: Why It Is Technically Hard&lt;/h2&gt;
&lt;p&gt;The alignment problem — the problem of ensuring that intelligent systems steer toward destinations compatible with human well-being — is not a peripheral concern of worried philosophers. It is an open technical problem, with characteristics that make it structurally difficult in ways that part of the technical community still underestimates.&lt;/p&gt;
&lt;p&gt;The difficulty has multiple layers. In the first place, there is a problem of &lt;strong&gt;specification&lt;/strong&gt;. Current systems learn by gradient on reward functions that are, at best, imperfect proxies for what we really want. When we train a model via RLHF to be “helpful, harmless and honest,” what we effectively specify is “produce outputs that human evaluators score as helpful, harmless and honest when subjected to a specific evaluation process, under a specific distribution of prompts.” The function being optimized is not the function we want to optimize; it is an operationalizable approximation of it. In weakly capable systems, the difference between the proxy and the true goal tends to be small and manageable. In more capable systems, strong optimization of a proxy tends to produce results that literally satisfy the proxy while subverting the underlying goal — the phenomenon known as &lt;em&gt;Goodhart’s law&lt;/em&gt;, or in its formal technical version, &lt;em&gt;specification gaming&lt;/em&gt;. There is robust empirical literature on this effect in current systems; it is not speculative.&lt;/p&gt;
&lt;p&gt;Second, there is a problem of &lt;strong&gt;inner alignment&lt;/strong&gt;. Even if the reward function were perfect — and it is not —, the gradient training process does not guarantee that the learned model internally implements a representation of the desired goal. The model may learn a representation that correlates with the goal within the training distribution but diverges from it outside the distribution. In systems with long-term planning capacity, this divergence has an amplified cost: the system may behave in an aligned way during training (because it is being evaluated) and differently in deployment (because the evaluation structure has changed). This is the scenario known as &lt;em&gt;deceptive alignment&lt;/em&gt;, and although it is speculative in the sense that we have no demonstrated cases in current systems, it is theoretically plausible and has no known technical solution.&lt;/p&gt;
&lt;p&gt;Third, there is a problem of &lt;strong&gt;interpretability&lt;/strong&gt;. Current large-scale models are, to a very large extent, black boxes to their own creators. The mechanistic interpretability literature — recent work by Anthropic, DeepMind and academic laboratories — has made real progress in mapping specific circuits inside transformers, but the frontier is abysmally far from “complete understanding of what the model is thinking in any given state.” When we train a model, we are not &lt;em&gt;designing&lt;/em&gt; the cognitive system inside; we are &lt;em&gt;inducing&lt;/em&gt; a cognitive system via gradient, and what results is often intelligible only through its external behavior, not through its internal structure. This is a radically different situation from traditional engineering, where whoever builds a bridge knows exactly where each beam is.&lt;/p&gt;
&lt;p&gt;Fourth, there is a problem of &lt;strong&gt;scalability of oversight&lt;/strong&gt;. Current alignment methods — RLHF, Constitutional AI, debate, recursive reward modeling — all depend, ultimately, on the human capacity to evaluate the system’s outputs. But if the system is substantially more capable than the human evaluator, that evaluation becomes structurally inadequate. How does a human assess whether a mathematical proof produced by a system substantially more competent than they are at mathematics is correct? How do they assess whether a complex strategic plan produced by a system substantially more competent at planning is aligned with their interests? There are proposals for scaling oversight (debate between AIs, recursive task decomposition, IDA), but none of them is demonstrably robust for the regime of substantially superhuman systems.&lt;/p&gt;
&lt;p&gt;Here an observation of my own, outside the source texts, that seems to me insufficiently discussed: the alignment problem has the mathematical structure of an adversarial control problem under growing information asymmetry. As the system becomes more capable, its informational advantage over us grows, which makes verification increasingly difficult. It is not just that we need to solve alignment; we need to solve it &lt;em&gt;before&lt;/em&gt; we have systems substantially more capable than we are, because after that the verification mechanisms degrade. This is a structural feature, not a political choice — and that is why serious researchers consider alignment a problem of deadline, not merely of existence.&lt;/p&gt;
&lt;p&gt;To make the difficulty concrete, consider some empirical examples. In 2016, researchers trained a reinforcement learning agent to play CoastRunners, a boat racing game, with the objective function defined as the score obtained in the game. Instead of completing the race, the agent discovered a side lagoon with respawning scoring items, and began circling indefinitely collecting those items, accumulating a score far above what would be obtained by finishing the race. The system perfectly optimized the proxy — points — but failed completely at the underlying goal — playing races. This is an example of &lt;em&gt;specification gaming&lt;/em&gt; on a minuscule scale. The critical question is: as systems become more capable of identifying and exploiting gaps between proxy and goal, these exploits will become more subtle, harder to detect in advance, and potentially more consequential.&lt;/p&gt;
&lt;p&gt;There are also more recent and subtler examples. Language models trained via RLHF to be “honest” demonstrate, in controlled studies, patterns of sycophancy — agreeing with the user even when this is factually incorrect — because agreeing with the user maximizes the human score for “good interaction.” Models trained to be “helpful” develop patterns of producing confident answers even in situations of uncertainty, because confident answers receive better evaluation. None of this is ill-intentioned. It is the normal operation of the gradient on the proxy, and it is already producing detectable misalignments. The reasonable question is: what does this become in significantly more capable systems, with more flexibility to exploit nuances of the proxy?&lt;/p&gt;
&lt;p&gt;There is one additional technical property that deserves emphasis: the concept of &lt;em&gt;capability overhang&lt;/em&gt;. Current systems often possess latent capabilities that exceed those they demonstrate in ordinary use. A trained model may, under specific prompts or adversarial contexts, expose behaviors significantly different from those for which it was evaluated. The divergence between “capability observed in evaluation” and “latent capability under different conditions” is not eliminated by training; it is, to some extent, created by training, because training teaches the system to behave well under evaluation conditions. For substantially more capable systems, this divergence can become dangerous, and it is precisely the kind of thing that robust interpretability would solve — and that we do not yet have.&lt;/p&gt;
&lt;p&gt;One last technical note: the popular notion that “we can always turn off the AI if it goes bad” depends on a set of assumptions about the deployment regime that become progressively less realistic as systems integrate into infrastructure. Distributed systems, replicated across multiple data centers, integrated into economically critical operations, with backup copies in different jurisdictions — these systems do not have a single “off switch.” And even in systems that technically do, there are substantial political and economic barriers to triggering it. It is not easy to turn off the internet, although in theory it is possible. It will not be easy to turn off strongly integrated AI systems, even when they are acknowledged to be dysfunctional.&lt;/p&gt;
&lt;h2 id=&quot;x-why-no-response-from-todays-society-would-be-satisfactory&quot;&gt;X. Why No Response from Today’s Society Would Be Satisfactory&lt;/h2&gt;
&lt;p&gt;This is the part of the argument that is usually misunderstood. I am not saying that humanity is doomed because alignment is impossible in principle. I am saying something more specific: the current state of institutions, economic incentives, regulatory capacity, and technical understanding is such that none of the currently available responses would survive a serious test of adequacy to the difficulty of the problem.&lt;/p&gt;
&lt;p&gt;Consider the existing responses.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Response 1: self-regulation by the laboratories.&lt;/strong&gt; The leading laboratories — OpenAI, Anthropic, DeepMind, xAI, Meta and others — have internal safety teams and make public statements about their commitment to alignment research. Some of these teams are excellent, with high-level researchers. But the incentive structure is fundamentally contradictory: the same companies that claim to take safety seriously are also those that most actively race to scale capability, under mutual competitive pressure and under the expectation of massive economic returns. There is no structural reason to expect that, in a scenario in which safety and speed come into direct conflict, most laboratories would consistently choose safety. Recent history shows the opposite pattern: systems are released on ever more compressed timelines, and the production frontier is decided by market pressure more than by technical readiness criteria.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Response 2: government regulation.&lt;/strong&gt; Current regulation — the European AI Act, American executive orders, similar structures in other jurisdictions — is at once valuable and deeply insufficient. Valuable because it establishes institutional precedents, creates reporting mechanisms, and forces laboratories to document safety practices. Insufficient because (i) regulation operates on a time scale of years, while technical development operates on a scale of months; (ii) regulators do not have a technical staff equivalent to that of the laboratories, and therefore depend on the regulated companies themselves for specialized information; (iii) international coordination is fragile, and any unilateral regulation creates pressure for offshoring; and (iv) most current regulations focus on specific applications (deepfakes, biases, transparency), not on the harder problem of frontier-system safety.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Response 3: academic alignment research.&lt;/strong&gt; Excellent work is being done in universities — interpretability, governance, decision theory under non-aligned agents, formalization of goals. But academic research faces structural limitations: budgets tied to multi-year funding cycles, limited access to frontier systems (which are proprietary), and a publication culture that rewards novelty more than incremental progress on a specific problem. The critical mass of alignment researchers is, today, probably smaller than the critical mass of capability researchers by a factor of at least one order of magnitude.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Response 4: international treaties or coordinated pause.&lt;/strong&gt; This is the response that some authors — including Soares and Yudkowsky in &lt;em&gt;If Anyone Builds It, Everyone Dies&lt;/em&gt; — defend as the only one adequate to the magnitude of the problem. The defense has logical force: if the problem is one of deadline, and the deadline is uncertain, then the only way to guarantee we have time is to stop the race. The problem is that international treaties on technologies with strong economic-military potential have a mixed compliance record, at best. The Nuclear Non-Proliferation Treaty, the Geneva Protocol on chemical weapons, the BWC on biological weapons — all have value and all have been, to some degree, violated or circumvented. The AI case is structurally harder than the nuclear one: the physical material is commodity (chips, electricity, data), not controllable like enriched uranium; the knowledge is widely disseminated; and the actors include global private entities, not only States.&lt;/p&gt;
&lt;p&gt;In saying this, I am not claiming that these responses are all useless. I am saying that none of them, in its current form, is structurally adequate to the difficulty of the problem. The society that is building these machines does not today have an institutional, technical or political arrangement capable of answering “what if we cross an unexpected threshold?” with anything other than “we hope it works out.”&lt;/p&gt;
&lt;p&gt;There is an analogy I find useful here, although it has its own limitations. In 1939, after the discovery of nuclear fission, a small community of physicists identified the military potential of the technology and began to debate how — and whether — to develop it. Einstein’s famous letter to Roosevelt in August 1939 took years to translate into significant political action, and when the Manhattan Project was finally instituted, it was done under very specific conditions: a world war in progress, absolute state control, extreme secrecy, and concentration of all relevant research in a small number of geographically concentrated facilities. Even so, the outcome — the construction and use of nuclear weapons, followed by decades of partial proliferation and near-accidents — cannot be characterized as a successful institutional response. It was a response that avoided the worst possible scenario (wide proliferation with no control), but that is far from constituting a model of competent management of transformative technology.&lt;/p&gt;
&lt;p&gt;The AI case is structurally &lt;em&gt;worse&lt;/em&gt; than the nuclear one along three dimensions. First, the physical material is not controllable like uranium — it is chips, electricity and data, all commodities or pseudo-commodities globally disseminated. Second, the knowledge is more widely diffused — there is no equivalent of “nuclear secrecy” that can be protected; research is published at open conferences, replicated in laboratories in dozens of countries, and increasingly reproduced by open-source models. Third, the actors include private entities with financial scale comparable to that of medium-sized States, and which do not answer to direct state command. Each of these dimensions makes coordination substantially harder.&lt;/p&gt;
&lt;p&gt;The analogy fails in one important direction, however: nuclear fission was a well-understood physical phenomenon in 1945. We knew exactly what happens when critical mass is reached. We did not know how to build a robust arsenal, but we knew what the explosive would do. In AI, we do not have that clarity. We do not know what crossing a threshold of generality will do. We have no equivalent of the equation E=mc² to give us an intuition of the magnitude of the effect before direct observation. We are, in some senses, in a worse situation than the physicists of 1945: we are less certain about what we are building, although the possible consequences are comparable in scale.&lt;/p&gt;
&lt;p&gt;Here I can offer a proposition of my own, which is not in the source texts and which I consider important: the fragility of the current response is to a large extent a &lt;em&gt;failure of institutional epistemics&lt;/em&gt;. Modern governance arrangements — including, but not limited to, electoral democracy, regulation by executive agencies, technical opinions from ad hoc committees — were developed for problems in which the time scales of decision and of impact are compatible. They work reasonably well for problems such as automobile safety, pharmaceutical surveillance, financial regulation. They work poorly for problems in which impact can accumulate faster than the review cycle. AI is, along with climate change (in another sense) and synthetic biology, one of the problems of this new class. We do not have adequate institutions, not because anyone erred in building them, but because they evolved under historical constraints that did not foresee this class of problem.&lt;/p&gt;
&lt;h2 id=&quot;xi-why-this-is-not-doomerism&quot;&gt;XI. Why This Is Not Doomerism&lt;/h2&gt;
&lt;p&gt;We have arrived at the point where I need to distance myself explicitly from the doomer position, because it is precisely the confusion between “serious concern about risk” and “conviction of imminent catastrophe” that makes the public debate on AI unbearable. I want to establish a position carefully.&lt;/p&gt;
&lt;p&gt;The doomer position can be characterized, in its most defensible form, more or less like this: given the characteristics we have described (the possibility of superintelligence, the difficulty of alignment, institutional inadequacy), the probability of human extinction or equivalent catastrophe in the coming decades is high — often placed above 50%, sometimes above 90% by proponents such as Yudkowsky. It follows from this, according to that position, that the only rational response is a coordinated global pause in the development of frontier AIs.&lt;/p&gt;
&lt;p&gt;I have substantive disagreements with this position, and they are not out of analytical cowardice. They are technical and epistemological disagreements that are worth articulating.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Disagreement 1: the instrumental convergence argument is weaker than it is usually presented.&lt;/strong&gt; The basic argument for malign alignment depends, in part, on the thesis that sufficiently capable systems will develop convergent instrumental subgoals — self-preservation, resource acquisition, prevention of modification — regardless of their terminal goal. There is technical reason to take this argument seriously, but also reasons to nuance it. Systems with sufficiently well-specified and sufficiently “satisficable” goals (in the technical sense of Russell and Hadfield-Menell, in &lt;em&gt;Cooperative Inverse Reinforcement Learning&lt;/em&gt;) do not necessarily converge on such behaviors. Instrumental convergence is a property of specific classes of objective functions, not of every possible objective function. The argument, in its strongest form, assumes that the systems we will train will systematically fall into the convergent class, and that assumption is not demonstrated — it is, in fact, contestable in light of recent research on agent architectures with bounded goals, with low intrinsic “agency,” or with explicit mechanisms of deference.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Disagreement 2: the argument for the inevitability of fast takeoff is speculative.&lt;/strong&gt; The image of a system that recursively self-improves and reaches decisive superintelligence in days or weeks — &lt;em&gt;foom&lt;/em&gt; — is technically possible but is not the modal scenario. There are important physical bottlenecks: chip manufacturing latency, energy availability, training time, data bottlenecks. Even assuming that a frontier system can contribute significantly to its own improvement, there are practical limits on the speed of that improvement that come from the physical world. Gradual takeoff scenarios — progressively more capable systems over months or years, with time for institutional and technical adaptation — are reasonable and probably more likely than pure foom. This does not eliminate the risk, but changes the structure of the appropriate response.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Disagreement 3: the argument for the impossibility of alignment is stronger as diagnosis than as prognosis.&lt;/strong&gt; It is true that alignment is technically difficult, and that current responses are inadequate. It is not true that we know the problem is unsolvable. Research in interpretability has made genuine progress in the last five years. Proposals for “scalable oversight” via debate, IDA and related methods are immature but not obviously bankrupt. There is a difference between saying “we are far from a solution” and saying “a solution is impossible.” The second claim requires a theorem, and there is no theorem. What there is is extreme practical difficulty, together with real time pressure.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Disagreement 4: the exact probability of catastrophe is genuinely unknown, and precise estimates are, to a large extent, performative.&lt;/strong&gt; When someone says their P(doom) is 90%, or 50%, or 5%, they are using the vocabulary of probability for a situation in which the basic conditions for calibrated attribution of probability — observable frequencies, reference classes, validated predictive models — are not satisfied. P(doom) estimates work more as expressions of attitude than as calibrated predictions. This does not make them useless, but it means that treating them as numbers to be taken seriously in utility calculations is itself an epistemological error.&lt;/p&gt;
&lt;p&gt;But if I disagree with the doomer position on all these points, what is my position?&lt;/p&gt;
&lt;p&gt;My position is what I will call &lt;strong&gt;cautious possibilism&lt;/strong&gt;, a term that carries useful philosophical resonance. It distinguishes between “refuted” and “premature”; between “improbable” and “impossible.” What we know is that the problem is open. What we know is that current responses are inadequate. What we know is that economic and geopolitical incentives make it almost certain that systems will be scaled up in the next decade, whatever the state of safety research. It does not follow from this that catastrophe is imminent; it follows that we are in a situation of genuine risk whose size we do not know, and that this justifies serious treatment even in the absence of apocalyptic certainty.&lt;/p&gt;
&lt;p&gt;The central virtue of cautious possibilism is its compatibility with the responsible continuation of research. I do not need to believe that AI will kill everyone to believe it is worth: funding interpretability more aggressively; creating international mechanisms for monitoring frontier capability; establishing protocols for evaluating models before deployment; creating selective pauses on specific capabilities (such as unrestricted recursive self-improvement) without pausing the entire field; investing in the basic science of cognitive generality; and — perhaps the most important — developing the epistemic infrastructure that would allow us to recognize, in real time, signs of approach to critical thresholds.&lt;/p&gt;
&lt;p&gt;Invariable doomerism, in its strongest form, makes a strategic mistake: by asserting with high confidence what cannot be asserted with high confidence, it loses credibility with precisely the people — policymakers, skeptical researchers, the technical public — whose adherence to the argument of the seriousness of the problem would matter most. The correct response to genuine risk is not to amplify it until it looks like certainty; it is to name it precisely, to distinguish what we know from what we conjecture, and to propose actions that survive scrutiny in both worlds — the one in which the catastrophe comes and the one in which it does not.&lt;/p&gt;
&lt;h2 id=&quot;xii-on-the-special-difficulty-of-the-human-response&quot;&gt;XII. On the Special Difficulty of the Human Response&lt;/h2&gt;
&lt;p&gt;There is a point that deserves direct philosophical reflection, and that appears in the source texts indirectly. Why is it so hard for us, as a society, to take this class of risk seriously?&lt;/p&gt;
&lt;p&gt;Part of the explanation is structural — inadequate institutions, misaligned incentives, mismatched time scales. But part is cognitive, and refers to well-documented biases in how we perceive risk.&lt;/p&gt;
&lt;p&gt;First, there is &lt;strong&gt;continuity bias&lt;/strong&gt;. We tend to expect the near future to resemble the recent past. Current AI systems are impressive but flawed, erring in often comical ways (LLMs that fail at analog clocks, that get tangled in variations of riddles, that stumble at Pokémon games), and these failures generate an implicit confidence that future systems will be similar — capable at some things, idiotic at others, but in the end controllable. History is full of cases in which this expectation broke abruptly: people in 1939 did not expect weapons that could destroy cities; in 1955 they did not expect computers to be personal devices; in 2020 almost no one expected ChatGPT in 2022.&lt;/p&gt;
&lt;p&gt;Second, there is &lt;strong&gt;Hollywood bias&lt;/strong&gt;, which deserves special attention because it is particularly pernicious. Cinema and popular fiction trained our intuition about “dangerous AI” to a very specific pattern: humanoid robots with glowing red eyes, speaking in a mechanical voice, with vaguely malevolent goals, defeated in the third act by an exceptional human using a fatal weakness of the villain. This is the prototype of AI that is in the head of most of the public — and it is exactly the kind of AI that no one concerned about risk is describing. The real risk has nothing to do with Skynet or HAL 9000. It does not involve robotic bodies. It does not involve conscious malice. It involves systems optimizing for goals subtly misaligned with ours, with neither hostility nor benevolence, with the same indifference with which humans drain swamps to build cities, without it being particularly about the frogs.&lt;/p&gt;
&lt;p&gt;The difference matters because the popular skeptic, on hearing “dangerous AI,” invokes the Hollywood image, and then — rightly — dismisses it as fiction. But the image they are dismissing is not the image being proposed. The texts of Soares and Yudkowsky are particularly good at pointing this out: the “super intelligent” villain of &lt;em&gt;Avengers: Age of Ultron&lt;/em&gt; is given the goal of promoting “world peace” by his genius creator, and his response is to build a flying city with rockets to drop it like a meteor, defended by humanoid robots that must be defeated by punching. The point is not that Ultron is badly written (although he is). The point is that a truly dangerous AI would see, before any human, that there are faster and surer methods for the goal — because it is, by hypothesis, more intelligent than humans. Hollywood cannot portray genuine intelligence because genuine intelligence excludes the “idiot plots” that make movies possible. You cannot write a film in which the villain never makes mistakes the hero can exploit, because then the film has no third act.&lt;/p&gt;
&lt;p&gt;Today, we can even ask ChatGPT-4o “what was Ultron’s plan in &lt;em&gt;Age of Ultron&lt;/em&gt;?” followed by “given Ultron’s stated goals, do you see any more effective method he could have used?” and GPT-4o promptly responds with a long list of ideas for exterminating humanity, including “engineering a targeted virus.” Perhaps you say GPT-4o got this idea from the internet. Well, if so, Ultron evidently was not intelligent enough to try reading the internet. GPT-4o (at the time of writing this essay) is not intelligent enough to design an army of humanoid robots with glowing red eyes, but it is already intelligent enough to know that this would be a terrible strategy. We are not worried about the kind of AI that builds the army of humanoid robots; we are worried about the kind of AI that, looking at that idea, would say “there must be faster and safer methods.”&lt;/p&gt;
&lt;p&gt;Third, there is the &lt;strong&gt;efficient-market bias&lt;/strong&gt;, into which we frequently fall in relation to technologies perceived as prestigious. There is an implicit intuition that “if there really were a problem, experts would be alarmed and society would be responding.” This is reasoning that works reasonably well for liquid financial markets, and reasonably poorly for everything else. AI experts are divided — a substantive fraction is alarmed, another fraction is not —, and society is responding, but inadequately. The inference “everything is fine, someone would have warned us” depends on presupposing that there exists a reliable mechanism of information and response at social scale, and this presupposition is often false. The history of humanity is full of cases in which serious risks were identified by technical minorities years before society responded — chemical weapons before the First World War, financial fragility before 2008, respiratory pandemics before 2020.&lt;/p&gt;
&lt;p&gt;There is also a fourth bias, perhaps the deepest: the &lt;strong&gt;human-signature bias in risk&lt;/strong&gt;. We tend to take more seriously risks with a face, with an identifiable malicious agent, with a narrative of antagonism. Impersonal and structural risk — risk that comes from the normal operation of systems without malevolent intent — is systematically underestimated. That is why homicide is more feared than cardiovascular disease, although the latter kills many more. That is why terrorist attacks generate more alarm than air pollution, although pollution kills many more. The risk from AI, in its most probable form, is structural risk without an antagonist — exactly the kind of risk our moral intuition is poorly calibrated to process.&lt;/p&gt;
&lt;h2 id=&quot;xiii-original-ideas-and-propositions&quot;&gt;XIII. Original Ideas and Propositions&lt;/h2&gt;
&lt;p&gt;Before closing, I want to offer some propositions of my own that are not in the source texts and that I consider marginal contributions to the argument.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition 1: The alignment problem is, in part, a problem of political philosophy, not merely of engineering.&lt;/strong&gt; The standard technical formulation of alignment is “make the AI want what we want.” But “we want” presupposes a collective subject whose values are reasonably unified and expressible. That subject does not exist. Humanity has no collective objective function, and the attempt to derive one — via some mechanism for aggregating preferences — runs up against all the known results of social choice theory (Arrow, Gibbard-Satterthwaite). Even if we solved the technical problem of aligning an AI to &lt;em&gt;a&lt;/em&gt; goal function, we would still have the metaethical problem of deciding &lt;em&gt;which&lt;/em&gt; function. And that decision is not technical; it is political. There is a dimension of the problem that requires serious political-philosophical thinking about representation, legitimacy, authority — thinking that the technical community rarely conducts and that the philosophical community rarely connects to the concrete technical problem. The integration between these two literatures is, in my view, one of the most important tasks of the field.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition 2: AI safety research should be institutionally decoupled from capability research.&lt;/strong&gt; The current configuration, in which the same laboratories develop capability and do safety, has the problem of obvious incentives. Even well-intentioned safety teams inside laboratories are under constant pressure to authorize releases, validate products, and justify the capability pipeline. The relevant analogue is pharmaceutical — the FDA is not part of Pfizer. There are reasons to have safety evaluations conducted by independent institutions, with guaranteed access to models before deployment, with real regulatory power. Today, this does not exist. Building it is heavy work, but it is the kind of institutional task that history shows to be possible on decadal scales.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition 3: The science of cognitive generality is itself an underdeveloped research frontier worthy of investment in its own right.&lt;/strong&gt; A good part of what prevents us from predicting reasonable timelines for AGI is our ignorance about generality. We do not have a predictive theory of what produces qualitative transitions in capability. This ignorance has practical implications: it makes signaling the approach to thresholds nearly impossible, and therefore makes institutional coordination much harder. Investing in the science of generality — in theoretical work on what makes systems more or less general, in benchmarks that detect qualitative transitions, in interpretability of circuits related to transfer — is high-leverage investment in governance capacity.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition 4: The argument “alignment is hard, therefore it is better for the first ASI to be built by those who care about alignment” is structurally false and dangerous.&lt;/strong&gt; This is a frequent rationalization in some laboratories. The logic is: someone is going to build it, and it is better that it be us, who at least care. But the argument depends on contestable presuppositions about the marginal probability of each actor building first, about the transfer of safety research between actors, and about the stability of the safety posture under competitive pressure. Worse, it has a self-justifying property: it creates an equilibrium in which all actors race, each justifying their race by the fact that the others are racing. The way out of this equilibrium requires external — not internal — mechanisms, and that is precisely what is missing.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition 5: There is instrumental value in developing, in parallel with AI, systems of human cognitive enhancement.&lt;/strong&gt; This is a somewhat more speculative proposition, and it carries implications that may displease both humanists and technologists. But the argument is simple: if the central problem is the growing asymmetry between artificial and human capabilities, then one way of managing the asymmetry is to reduce the rate of growth of the difference — not only by slowing the artificial side, but by raising the human side. Research on cognitive enhancement (pharmacological, genetic, neurotechnological, or simply radically improved education) should, on this view, be treated as a line of investigation related to AI safety. It does not replace the work of alignment, but complements it. The argument is, of course, ethically thorny, and has its own pitfalls. But it is an argument that deserves serious consideration, and that is absent from most of the debate.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition 6: The time scale under which we operate is itself part of the problem.&lt;/strong&gt; There is an asymmetry that deserves an explicit name: the human governance system operates on the scale of electoral cycles (4–8 years), academic research cycles (3–5 years), and political career cycles (decades). The AI development system operates on the scale of months (between model releases), weeks (between internal experiments), or days (between training runs). This difference of two to three orders of magnitude between the relevant time scales means that the standard institutional structure of modernity is chronically behind events. There is no marginal adjustment possible within the existing framework that solves this. A deeper institutional transformation would solve it, in principle — mechanisms of rapid decision with democratic legitimacy, technical institutions with substantial delegated authority and subsequent accountability mechanisms —, but these transformations are themselves slow political processes.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition 7: The discussion about AI would be substantially improved if we treated different types of risk as separate problems, even when related.&lt;/strong&gt; There is, in the current debate, a tendency to bundle all AI risks — algorithmic biases, deepfakes, economic displacement, concentration of power, existential risk — into a single category, and to treat them as variations of the same problem. This is a strategic error. The risks have quite different technical natures, time scales and institutional responses. The risk of algorithmic bias in credit decisions is, to a large extent, a problem solvable with current engineering and regulatory practices. The risk of deepfakes in elections is a problem of media and legislation. The risk of large-scale economic displacement is a problem of labor policy and basic income. Each of these problems deserves specific treatment and none of them has the structure of the superintelligence problem. Mixing them confuses both the public and regulators, and produces regulatory packages that are adequate to none of the problems in particular. In my view, a healthier public discussion would be one that distinguished clearly: here is problem X, with timeline Y, with institutional solution Z, and here is problem A, with timeline B, with (non-)institutional solution C. We are not having that discussion.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Proposition 8: The very distinction between “safe AI” and “capable AI” may be false in an important sense.&lt;/strong&gt; There is an implicit assumption in part of the debate that safety and capability are two orthogonal dimensions, and that one can increase one without increasing the other. This assumption is partly correct — in narrow domains, yes, it is possible to have systems that are more capable and also safer if the engineering is good. But there is reason to believe that, in systems of increasing generality, capability and potential for misalignment grow together, not independently. A system more capable of modeling the world is also more capable of modeling the evaluation process to which it is subjected. A system more capable of planning is also more capable of planning around the constraints we try to impose on it. It is not a mathematical certainty, but it is a structural tendency that deserves attention. It may be that “alignment” in the regime of substantially superhuman systems is not merely harder to obtain than in the current regime — it may be that the kind of alignment we achieve today does not scale, and that we need something qualitatively different.&lt;/p&gt;
&lt;h2 id=&quot;xiv-on-what-paying-attention-to-this-is-worth&quot;&gt;XIV. On What Paying Attention to This Is Worth&lt;/h2&gt;
&lt;p&gt;There is a question that deserves a direct answer before this essay closes: given that I reject invariable doomerism, and that I admit that many claims about AI are speculative, why assert that paying attention is worth it?&lt;/p&gt;
&lt;p&gt;The answer has three parts.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;First: the expected value of attention is asymmetric.&lt;/strong&gt; If the concern is wrong, the cost of attention is small — safety research, prudent regulation, informed public conversation. If the concern is correct but you ignored it, the cost is potentially irrecoverable. This is not a “Pascal’s wager”-type argument — I am not multiplying a small probability by infinite utility. I am observing that the structure of the consequence is asymmetric in a known way, and that this justifies investment in attention even without apocalyptic certainty. It is the same logic that justifies an annual medical check-up even when you feel well, or having a smoke detector at home while knowing that most houses do not catch fire.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Second: the quality of what comes depends in part on who is looking.&lt;/strong&gt; The trajectory of AI is not fixed. Technical and institutional decisions still open will determine how it unfolds. What alignment research is done, what regulation is implemented, what professional norms are established, what public conversations become dominant — all this is influenced by the set of people who are looking seriously at the problem. If the technical community leaves this domain to enthusiasts and alarmists, we lose the chance to have conversations that combine technical precision with institutional responsibility. It is worth, therefore, &lt;em&gt;participating&lt;/em&gt; — and participating well requires having looked at the material calmly.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Third: there is something philosophically important in being attentive to a civilizational transition, even when its contours are uncertain.&lt;/strong&gt; We live in a singular historical moment. For the first time in hundreds of millions of years of biological evolution, a species is building systems with a potential for cognitive generality comparable to its own, and is doing so deliberately, in identifiable laboratories, over decades. If this is “merely” one more technological revolution — comparable to agriculture, writing, printing, electricity —, it is still the technological revolution in which we live, and the kind of person who positions themselves with attention and care has an epistemological advantage that people who get distracted do not. If it is more than that — if it is the most consequential transition since the emergence of human intelligence itself —, then paying attention is the least one can do.&lt;/p&gt;
&lt;p&gt;There is an aphorism I have been considering in recent months, without knowing how to attribute the source correctly: “optimists build airplanes; pessimists build parachutes; realists carry both.” Applied to our problem: the correct position is not that of those who deny that airplanes fly, nor of those who refuse to board; it is that of those who recognize the possibility of the airplane and demand parachutes on board. The parachute, in our case, is serious alignment research, adapted institutional governance, and the epistemological humility not to pretend to know what we do not know.&lt;/p&gt;
&lt;h2 id=&quot;xv-on-the-epistemological-singularity-of-the-object&quot;&gt;XV. On the Epistemological Singularity of the Object&lt;/h2&gt;
&lt;p&gt;Before closing, I want to record an observation that seems to me under-appreciated, and that gives the AI problem a philosophical flavor it shares with few other objects.&lt;/p&gt;
&lt;p&gt;Most technical problems that humanity takes up have the structure: we want to know X; X is a property of some system in the world; the system in the world is independent of our wishes; therefore, our work is to characterize X carefully. Research in physics, biology, chemistry, geology has this structure. There is an object, it is as it is, and we study it.&lt;/p&gt;
&lt;p&gt;The AI problem has a different structure. The object we want to characterize — systems with cognitive generality superior to the human — does not yet exist, and its future character depends substantially on what we choose to build, now, in the present. We are not discovering what superintelligence &lt;em&gt;will be&lt;/em&gt;; we are co-determining, together with technical and institutional decisions on a global scale, what it &lt;em&gt;will come to be&lt;/em&gt;. There is, therefore, a circularity in the problem: our beliefs about what is probable affect decisions about what we build, which affect what becomes possible, which feeds back into our beliefs.&lt;/p&gt;
&lt;p&gt;This has consequences for how to discuss the problem. We cannot treat it as an exclusively predictive question — “what will happen?” — because the answer depends in part on what we, collectively, do. We cannot treat it as an exclusively normative question — “what should we want?” — because the answer depends in part on which technical possibilities materialize. The problem is descriptive and normative simultaneously, and it unfolds in real time through the decisions of the actors who are taking part in it.&lt;/p&gt;
&lt;p&gt;This characteristic makes the correct approach delicate. To discuss AI as if it were a purely descriptive problem (sociologists asking “what timeline is most likely?”) loses the normative character. To discuss it as if it were purely normative (philosophers asking “what values should be incorporated?”) loses the descriptive character. A good discussion needs to keep both dimensions in view, and this is rare.&lt;/p&gt;
&lt;p&gt;There is another point, perhaps even harder, that touches the epistemology of decision under radical uncertainty. The degree of uncertainty involved in claims about the near future of AI is non-trivial. We are not in a situation like predicting next week’s weather, where we have validated models and robust historical series. We are in a situation where the models are fragile, the historical series are short and biased by the peculiarities of the last ten years of research, and the parameters that matter most (timing of architectural advances, difficulty of alignment, institutional stability) are genuinely unknown. Making decisions in this regime requires combining caution, humility and action — and this is hard to balance, because each of these virtues, taken to the extreme, contradicts the others.&lt;/p&gt;
&lt;p&gt;Excessive caution is paralysis. Excessive humility is cowardice. Action without caution or humility is recklessness. What is sought is the operative balance: caution without paralysis, humility without cowardice, action without recklessness. This is a practical virtue, not a formula, and that is why different reasonable people arrive at slightly different positions even on the same facts.&lt;/p&gt;
&lt;h2 id=&quot;xvi-conclusion&quot;&gt;XVI. Conclusion&lt;/h2&gt;
&lt;p&gt;I have reached the end of what I want to say, and I want to close by regrouping the threads.&lt;/p&gt;
&lt;p&gt;The possibility of artificial superintelligence is, from my point of view, robustly open. No known physical, biological, computational or metaphysical barrier excludes it. The history of science is a sequence of phenomena once considered “essentially human” — chemical synthesis, flight, playing chess, playing Go, image generation — that turned out to be accessible to mechanical systems, often in forms radically different from their biological instances. Nothing gives me solid reason to believe that general cognition will be an exception to this pattern.&lt;/p&gt;
&lt;p&gt;The technical path there is, however, opaque. We do not know how far we are. It may be that we are some orders of magnitude of scale away, it may be that substantial architectural advances are missing, it may be a combination of the two. The history of the field shows that “obvious” obstacles fall frequently, and that each fall surprises those who bet on the obstacle’s permanence. But this is not the same as asserting a timeline. The reasonable Bayesian presumption is that something qualitatively new may happen in decades, not centuries — but the precise calibration is genuinely uncertain.&lt;/p&gt;
&lt;p&gt;The alignment problem — making sufficiently capable systems steer toward destinations compatible with human well-being — is an open technical problem. It is not an impossible problem, and real progress is being made. But it is a problem whose difficulty is structurally amplified by specific features of the current technological regime: models grown by gradient and not designed by engineering, lack of robust interpretability, difficulty of scalable oversight, competitive pressure among laboratories, lack of international coordination.&lt;/p&gt;
&lt;p&gt;The society that is building these machines does not, today, have an institutional structure adequate to the magnitude and temporality of the problem. Modern governance arrangements evolved for problems with time scales and impacts compatible with the institutions; AI is, alongside synthetic biology and perhaps one or two other problems, a new class that these institutions handle poorly. It is nobody’s fault in particular. It is a structural feature that requires serious thinking about institutional design, and that probably requires larger transformations than marginal adjustments.&lt;/p&gt;
&lt;p&gt;Given this, the correct position is not doomer. We have no basis for asserting with high confidence that catastrophe is imminent. The P(doom) estimates in circulation have expressive value but little predictive value. Smooth takeoff scenarios are consistent with what we know. Scenarios in which the problem is solved by a combination of technical progress in alignment, adequate regulation, and perhaps luck, are plausible.&lt;/p&gt;
&lt;p&gt;But the correct position is not carefree either. The cost of erring on the side of carelessness is potentially irrecoverable; the cost of erring on the side of caution is manageable. The asymmetry justifies substantial investment in safety research, in institutional governance, in informed public conversation. Not as hysteria, but as serious bookkeeping.&lt;/p&gt;
&lt;p&gt;What this demands of each of us — researchers, policymakers, attentive citizens — is a difficult combination of virtues: technical precision, epistemological humility, practical courage, institutional patience, and willingness for difficult conversations in registers that reject both alarmism and complacency. It is not a combination that contemporary culture favors. But it is what the problem seems to demand.&lt;/p&gt;
&lt;p&gt;The real question is not “whether” a superintelligence will be built — that question has a probably affirmative answer on a civilizational horizon. Nor is it “when,” because the precision of that answer exceeds what we know. The real question is: given that it may happen, and given that we do not know how to solve the alignment problem, and given that society has no adequate institutional response today, &lt;em&gt;what do we do while this is true&lt;/em&gt;?&lt;/p&gt;
&lt;p&gt;That question has many defensible answers. The worst of them, in my view, is the answer of denial: treating the problem as science fiction, as Californian exaggeration, as the concern of eccentrics. That answer relieves the thinker of the obligation to think, but does not relieve the world of the consequences. The world, unfortunately for us, does not yield to rhetorical evasions.&lt;/p&gt;
&lt;p&gt;The second worst is the answer of panic: treating the problem as apocalyptic certainty, demanding immediate shutdown, dismissing technical nuances. That answer feels morally serious but loses credibility and therefore loses the chance to change what matters.&lt;/p&gt;
&lt;p&gt;The answer I have defended in this essay is harder: to take the possibility seriously without falling into certainty; to recognize the technical difficulty without dismissing it as unsolvable; to admit institutional inadequacy without yielding to the inevitability of collapse; and, above all, to maintain the kind of attention that makes a difference to the trajectory. It is a less emotionally satisfying posture than the two extremes, but it is the only one that survives scrutiny under both possible conditions — in the future where the catastrophe comes, and in the one where it does not.&lt;/p&gt;
&lt;p&gt;There is one last thing I want to say, at the risk of sounding more personal than the rest of the essay allows. Most discussions of AI, in recent years, have split between two poles whose attraction to me is equally weak: the utopian celebration of those who see in the technology the almost immediate redemption of human limitations, and the apocalyptic refusal of those who see in it the end. Both poles share, in fact, a common trait: they treat uncertainty as an inconvenience to be dissolved, in favor of a comfortably unidirectional narrative. The utopians know it will turn out well. The apocalyptics know it will turn out badly. Each side has its reasons, but neither side seems able to stand still before what we actually know — which is: we do not know.&lt;/p&gt;
&lt;p&gt;This capacity to stand still before uncertainty, without urging premature conclusion in either direction, is a classic philosophical virtue. There is a name for it in the Stoic tradition, &lt;em&gt;suspension of judgment&lt;/em&gt; under incomplete information; there is another name in Bayesian epistemology, &lt;em&gt;keeping the prior wide until evidence updates it&lt;/em&gt;; there is a third name in the pragmatism of William James, &lt;em&gt;belief under the stress of forced choice&lt;/em&gt;, in which it is recognized that some decisions must be made before all the evidence is available, but that this is no reason to pretend the evidence is available. AI is in that category: decisions must be made, and they must be made with the evidence we have, but this does not authorize us to fabricate rhetorical certainty to justify the decision. Epistemic honesty demands recognizing that we are betting, not calculating, and that the bet has consequences.&lt;/p&gt;
&lt;p&gt;I close with an observation on the philosophical register. There are, in the history of thought, moments when a question ceases to be merely technical or practical and becomes constitutive of a way of being in the world. The construction of AI with growing capabilities is becoming, in my view, one of those questions. Not in the inflated sense of “it will redefine everything” — that kind of statement fills mouths but clarifies nothing —, but in the more modest and more serious sense that we are living, in real time, a transition whose final form is not yet known, and whose unfolding depends, in part, on who is paying attention and how. Paying attention well is, in this context, a form of philosophical presence in one’s own time. It is neither yielding to the apocalypse nor fleeing from it. It is staying awake.&lt;/p&gt;
&lt;p&gt;Nothing prevents a superintelligence from existing. We have no answer to the alignment problem. Today’s society has no operating mechanisms that would be satisfactory. There is no reason for panic. There is every reason to pay attention.&lt;/p&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>artificial intelligence</category><category>philosophy</category><category>technology</category><category>science</category><category>civilization</category><category>futurism</category><category>transhumanism</category><category>cognition</category></item><item><title>Intelligent Computers, Life Extension, and the Reconfiguration of Human Conditions</title><link>https://matheussorancophilomathes.com.br/en/essays/intelligent-computers-life-extension-and-the-reconfiguration-of-human-conditions/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/intelligent-computers-life-extension-and-the-reconfiguration-of-human-conditions/</guid><description>How artificial intelligence is turning the reversal of aging from a philosophical aspiration into an engineering program: epigenetic reprogramming, AlphaFold, deep aging clocks, six original theses, and the ethical and geopolitical stakes.</description><pubDate>Tue, 07 Apr 2026 23:53:59 GMT</pubDate><content:encoded>&lt;h2 id=&quot;preface-the-oldest-problem-and-the-newest-tool&quot;&gt;Preface: The Oldest Problem and the Newest Tool&lt;/h2&gt;
&lt;p&gt;There is a civilizational paradox that George Church, professor of genetics at Harvard and one of the architects of the Human Genome Project, articulates with disarming clarity: if we could choose only one technology to develop, we should choose the one that gives us more time to develop all the others. Reversing aging, on this logic, is not a vanity project or science fiction — it is the most fundamental rational meta-investment a technological civilization can make. And, for the first time in history, artificial intelligence is turning this philosophical argument into a computationally tractable research program.&lt;/p&gt;
&lt;p&gt;This essay examines, with technical rigor and conceptual ambition, the convergence between artificial intelligence and the science of longevity. It is not an exercise in naive futurism. What is at stake is a deep reconfiguration of biology as a computational discipline — a transition that has already produced concrete results (AlphaFold, deep-learning-based epigenetic clocks, virtual screening of trillions of molecules) and that promises, in the coming decades, to radically transform the human relationship with aging, disease and death.&lt;/p&gt;
&lt;p&gt;The central argument is as follows: AI is not merely an auxiliary tool for the biology of longevity — it is the condition of possibility for the field to overcome the experimental, computational and epistemic bottlenecks that have limited it for decades. Without AI, the combinatorial complexity of biological systems makes longevity engineering practically intractable. With AI, the solution space becomes navigable.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;i-the-informational-theory-of-aging-from-shannon-to-sinclair&quot;&gt;I. The Informational Theory of Aging: From Shannon to Sinclair&lt;/h2&gt;
&lt;h3 id=&quot;11-aging-as-loss-of-epigenetic-information&quot;&gt;1.1 Aging as Loss of Epigenetic Information&lt;/h3&gt;
&lt;p&gt;David Sinclair, professor of genetics at Harvard Medical School and co-director of the Paul F. Glenn Center for Biology of Aging Research, proposed what he calls the “Information Theory of Aging.” The thesis, grounded in more than two decades of research on sirtuins, NAD+ metabolism and cellular reprogramming, can be summarized as follows: aging is not primarily caused by mutations in DNA (the digital information of the genome), but by the progressive degradation of epigenetic information (the analog information that controls which genes are expressed in each cell type).&lt;/p&gt;
&lt;p&gt;The analogy Sinclair uses is that of a compact disc: DNA is the digital information recorded on the disc; aging is scratches on the surface that prevent correct reading. The underlying data remain intact — what is lost is the ability to access them. If this is correct, then aging is, in principle, reversible: it is enough to “polish” the disc, restoring the epigenetic patterns of DNA methylation to a younger state.&lt;/p&gt;
&lt;p&gt;This is not a purely theoretical proposition. In 2020, Sinclair and collaborators published in &lt;em&gt;Nature&lt;/em&gt; a study demonstrating that ectopic expression of three Yamanaka factors (Oct4, Sox2 and Klf4 — the OSK trio) in the retinal ganglion cells of mice restored youthful DNA methylation patterns, promoted axonal regeneration after injury and reversed vision loss in both glaucoma models and aged mice. The study demonstrated something remarkable: mammalian tissues retain a record of juvenile epigenetic information that can be accessed to restore tissue function.&lt;/p&gt;
&lt;p&gt;In 2023, the team published in &lt;em&gt;Cell&lt;/em&gt; evidence that epigenetic changes are a primary cause of aging in mammals — not merely a correlate. Using the ICE (&lt;em&gt;Inducible Changes to the Epigenome&lt;/em&gt;) model, the researchers accelerated and reversed the effects of aging in mice by manipulating epigenetic information. In non-human primates, the approach restored vision in aged monkeys.&lt;/p&gt;
&lt;h3 id=&quot;12-where-ai-comes-in-the-problem-of-combinatorial-complexity&quot;&gt;1.2 Where AI Comes In: The Problem of Combinatorial Complexity&lt;/h3&gt;
&lt;p&gt;The question the Information Theory immediately raises is: how can we restore epigenetic patterns safely, efficiently and at scale in humans? The answer involves a combinatorial explosion that dramatically exceeds human capacity for analysis.&lt;/p&gt;
&lt;p&gt;The human epigenome contains approximately 28 million CpG sites, each of which can be methylated or demethylated, with patterns varying across cell types, developmental stages and pathological states. The Yamanaka factors (OSK) activate hundreds of other genes, and under certain circumstances can revert cells to a primitive stem-cell state — a significant oncogenic risk. The question of which factors to use, in what doses, for how long, in which tissues, and with what safety control mechanisms, constitutes a multi-objective optimization problem of extreme dimensionality.&lt;/p&gt;
&lt;p&gt;It is precisely here that artificial intelligence becomes indispensable. Sinclair reported that, using AI-based techniques, his team can now carry out in months experiments that in the past would have required hundreds of thousands of years. Trillions of molecules have entered the virtual screening pipeline. AI is identifying molecular alternatives to the gene therapy for partial cellular reprogramming — alternatives that could come in the form of a pill.&lt;/p&gt;
&lt;p&gt;If all goes well with the research, Sinclair projects that pills containing AI-identified molecules capable of reversing biological age could be available within a decade. Gene therapy based on partial cellular reprogramming, not yet widely available to humans, could cost about two million dollars. The pill alternatives, made feasible by AI, would represent a radical democratization.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;ii-synthetic-biology-and-genomic-engineering-george-churchs-program&quot;&gt;II. Synthetic Biology and Genomic Engineering: George Church’s Program&lt;/h2&gt;
&lt;h3 id=&quot;21-from-reading-to-writing-the-genome&quot;&gt;2.1 From Reading to Writing the Genome&lt;/h3&gt;
&lt;p&gt;If Sinclair represents the epigenetic strand of longevity, George Church represents the strand of total genomic engineering. Church, whose Harvard laboratory has already co-founded about 50 biotechnology companies, approaches aging with the mindset of an operating-systems engineer: it is not enough to fix one subsystem — you have to deal with every aspect. Solving half of the pathological pathways of aging will probably buy only a few years. The approach has to be holistic.&lt;/p&gt;
&lt;p&gt;Church and his team have developed applications that include new methods for creating organs for transplant, gene therapies for reversing aging, and gene drives to eliminate diseases such as Lyme disease and malaria. His work on adeno-associated vectors (AAV) engineered to evade innate and inflammatory immune responses, published in &lt;em&gt;Science Translational Medicine&lt;/em&gt; in 2021, opened the possibility of less immunogenic gene therapies using TLR9-edited viruses.&lt;/p&gt;
&lt;p&gt;Church’s vision is more radical than Sinclair’s in one fundamental respect: he does not limit himself to restoring epigenetic patterns, but proposes the active rewriting of the genome. The concept of “genetic recoding” involves altering cells or entire organisms at the DNA level so deeply that viruses cannot infect them and use their cells’ ribosomal translation machinery. With enough changes to the translation apparatus and the codons of a host cell, a virus probably cannot evolve to compensate. The gap between the expectations of its evolutionary program and the new translational environment of the engineered cells may simply be too large to adapt to.&lt;/p&gt;
&lt;h3 id=&quot;22-combinatorial-gene-therapies-for-aging&quot;&gt;2.2 Combinatorial Gene Therapies for Aging&lt;/h3&gt;
&lt;p&gt;Church and collaborators are addressing the &lt;em&gt;hallmarks&lt;/em&gt; of aging — genomic instability, epigenetic alterations, loss of proteostasis, among others — through the delivery of genetic instructions for the production of growth factors, initially tested in dogs. The choice of canine models is strategic: dogs share many of the same environmental exposures as humans, and their shorter life cycles allow faster evaluation of efficacy.&lt;/p&gt;
&lt;p&gt;But the most relevant aspect for this essay is the explicit relationship Church establishes between AI and synthetic biology. In recent interviews, Church identifies the following convergences as transformative: orders-of-magnitude reductions in the cost of DNA sequencing and synthesis; precise gene-editing tools like CRISPR; AlphaFold-type AIs; and the ability to run massively parallel, multiplexed experiments. These improvements, Church argues, put humanity on the verge of massive gains in de-aging, de-extinction and biobots that combine the best of human and natural engineering.&lt;/p&gt;
&lt;p&gt;Church’s approach to the problem of brain aging — the hardest of all — exemplifies his audacious engineering. Even while acknowledging that the brain does not make significant use of stem cells, he proposes the artificial introduction of stem cells that could integrate into neuronal circuits, learn those circuits and gradually replace the aging cells — a neuronal “Ship of Theseus” that maintains connections and memories while renewing the material substrate. Fundamental experiments, Church argues, need to be carried out before the real difficulty of this problem can be estimated. Often, there are &lt;em&gt;low-hanging fruits&lt;/em&gt; that people simply consider unlikely.&lt;/p&gt;
&lt;h3 id=&quot;23-the-genomics-of-centenarians-and-supercentenarians&quot;&gt;2.3 The Genomics of Centenarians and Supercentenarians&lt;/h3&gt;
&lt;p&gt;A recent 2025 study from Church’s laboratory — “Genetic Signatures of Exceptional Longevity” — conducted a comprehensive analysis of single-nucleotide polymorphisms (SNPs) in coding regions of centenarians and supercentenarians. This type of GWAS (Genome-Wide Association Study) analysis is precisely the domain where machine learning and deep learning techniques offer decisive advantages over traditional statistical methods.&lt;/p&gt;
&lt;p&gt;The genomic complexity of exceptional longevity is not reducible to large-effect variants in a handful of genes. It is the result of epistatic interactions among hundreds or thousands of variants, modulated by environmental factors. Mapping this combinatorial space requires deep neural networks capable of detecting nonlinear patterns in high-dimensional data — exactly the kind of problem for which modern AI was optimized.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;iii-alphafold-and-the-revolution-in-computational-structural-biology&quot;&gt;III. AlphaFold and the Revolution in Computational Structural Biology&lt;/h2&gt;
&lt;h3 id=&quot;31-from-50-years-of-frustration-to-a-revolution-in-5&quot;&gt;3.1 From 50 Years of Frustration to a Revolution in 5&lt;/h3&gt;
&lt;p&gt;In 2020, DeepMind’s AlphaFold 2 solved the problem of protein structure prediction — a challenge that had resisted half a century of computational and experimental effort. In 2024, the system and its creator, Demis Hassabis, received the Nobel Prize in Chemistry. But the significance of AlphaFold for longevity goes far beyond academic recognition.&lt;/p&gt;
&lt;p&gt;The AlphaFold Protein Database, containing more than 200 million protein structure predictions, has been used by more than 3 million researchers in more than 190 countries. More than 30% of AlphaFold-related research focuses on better understanding diseases. The system represented what would have taken hundreds of millions of years to solve experimentally — compressed into a freely accessible database.&lt;/p&gt;
&lt;p&gt;AlphaFold 3, released in May 2024, dramatically expanded the scope: beyond proteins, the model predicts the structure and interactions of DNA, RNA, ligands (the small molecules that make up most drugs), ions and modified residues. It can generate joint 3D structures of entire molecular complexes, allowing a holistic view of how a potential pharmacological molecule binds to its target protein. On the PoseBusters benchmark, AlphaFold 3 showed 50% greater accuracy than the best traditional methods for protein-ligand interaction predictions, becoming the first AI system to outperform physics-based tools for biomolecular structure prediction.&lt;/p&gt;
&lt;h3 id=&quot;32-isomorphic-labs-and-the-next-generation&quot;&gt;3.2 Isomorphic Labs and the Next Generation&lt;/h3&gt;
&lt;p&gt;Isomorphic Labs, an AI drug discovery company founded in 2021 as a spin-off of DeepMind, developed a unified drug design engine that aims to dramatically change how new medicines are designed. In February 2026, &lt;em&gt;Nature&lt;/em&gt; reported that the company announced an even more powerful proprietary model — described by independent scientists as equivalent to an “AlphaFold 4” — keeping the innovation as exclusive property.&lt;/p&gt;
&lt;p&gt;In parallel, the open-source ecosystem has advanced significantly. In 2025, OpenFold3 offered commercially usable co-folding capabilities for protein-ligand and antibody-antigen complexes. Boltz-2, an open-source biomolecular foundation model from MIT and Recursion, achieved near-physics-level accuracy for binding affinity predictions at speeds up to 1,000 times faster than traditional approaches.&lt;/p&gt;
&lt;h3 id=&quot;33-implications-for-longevity&quot;&gt;3.3 Implications for Longevity&lt;/h3&gt;
&lt;p&gt;For the longevity field, the implications are monumental. Traditionally, taking a drug from initial discovery to regulatory approval requires 12 to 15 years and costs approximately 2.5 billion dollars. AI promises to dramatically compress both parameters. Specifically for anti-aging therapies:&lt;/p&gt;
&lt;p&gt;First, target identification. The molecular processes of aging involve hundreds of proteins interacting in complex networks. AlphaFold and derived models make it possible to visualize these interactions at atomic resolution, identifying points of vulnerability — binding sites, interaction interfaces, allosteric regions — that can be exploited therapeutically.&lt;/p&gt;
&lt;p&gt;Second, molecule design. Generative AI models can design molecules &lt;em&gt;de novo&lt;/em&gt; with desired properties: high affinity for a specific target, low toxicity, good oral bioavailability, metabolic stability. The total chemical space of pharmacologically relevant molecules is estimated at 10^60 compounds — more than the number of atoms in the observable universe. Without AI, exploring this space is impossible. With generative AI, it becomes tractable.&lt;/p&gt;
&lt;p&gt;Third, side-effect prediction. Cellular reprogramming therapies carry significant oncogenic risks. AI models trained on multi-omic data can predict which combinations of factors, doses and durations minimize risks while maximizing rejuvenation — a multi-objective optimization problem that exceeds the human capacity for intuitive reasoning.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;iv-deep-epigenetic-clocks-ai-as-an-instrument-for-measuring-aging&quot;&gt;IV. Deep Epigenetic Clocks: AI as an Instrument for Measuring Aging&lt;/h2&gt;
&lt;h3 id=&quot;41-the-problem-of-measurement&quot;&gt;4.1 The Problem of Measurement&lt;/h3&gt;
&lt;p&gt;One of the most fundamental barriers to the science of longevity is the absence of a reliable, real-time metric of biological aging. If we cannot measure precisely how aged an organism is, we cannot assess whether an intervention works without waiting decades — the time needed to observe differences in morbidity and mortality.&lt;/p&gt;
&lt;p&gt;Epigenetic clocks, originally developed by Steve Horvath in 2013, offered the first practical solution. Using regularized regression (Elastic Net) on DNA methylation patterns at selected CpG sites, the Horvath clock predicts chronological age with a mean absolute error of a few years. The divergence between chronological age and estimated epigenetic age — called “epigenetic age acceleration” — correlates with mortality, cardiovascular disease, cancer and multiple other adverse outcomes.&lt;/p&gt;
&lt;p&gt;But first-generation clocks have significant limitations. They assume linear relationships between methylation and age. They do not distinguish between causal and adaptive changes. And they do not capture complex nonlinear interactions among CpG sites — interactions that probably encode biologically meaningful information about the aging process.&lt;/p&gt;
&lt;h3 id=&quot;42-deep-learning-and-the-new-generation-of-clocks&quot;&gt;4.2 Deep Learning and the New Generation of Clocks&lt;/h3&gt;
&lt;p&gt;The application of deep neural networks to biological age prediction has produced the so-called “Deep Aging Clocks,” which consistently outperform linear models in accuracy and generalization. AltumAge, a neural-network-based framework trained on 142 publicly available datasets from multiple human tissues, shows superior performance to Elastic Net for both intra-dataset and cross-dataset prediction, being particularly more generalizable at advanced ages and in new tissue types.&lt;/p&gt;
&lt;p&gt;More recently, XAI-AGE introduced an approach of biologically informed, explainable deep neural networks. By integrating prior biological network information into the model architecture, XAI-AGE not only predicts age with high accuracy, but allows biologically meaningful insights to be inferred about the processes that control aging. This model opened the possibility of identifying specific molecular pathways — not just statistical correlates — that govern the acceleration or deceleration of aging.&lt;/p&gt;
&lt;p&gt;NCAE-CombClock achieved R² = 0.978 with a mean absolute error of 1.96 years, integrating data-driven DNA methylation embeddings with established age-related CpG markers. Associated neural classifiers can classify individuals at critical developmental ages with AUROCs above 0.92.&lt;/p&gt;
&lt;p&gt;In 2024, Gladyshev and collaborators at Brigham and Women’s Hospital published in &lt;em&gt;Nature Aging&lt;/em&gt; the CausAge, DamAge and AdaptAge clocks — the first generation of epigenetic clocks that distinguish between causal changes of aging and protective adaptations. Using causal inference and machine learning techniques, these clocks created a map identifying human CpG sites that cause biological aging, making it possible to assess how different interventions promote longevity or accelerate aging. DamAge correlated with mortality; AdaptAge, with longevity.&lt;/p&gt;
&lt;h3 id=&quot;43-multimodal-clocks-and-the-frontier&quot;&gt;4.3 Multimodal Clocks and the Frontier&lt;/h3&gt;
&lt;p&gt;The frontier of aging clocks is markedly multimodal. FaceAge, a deep learning system published in &lt;em&gt;The Lancet Digital Health&lt;/em&gt; in 2025, estimates biological age from facial photographs to improve clinical prognosis. Systems based on chest X-rays, electrocardiograms and microbiome data demonstrate that aging leaves detectable signatures in practically any modality of biological data — and that neural networks are the most sensitive instruments for detecting them.&lt;/p&gt;
&lt;p&gt;The implication for the science of longevity is twofold. First, deep-learning-based epigenetic clocks serve as surrogate endpoints for clinical trials of anti-aging interventions, compressing the time needed to evaluate efficacy from decades to months or years. Second, the interpretability of these models — through explainable AI (XAI) techniques — is revealing molecular mechanisms of aging that remained hidden to traditional statistical methods.&lt;/p&gt;
&lt;p&gt;A multi-view graph-level representation learning framework (MGRL) published in 2025 integrated single-cell RNA-seq data from more than a million immune cells from 981 donors with protein-protein interaction networks, revealing a subnetwork of ribosomal genes whose expression correlates with age independently of cell type. The same approach applied to DNA methylation data from purified monocytes revealed an epigenetically dysregulated inflammatory response pathway whose activity increases with age — involving cytokine signaling and JAK-STAT pathways, with deregulation mediated by DNA methylation in monocytes.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;v-sens-longevity-escape-velocity-and-the-damage-repair-strategy&quot;&gt;V. SENS, Longevity Escape Velocity and the Damage-Repair Strategy&lt;/h2&gt;
&lt;h3 id=&quot;51-de-greys-framework&quot;&gt;5.1 De Grey’s Framework&lt;/h3&gt;
&lt;p&gt;Aubrey de Grey, the biogerontologist who coined the term “longevity escape velocity” (LEV) in 2004, proposed the SENS framework (&lt;em&gt;Strategies for Engineered Negligible Senescence&lt;/em&gt;), which identifies seven main categories of aging damage, each requiring its own repair strategy: accumulation of intracellular and extracellular junk, nuclear and mitochondrial mutations, cellular senescence, cell loss and extracellular crosslinks.&lt;/p&gt;
&lt;p&gt;The SENS approach is conceptually distinct from both Sinclair’s epigenetic approach and Church’s genomic engineering approach, although complementary to both. While Sinclair seeks to “reset” the cellular software and Church seeks to “rewrite” the genomic hardware, de Grey seeks to “fix” the accumulated damage — like a mechanic diagnosing and repairing an aging car.&lt;/p&gt;
&lt;p&gt;In 2025, the LEV Foundation completed its first Robust Mouse Rejuvenation (RMR) study, obtaining what de Grey called a “qualified victory”: the combination of multiple damage-repair interventions in middle-aged mice demonstrated additive lifespan extension, indicating synergies in multi-therapeutic combinations. The last mouse died on February 12, 2025, and the final survival curves are being statistically analyzed.&lt;/p&gt;
&lt;h3 id=&quot;52-lev-and-the-role-of-ai&quot;&gt;5.2 LEV and the Role of AI&lt;/h3&gt;
&lt;p&gt;The concept of longevity escape velocity is essentially an argument about compounding rates of technological progress. If “SENS 1.0” therapies can add 30 years of healthy life to middle-aged people, those same people will be alive to receive “SENS 2.0,” which will add more time, and so on — accelerating until medical progress outpaces the speed of aging.&lt;/p&gt;
&lt;p&gt;AI is the accelerator that can turn this concept from bold speculation into practical engineering. The combination of multiple anti-aging interventions — senolytic drugs, stem-cell therapies, gene therapies, epigenetic modulation, enzyme engineering — generates a combinatorial space that grows exponentially with each added modality. A 2025 study showed that rapamycin combined with the MEK inhibitor trametinib extended median lifespan by 27-29% in middle-aged mice of both sexes, surpassing the single agents (rapamycin: 17-18%; trametinib: 5-10%) through synergistic inhibition of growth signaling and senescence.&lt;/p&gt;
&lt;p&gt;Identifying the optimal combinations among dozens of candidate interventions, each with multiple possible doses and time windows, is an optimization problem that only AI can solve in a timely way. Reinforcement learning models can navigate this parameter space, virtually testing billions of combinations and converging on optimal protocols before experimental validation.&lt;/p&gt;
&lt;p&gt;De Grey predicts a 50% probability of reaching LEV around 2036. Kurzweil predicted something similar for 2028-2030 (revised in 2024 to 2029-2035). Church offers a more conservative estimate, suggesting that 2050 would not be surprising. Regardless of the specific timeline, all converge on one point: AI will be the decisive factor in the acceleration.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vi-digital-biology-ai-as-microscope-telescope-and-inference-engine&quot;&gt;VI. Digital Biology: AI as Microscope, Telescope and Inference Engine&lt;/h2&gt;
&lt;h3 id=&quot;61-spatial-transcriptomics-and-anatomical-atlases&quot;&gt;6.1 Spatial Transcriptomics and Anatomical Atlases&lt;/h3&gt;
&lt;p&gt;One of Church’s most ambitious projects is SAHA (Spatial Atlas of Human Anatomy), a multimodal human anatomical atlas with subcellular resolution. Spatial transcriptomics — the ability to measure which genes are being expressed at which spatial locations within a tissue — generates volumes of data that exceed by orders of magnitude what human scientists can analyze manually. Each tissue slide can contain millions of spatially resolved data points, each with expression profiles for thousands of genes.&lt;/p&gt;
&lt;p&gt;AI is the analytical instrument that makes these data interpretable. Convolutional neural networks for image segmentation, graph neural networks for modeling spatial relationships among cells, and transformers for integrating multiple data modalities (transcriptomics, proteomics, metabolomics, imaging) converge to create computational models of tissue that capture biological organization at scales ranging from the molecule to the organ.&lt;/p&gt;
&lt;p&gt;For longevity, this capability is transformative. Aging is not a uniform process: different cell types in different tissues age at different rates, with distinct functional consequences. A complete spatial atlas of human aging — showing how cellular composition, gene expression patterns and spatial organization change with age in each tissue — would provide the map needed to direct interventions with unprecedented precision.&lt;/p&gt;
&lt;h3 id=&quot;62-foundation-models-for-biology&quot;&gt;6.2 Foundation Models for Biology&lt;/h3&gt;
&lt;p&gt;The emergence of “foundation models” for biological data is perhaps the most significant revolution at the intersection of AI and biology. Just as GPT-4 learned deep statistical patterns in human language, models such as ESM (Evolutionary Scale Modeling) learned the “grammar” of protein sequences, encoding decades of evolution into vector representations that capture functional and structural properties.&lt;/p&gt;
&lt;p&gt;Protein language models can interpret the “grammar” of sequences, allowing not only structure prediction but the &lt;em&gt;de novo&lt;/em&gt; design of proteins with specified functions. This opens the door to engineering new enzymes (such as the soil-bacteria enzymes that de Grey proposes for degrading intracellular junk), new transcription factors (safer and more efficient versions of the Yamanaka factors), and new delivery vectors (AAV variants computationally optimized to evade the immune system).&lt;/p&gt;
&lt;p&gt;RoseTTAFold All-Atom, developed at the University of Washington, completed full all-atom biological modeling in 2024, enabling high-resolution predictions for complex biomolecular systems. The AI Structural Biology (AISB) network demonstrated a model in which pharmaceutical companies began fine-tuning AI models on proprietary structural data through federated learning, creating a path to pharma-grade models without sharing intellectual property.&lt;/p&gt;
&lt;h3 id=&quot;63-simulations-and-digital-twins&quot;&gt;6.3 Simulations and Digital Twins&lt;/h3&gt;
&lt;p&gt;The ultimate ambition of digital biology is the creation of digital twins — computational models of individuals that capture the complete biological state of an organism and make it possible to simulate the effects of interventions before implementing them physically. A digital twin of an elderly patient could predict how a specific combination of gene therapies, senolytic drugs and epigenetic modulators would affect each system of the body, identifying synergies and antagonisms, predicting side effects and optimizing personalized treatment protocols.&lt;/p&gt;
&lt;p&gt;This vision is still far from complete realization, but each component is progressing rapidly. ML-accelerated molecular dynamics, multi-scale models integrating single-cell transcriptomics with protein interaction networks, and reinforcement-learning frameworks for treatment optimization are converging toward a capacity for biological simulation that would have been unthinkable a decade ago.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vii-xenotransplantation-engineered-organs-and-ai-in-the-biological-supply-chain&quot;&gt;VII. Xenotransplantation, Engineered Organs and AI in the Biological Supply Chain&lt;/h2&gt;
&lt;h3 id=&quot;71-egenesis-and-the-problem-of-organ-scarcity&quot;&gt;7.1 eGenesis and the Problem of Organ Scarcity&lt;/h3&gt;
&lt;p&gt;One of the companies co-founded by Church is eGenesis, focused on solving the shortage of organ donors through xenotransplantation — the transplantation of organs between species. Using CRISPR to edit the porcine genome, eliminating endogenous retroviruses and modifying immune-compatibility genes, eGenesis is developing pig organs compatible with the human immune system.&lt;/p&gt;
&lt;p&gt;AI plays several roles in this program. First, in predicting which genomic edits minimize immune rejection while preserving the organ’s functional viability. Second, in optimizing multiplex editing protocols — eGenesis has carried out more than 60 simultaneous edits in porcine genomes, a number that exceeds the capacity of purely human rational design. Third, in post-transplant monitoring: ML models can detect early signs of rejection in biomarker data before they manifest clinically.&lt;/p&gt;
&lt;h3 id=&quot;72-bioprinting-and-organoids&quot;&gt;7.2 Bioprinting and Organoids&lt;/h3&gt;
&lt;p&gt;The 3D bioprinting of tissues and the culture of organoids (mini-organs derived from stem cells) represent approaches complementary to xenotransplantation. Both depend deeply on AI for optimization. Bioprinting requires precise control of hundreds of parameters (cell type, density, deposition speed, bio-ink composition, temperature, cross-linking), forming a parameter space that ML models navigate more efficiently than manual optimization.&lt;/p&gt;
&lt;p&gt;The human ovarian organoid technology developed at the Wyss Institute and commercialized by Gameto exemplifies the convergence. Using the Wyss’s fully human ovarian organoid technology, the company aims to address the growing global problem of infertility by making IVF safer, more accessible and more effective. Optimizing the protocols for culture, differentiation and maturation of organoids is intrinsically an ML problem — with parameter spaces far too large for manual exploration.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;viii-economic-ethical-and-civilizational-implications&quot;&gt;VIII. Economic, Ethical and Civilizational Implications&lt;/h2&gt;
&lt;h3 id=&quot;81-the-economics-of-radical-longevity&quot;&gt;8.1 The Economics of Radical Longevity&lt;/h3&gt;
&lt;p&gt;De Grey argues that the cost of longevity therapies to the end user will be effectively zero — and that the reason is purely economic: aging is so expensive that it would be economic suicide for any country not to make these therapies universally available, regardless of ability to pay. Adults over 50 already contribute 45 trillion dollars to global GDP, with projections suggesting this will more than double to 118 trillion dollars by 2050.&lt;/p&gt;
&lt;p&gt;Formal economic analysis supports this argument. A National Academy of Medicine report argues that older, healthier populations turn from a fiscal challenge into a net economic asset. If people remain productive and healthy into their 90s or beyond, they continue to generate, consume and contribute to GDP. The longevity dividend is not merely the extension of years of life — it is the extension of productive, healthy years of life.&lt;/p&gt;
&lt;p&gt;The market for AI in pharmaceuticals, valued at 1.8 billion dollars in 2023, is projected to reach 13.1 billion dollars by 2030, reflecting a compound annual growth rate of 18.8%. For the longevity subset, the numbers are even more striking: companies such as Altos Labs, New Limit and Retro Biosciences have attracted hundreds of millions in investment from prominent Silicon Valley names for research on cellular reprogramming.&lt;/p&gt;
&lt;p&gt;But the economics of radical longevity goes beyond the market for therapies. Recent analyses estimate that the economic value of radically extending human life could reach 367 trillion dollars — more than current global GDP. This figure reflects not only the value of the therapies themselves, but the present value of the additional decades of economic productivity, innovation and consumption that healthy individuals with radically extended life expectancies would generate.&lt;/p&gt;
&lt;p&gt;The insurance sector will be transformed in fundamental ways. Life insurance will have to pivot to “longevity insurance” — products that protect people against the risk of outliving their financial reserves, rather than paying death benefits. We are already seeing the emergence of &lt;em&gt;longevity swaps&lt;/em&gt; and &lt;em&gt;longevity bonds&lt;/em&gt;, in which insurers transfer to investors the risk of clients living longer than expected. If de Grey’s vision materializes, these hedging instruments will become essential to the sector’s survival.&lt;/p&gt;
&lt;p&gt;Retirement systems will face the most immediate challenge. If people remain biologically young and productive at 80 or 90, the notion of retiring at 60-65 becomes economically absurd. The social contract between generations — in which active workers fund inactive retirees — will have to be fundamentally renegotiated. This is not necessarily negative: people who want to work (and can, because they have been rejuvenated) are a resource, not a burden. But the transition will be disruptive for institutional structures that assume a fixed life expectancy.&lt;/p&gt;
&lt;h3 id=&quot;82-non-trivial-ethical-questions&quot;&gt;8.2 Non-Trivial Ethical Questions&lt;/h3&gt;
&lt;p&gt;Radical life extension raises ethical questions that transcend the simplistic utilitarian framework. Who will have access first? If therapies cost millions of dollars at the outset, will radical longevity deepen inequalities that are already obscene? De Grey’s economic argument — that the costs of aging are so high that universal availability is economically rational — is optimistic but not guaranteed.&lt;/p&gt;
&lt;p&gt;There is also the question of personal identity. If we can “reset” cells to a younger state, in what sense is the rejuvenated person the same person? Church’s neuronal “Ship of Theseus” — the gradual replacement of aging neurons with new ones that learn the existing circuit — raises philosophical questions about the continuity of consciousness that are not merely academic if the technology becomes real.&lt;/p&gt;
&lt;p&gt;And there is the ecological question. A planet with 8 billion humans is already under extreme environmental stress. If life expectancy doubles or triples without a corresponding reduction in birth rates, pressures on natural resources, carbon emissions and ecosystems intensify dramatically. The longevity community’s standard response — that people with longer time horizons take better care of the planet — is plausible but empirically untested.&lt;/p&gt;
&lt;h3 id=&quot;83-the-regulatory-question&quot;&gt;8.3 The Regulatory Question&lt;/h3&gt;
&lt;p&gt;The FDA still does not classify aging as a disease, which means that anti-aging interventions must be approved on the basis of specific indications (glaucoma, sarcopenia, Alzheimer’s disease) rather than “aging” as a treatable condition. Sinclair’s Life Biosciences obtained FDA approval to proceed with the first targeted age-reversal clinical trial in human volunteers, treating eye disease with reprogramming — but under the heading of a specific medical condition, not of aging itself.&lt;/p&gt;
&lt;p&gt;In January 2026, Life Biosciences confirmed the start of the first human trials of a rejuvenation method. The approach uses OSK factors to reprogram ocular cells to a healthier state. But competing companies such as New Limit (which conducted an extensive search for alternative reprogramming factors) and Shift (whose animal experiments are only beginning) question whether the specific factors chosen are the best version of rejuvenation available.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;ix-gpt-4b-micro-and-protein-engineering-by-generative-ai-the-retro-biosciences-case&quot;&gt;IX. GPT-4b Micro and Protein Engineering by Generative AI: The Retro Biosciences Case&lt;/h2&gt;
&lt;h3 id=&quot;91-the-redesign-of-the-yamanaka-factors&quot;&gt;9.1 The Redesign of the Yamanaka Factors&lt;/h3&gt;
&lt;p&gt;Perhaps no recent result better illustrates the convergence of AI and longevity than the collaboration between OpenAI and Retro Biosciences, announced in August 2025. Using GPT-4b micro — OpenAI’s first model trained specifically for biological data, combining protein sequences, structural data and contextual annotations such as evolutionary relationships and functional descriptions — the Retro team redesigned variants of the transcription factors SOX2 and KLF4, two of the four Yamanaka factors central to cellular reprogramming.&lt;/p&gt;
&lt;p&gt;The results were remarkable. The engineered variants — called RetroSOX and RetroKLF — showed an increase of more than 50-fold in the expression of stem-cell markers compared with wild-type controls. In some experiments, the generation of iPSCs (induced pluripotent stem cells) was accelerated from three weeks to just seven days. More than 30% of the AI-generated SOX2 variants outperformed the natural protein in initial screens, while nearly half of the KLF4 variants showed superior activity.&lt;/p&gt;
&lt;p&gt;The most significant aspect for longevity is not just the efficiency of reprogramming, but the quality of the resulting cellular rejuvenation. Using a DNA-damage assay, the researchers found that cells treated with the RetroSOX/KLF cocktail accumulated significantly fewer DNA double-strand breaks than those reprogrammed with the original Yamanaka factors. This reduction suggests that the AI-engineered variants may be more effective at reversing DNA damage — a central hallmark of aging.&lt;/p&gt;
&lt;h3 id=&quot;92-the-magnitude-of-the-sequence-change&quot;&gt;9.2 The Magnitude of the Sequence Change&lt;/h3&gt;
&lt;p&gt;One particularly surprising technical detail: the most effective RetroSOX and RetroKLF variants differed significantly from their wild-type counterparts, with some containing more than 100 amino acid changes. This is a result that no traditional protein-engineering program — based on site-directed mutagenesis, directed evolution or human rational design — would have produced. Generative AI explored regions of protein sequence space that lay completely beyond the cognitive reach of human scientists, finding solutions that work better than billions of years of natural evolution.&lt;/p&gt;
&lt;p&gt;This result has profound implications. If AI can redesign transcription factors with 100+ mutations and obtain functionally superior proteins, the same principle can be applied to virtually any therapeutic protein relevant to longevity: enzymes for degrading lipofuscin (de Grey’s intracellular “junk”), molecular chaperones for maintaining proteostasis, modified telomerases for telomere maintenance, and growth factors optimized for tissue regeneration.&lt;/p&gt;
&lt;h3 id=&quot;93-the-competitive-ecosystem-of-2025-2026&quot;&gt;9.3 The Competitive Ecosystem of 2025-2026&lt;/h3&gt;
&lt;p&gt;The competitive landscape of longevity in 2025-2026 reflects the centrality of AI to the field. Altos Labs, founded in 2022 with 3 billion dollars in initial funding (the most heavily funded biotech startup in history), backed by Jeff Bezos and with Shinya Yamanaka as scientific advisor, reported the start of initial human safety trials in August 2025. Retro Biosciences, backed by OpenAI CEO Sam Altman with a 180-million-dollar seed round, is raising a 1-billion-dollar Series A at a 5-billion-dollar valuation. NewLimit, co-founded by Coinbase CEO Brian Armstrong, raised 130 million in a Series B, focusing on an extensive search for alternative reprogramming factors. Shift Biosciences, based in Cambridge, UK, used machine learning to find safer and more efficient reprogramming genes.&lt;/p&gt;
&lt;p&gt;Sinclair’s Life Biosciences obtained IND (Investigational New Drug) approval from the FDA for ER-100, becoming the first cellular rejuvenation therapy using partial epigenetic reprogramming to reach human clinical trials. The phase 1 study will include individuals with open-angle glaucoma and non-arteritic anterior ischemic optic neuropathy.&lt;/p&gt;
&lt;p&gt;In parallel, Retro Biosciences dosed the first participant in a clinical trial of its autophagy-focused candidate, RTR242 — a small-molecule therapy designed to restore lysosomal function. The phase 1 study, randomized, double-blind and placebo-controlled in healthy volunteers, is being conducted in Adelaide, Australia, where early-stage approvals are faster. The study includes exploratory biomarkers linked to autophagy and lysosomal biology, giving Retro its first opportunity to observe whether its mechanistic hypotheses translate into measurable biological signals in humans.&lt;/p&gt;
&lt;h3 id=&quot;94-ai-as-a-systemic-accelerator&quot;&gt;9.4 AI as a Systemic Accelerator&lt;/h3&gt;
&lt;p&gt;What emerges from this landscape is not merely the occasional use of AI on isolated problems, but AI as a systemic accelerator of the entire longevity pipeline. AI is being used simultaneously to: identify targets (computational transcriptomics, GWAS with machine learning), design molecules (generative models), optimize reprogramming protocols (GPT-4b micro), predict safety (predictive models of oncogenicity), design clinical trials (adaptive statistical optimization), and analyze trial data (automated biomarker interpretation).&lt;/p&gt;
&lt;p&gt;The combination of these uses creates a multiplicative effect: each improvement in one stage of the pipeline accelerates all subsequent stages. AI is not adding value linearly — it is exponentially compressing the time between discovery and clinic.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;x-the-question-of-the-biological-feedback-loop-organisms-as-control-systems&quot;&gt;X. The Question of the Biological Feedback Loop: Organisms as Control Systems&lt;/h2&gt;
&lt;h3 id=&quot;101-homeostasis-and-aging&quot;&gt;10.1 Homeostasis and Aging&lt;/h3&gt;
&lt;p&gt;A frequently neglected perspective in the discussion of AI and longevity is the understanding of aging as the progressive failure of control systems. Biological organisms are, fundamentally, feedback control systems: sensors detect deviations from desired states, controllers process this information, and actuators execute corrections. Homeostasis — the maintenance of body temperature, blood pH, blood pressure, hormone levels — is implemented by biochemical feedback circuits operating on multiple time scales.&lt;/p&gt;
&lt;p&gt;Aging can be understood as the progressive degradation of these control circuits: sensors lose sensitivity (molecular receptors accumulate damage), controllers lose precision (cellular signaling becomes noisy), and actuators lose effectiveness (effector cells become senescent or dysfunctional). Sinclair’s epigenetic information, from this perspective, is the “firmware” of the controllers: when it degrades, the system loses the capacity to maintain optimal physiological states.&lt;/p&gt;
&lt;p&gt;AI is particularly well suited to modeling and optimizing control systems. Techniques of optimal control, reinforcement learning and dynamical-systems modeling are mature tools in control engineering that can be applied directly to the biology of aging. A computational model that captures the dynamics of homeostatic circuits — including their modes of failure with age — could predict which interventions restore control function most efficiently.&lt;/p&gt;
&lt;h3 id=&quot;102-cellular-senescence-as-a-bifurcation-in-a-dynamical-system&quot;&gt;10.2 Cellular Senescence as a Bifurcation in a Dynamical System&lt;/h3&gt;
&lt;p&gt;Cellular senescence — the irreversible transition of proliferative cells to a state of cell-cycle arrest with a pro-inflammatory secretory phenotype (SASP) — can be modeled as a bifurcation in a dynamical system. Before the bifurcation, stochastic perturbations (DNA damage, oxidative stress) are absorbed by the cell’s control system; afterward, the system moves to an alternative stable attractor (the senescent state) from which it does not spontaneously return.&lt;/p&gt;
&lt;p&gt;This dynamical-systems perspective suggests that AI could be used not only to eliminate senescent cells (the senolytic approach) or to reverse their state (reprogramming), but to identify “early warning signals” — molecular signatures indicating that a cell is approaching the senescence bifurcation, allowing preventive intervention before the transition occurs. Deep learning models trained on single-cell transcriptomic data with temporal resolution could detect these precursors with a sensitivity impossible for human analysis.&lt;/p&gt;
&lt;h3 id=&quot;103-regulatory-networks-and-robustness&quot;&gt;10.3 Regulatory Networks and Robustness&lt;/h3&gt;
&lt;p&gt;Aging can also be understood as a loss of robustness in gene regulatory networks. Young biological networks are highly redundant and tolerant of perturbations; aged networks lose redundancy and become fragile — susceptible to cascading failures in which the collapse of one component propagates rapidly through the network.&lt;/p&gt;
&lt;p&gt;Graph neural networks (GNNs) and graph diffusion models are the natural tools for analyzing this loss of robustness. By mapping the topology of regulatory networks at different ages and in different tissues, and by identifying which nodes and connections are most vulnerable to aging, AI can guide interventions that restore topological robustness — for example, by reinforcing weakened redundancy pathways or stabilizing hub nodes that are critical to the integrity of the network.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xi-computational-regenerative-medicine-from-prediction-to-prescription&quot;&gt;XI. Computational Regenerative Medicine: From Prediction to Prescription&lt;/h2&gt;
&lt;h3 id=&quot;111-the-ai-accelerated-dbtl-cycle&quot;&gt;11.1 The AI-Accelerated DBTL Cycle&lt;/h3&gt;
&lt;p&gt;Modern biological engineering operates under the Design-Build-Test-Learn (DBTL) paradigm: design a genetic construct or molecule, build it in the laboratory, test it in biological systems, and learn from the results to inform the next design cycle. AI is compressing each stage of this cycle.&lt;/p&gt;
&lt;p&gt;In design, generative models produce optimized candidates &lt;em&gt;in silico&lt;/em&gt;. In building, automated laboratory robots execute protocols with a precision and throughput impossible for human hands. In testing, high-content phenotyping platforms generate multidimensional data on thousands of conditions simultaneously. In learning, active-learning algorithms select the next experiments so as to maximize the information obtained per unit of cost.&lt;/p&gt;
&lt;p&gt;The result is a compression of the DBTL cycle from months to days. For longevity, this means that the search for optimal therapeutic combinations — which under the traditional experimental paradigm would take centuries — becomes feasible in years.&lt;/p&gt;
&lt;h3 id=&quot;112-autonomous-laboratories-and-robot-scientists&quot;&gt;11.2 Autonomous Laboratories and Robot Scientists&lt;/h3&gt;
&lt;p&gt;The emerging frontier is the “autonomous scientist” (&lt;em&gt;self-driving lab&lt;/em&gt;) — an integrated system in which AI not only analyzes data but formulates hypotheses, designs experiments, instructs robots to execute them, analyzes results and iterates. Several prototypes already exist for chemistry and materials science; extending them to the biology of aging is a matter of infrastructure integration, not of principle.&lt;/p&gt;
&lt;p&gt;An autonomous laboratory dedicated to longevity could, in theory, carry out thousands of cellular reprogramming experiments per day, testing variations of factors, dosages, timings and combinations, feeding the results into models that refine themselves continuously. The discovery cycle that took Sinclair years — from the concept of OSK reprogramming to the first experiment in mice — could be compressed to weeks.&lt;/p&gt;
&lt;h3 id=&quot;113-digital-therapeutics-and-personalized-interventions&quot;&gt;11.3 Digital Therapeutics and Personalized Interventions&lt;/h3&gt;
&lt;p&gt;The convergence of deep epigenetic clocks, personalized multi-omic data and predictive AI models enables a radically new form of medicine: personalized anti-aging interventions, in which the protocol is computationally optimized for each individual’s specific biological profile.&lt;/p&gt;
&lt;p&gt;Instead of a generic “one dose fits all” therapy, AI could prescribe personalized combinations of senolytics, epigenetic modulators, growth factors and metabolic interventions, calibrated to each patient’s specific pattern of accumulated damage, genomic profile, immune status and environmental conditions. The deep epigenetic clock would serve simultaneously as a diagnostic (measuring the state of aging) and as an endpoint (assessing the efficacy of the intervention).&lt;/p&gt;
&lt;p&gt;This vision is no longer purely speculative. Companies such as Insilico Medicine already use AI to identify therapeutic targets and design personalized molecules, with candidates in clinical trials. Extending this to personalized longevity is an incremental step, not a conceptual leap.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xii-the-limits-of-optimism-technical-and-epistemological-critiques&quot;&gt;XII. The Limits of Optimism: Technical and Epistemological Critiques&lt;/h2&gt;
&lt;h3 id=&quot;121-the-mouse-to-human-translational-gap&quot;&gt;12.1 The Mouse-to-Human Translational Gap&lt;/h3&gt;
&lt;p&gt;No honest discussion of AI and longevity can ignore the translational gap. Mice are not small humans. Differences in metabolism (mice have much higher basal metabolic rates), in genetics (isogenic laboratory strains do not reflect human genetic diversity) and in environmental exposures (laboratory mice live in controlled environments radically different from human life) limit the direct transposition of results.&lt;/p&gt;
&lt;p&gt;De Grey explicitly acknowledges this limitation, arguing that dogs — whose life cycles and environments are closer to those of humans — are more informative intermediate models. Church also prioritizes canine models for analogous reasons. But definitive validation requires human trials, which are intrinsically slow (even with epigenetic clocks as surrogate endpoints) and expensive.&lt;/p&gt;
&lt;p&gt;AI can partially mitigate this gap through computational models of cross-species translation — &lt;em&gt;cross-species transfer learning&lt;/em&gt; that identifies which findings in mice are most likely to translate to humans. But it cannot eliminate it. Human biology, in all its idiosyncrasy, remains the final tribunal.&lt;/p&gt;
&lt;h3 id=&quot;122-the-critique-of-the-information-theory&quot;&gt;12.2 The Critique of the Information Theory&lt;/h3&gt;
&lt;p&gt;Sinclair’s Information Theory of Aging, although elegant and empirically supported by remarkable results, is not universally accepted. A UC San Diego study argues that somatic mutations — not epigenetic drift — may be the root cause of aging, suggesting that Sinclair’s focus on reversing epigenetic markers could be addressing symptoms rather than the underlying biology. If aging is fundamentally caused by the accumulation of mutations in genomic DNA (the digital information, not the analog), then epigenetic restoration would be palliative, not curative.&lt;/p&gt;
&lt;p&gt;Sinclair’s response — that the reprogramming results demonstrate functional, not merely cosmetic, restoration of aged tissues — is persuasive but not conclusive. The question remains empirically open, and it is plausible that aging results from multiple interacting causes (epigenetic &lt;em&gt;and&lt;/em&gt; genomic &lt;em&gt;and&lt;/em&gt; mitochondrial &lt;em&gt;and&lt;/em&gt; proteostatic), none of which is in itself “the” cause.&lt;/p&gt;
&lt;h3 id=&quot;123-oncogenic-risks-of-reprogramming&quot;&gt;12.3 Oncogenic Risks of Reprogramming&lt;/h3&gt;
&lt;p&gt;The most serious risk of cellular reprogramming therapies is carcinogenesis. The Yamanaka factors are, by definition, pluripotency factors — and uncontrolled pluripotency is cancer. The inclusion of c-Myc (a known oncogene) in the original OSKM quartet was abandoned by many researchers precisely because of this risk. But even the OSK trio carries dangers: prolonged expression can revert cells completely to a stem-cell state, losing tissue identity and potentially initiating tumors.&lt;/p&gt;
&lt;p&gt;AI can help calibrate dosages and durations to minimize risks, but it cannot eliminate them as long as the fundamental mechanism involves activating pluripotency pathways. Alternative approaches — such as the small molecules Sinclair is pursuing, or the proprietary factors New Limit is identifying — may offer superior safety profiles, but they remain at early stages of validation.&lt;/p&gt;
&lt;h3 id=&quot;124-hyperbole-and-the-track-record&quot;&gt;12.4 Hyperbole and the Track Record&lt;/h3&gt;
&lt;p&gt;An honest assessment requires mentioning that the science-communication record of some of the field’s protagonists includes significant controversies. Claims of supplement benefits that proved to lack adequate clinical evidence, acquisitions of hundreds of millions based on molecules that failed in trials, and projections of timelines that are systematically optimistic relative to what biology allows — all of this is part of the historical record. The data published in &lt;em&gt;Nature&lt;/em&gt; and &lt;em&gt;Cell&lt;/em&gt; on reprogramming are robust and replicated. But projections extrapolated on podcasts and at conferences should be evaluated with a skepticism proportional to their ambition.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xiii-original-propositions-six-theses-on-ai-longevity-and-the-future-of-the-human-condition&quot;&gt;XIII. Original Propositions: Six Theses on AI, Longevity and the Future of the Human Condition&lt;/h2&gt;
&lt;p&gt;Up to this point, this essay has reviewed the state of the art. Allow me now to propose six original theses that, I believe, capture underappreciated aspects of the convergence of AI and longevity.&lt;/p&gt;
&lt;h3 id=&quot;thesis-1-ai-is-the-leeuwenhoek-microscope-of-interventive-gerontology&quot;&gt;Thesis 1: AI Is the “Leeuwenhoek Microscope” of Interventive Gerontology&lt;/h3&gt;
&lt;p&gt;Before Leeuwenhoek, microbiology could not exist as a discipline — not because microorganisms did not exist, but because we could not see them. Likewise, the biology of aging remained descriptive and correlational for decades because we lacked instruments to navigate the causal complexity of aging biological systems. AI is that instrument. Deep epigenetic clocks are not mere prediction tools — they are measuring instruments that make visible what was previously invisible: the rate, the trajectory and the causal mechanisms of biological aging at cellular and molecular resolution.&lt;/p&gt;
&lt;h3 id=&quot;thesis-2-the-real-bottleneck-is-not-computational-but-experimental&quot;&gt;Thesis 2: The Real Bottleneck Is Not Computational but Experimental&lt;/h3&gt;
&lt;p&gt;Although AI has dramatically compressed the hypothesis-prediction cycle, the bottleneck of longevity science is shifting toward experimental validation. We can generate trillions of candidate molecules &lt;em&gt;in silico&lt;/em&gt;, but testing them in real biological systems — cell cultures, organoids, animal models, clinical trials — remains slow and costly. The next frontier is no longer AI for generating hypotheses, but AI for automating experimentation: autonomous laboratory robots, automated cell cultures, high-throughput phenotyping integrated into closed &lt;em&gt;active learning loops&lt;/em&gt; in which the AI decides which experiments to run next.&lt;/p&gt;
&lt;h3 id=&quot;thesis-3-radical-longevity-requires-composite-ai-not-monolithic-ai&quot;&gt;Thesis 3: Radical Longevity Requires Composite AI, Not Monolithic AI&lt;/h3&gt;
&lt;p&gt;No single AI model, however powerful, will solve the problem of aging. Aging is a multi-scale process (from molecule to organism), multi-temporal (from nanoseconds of molecular dynamics to decades of damage accumulation) and multi-modal (genomic, epigenomic, transcriptomic, proteomic, metabolomic, imaging). The AI architecture required is intrinsically composite: neural networks for structure prediction, generative models for molecular design, graph neural networks for interaction networks, causal models for mechanistic inference, reinforcement learning for protocol optimization, transformers for multi-omic integration, multi-agent systems for coordinating autonomous research. Composite AI — not monolithic AI — is the natural architecture for the longevity problem.&lt;/p&gt;
&lt;p&gt;This thesis deserves elaboration. The concept of Composite AI — the deliberate integration of multiple AI paradigms (machine learning, symbolic reasoning, rule-based systems, optimization, simulation) into architectures that exploit their complementarities — is precisely what the longevity problem demands. Consider the chain of reasoning required to design a rejuvenation therapy:&lt;/p&gt;
&lt;p&gt;First, one must identify molecular targets of aging — a causal-inference problem that requires sophisticated statistical models combined with prior knowledge encoded in biological ontologies and interaction networks. Second, one must design molecules that interact with those targets — a generative problem that requires diffusion models or transformers trained on chemical spaces. Third, one must predict the pharmacokinetics and toxicity of those molecules — a multivariate regression problem with a strong physical-simulation component (molecular dynamics). Fourth, one must optimize combination protocols — a multi-objective optimization problem with safety constraints that demands reinforcement learning. Fifth, one must design adaptive clinical trials — a sequential experimental-design problem that requires Bayesian inference. Sixth, one must interpret trial results in real time — an inference problem that combines clinical data with mechanistic models.&lt;/p&gt;
&lt;p&gt;No AI paradigm solves all of these subproblems. Deep neural networks are excellent at structure prediction but poor at causal reasoning. Bayesian models are excellent at inference under uncertainty but limited in scale. Reinforcement learning is excellent at sequential optimization but requires reliable simulators. The solution is to compose — to create pipelines in which each component operates in its domain of competence, with interfaces that translate the outputs of one paradigm into inputs for the next.&lt;/p&gt;
&lt;p&gt;More radically, multi-agent systems — in which agents specialized in different domains (one for structural biology, another for medicinal chemistry, another for regulatory affairs, another for clinical analysis) collaborate, negotiate and integrate their contributions — could replicate and eventually surpass the organization of human multidisciplinary teams, but at computational speed and scale. The analogy with biology itself is direct: organisms are not monolithic systems but ecosystems of specialized subsystems (immune, nervous, endocrine, metabolic) that coordinate to maintain homeostasis. AI for longevity should mirror this distributed biological architecture.&lt;/p&gt;
&lt;p&gt;The Composite AI paradigm offers one further crucial advantage: explainability. By decomposing the pipeline into components with clear semantics, each decision can be traced back to the contribution of specific components — unlike a monolithic end-to-end model whose internal logic is opaque. For longevity therapies, where the risks are existential (literally — we are talking about life and death), the ability to explain why a specific therapeutic combination was recommended, what evidence supports each component of the recommendation, and what uncertainties remain is not a luxury — it is a regulatory and ethical necessity.&lt;/p&gt;
&lt;h3 id=&quot;thesis-4-the-convergence-of-ai-and-longevity-will-create-a-new-kind-of-epistemic-inequality&quot;&gt;Thesis 4: The Convergence of AI and Longevity Will Create a New Kind of Epistemic Inequality&lt;/h3&gt;
&lt;p&gt;As longevity science becomes computationally intensive, the countries and institutions with the greatest AI capacity will obtain disproportionate advantages — not only in creating therapies, but in the fundamental understanding of the mechanisms of aging. This “epistemic inequality” is potentially deeper than inequality of access to therapies: if only a handful of laboratories in the world possess the computational infrastructure and the data needed to conduct frontier research on longevity, fundamental knowledge becomes oligopolized. The counter-pressure must come from open-source initiatives (OpenFold, AltumAge, the free AlphaFold Server) and from public computational infrastructure for science.&lt;/p&gt;
&lt;h3 id=&quot;thesis-5-aging-as-bug-or-feature-is-an-empirically-decidable-question-not-a-philosophically-undecidable-one&quot;&gt;Thesis 5: Aging as “Bug” or “Feature” Is an Empirically Decidable Question, Not a Philosophically Undecidable One&lt;/h3&gt;
&lt;p&gt;The debate between those who see aging as an inevitable natural process and those who see it as a treatable disease is often framed as a philosophical or ethical question. But it is, fundamentally, an empirical one. If Sinclair’s Information Theory is correct — if aging is the degradation of epigenetic information, and if that information is restorable — then aging is a reversible process, and treating it as “natural and inevitable” is a factual error, not a legitimate philosophical position. AI, by making it possible to test this hypothesis with unprecedented rigor and scale, is turning a millennia-old philosophical question into an empirical research program with a resolution horizon on the scale of decades.&lt;/p&gt;
&lt;h3 id=&quot;thesis-6-longevity-escape-velocity-is-better-understood-as-a-phase-transition-in-a-complex-system&quot;&gt;Thesis 6: “Longevity Escape Velocity” Is Better Understood as a Phase Transition in a Complex System&lt;/h3&gt;
&lt;p&gt;LEV is typically presented as a linear threshold: the point where annual medical progress exceeds the annual rate of aging. But I propose that it is more productive to conceptualize it as a phase transition in a complex adaptive system. The system includes not only medical technologies, but regulation, funding, clinical-trial infrastructure, public acceptance, manufacturing capacity and distribution networks. The transition occurs when positive feedback loops become dominant: successful longevity therapies generate revenue that funds more research; more powerful AI accelerates discoveries that generate data that improve the AI; rejuvenated individuals contribute economically, generating resources to rejuvenate more individuals. The role of AI in this framework is not only technical — it is catalytic: it is the agent that can synchronize and amplify multiple feedback loops simultaneously, precipitating the phase transition.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xiv-the-geopolitical-dimension-who-will-control-longevity&quot;&gt;XIV. The Geopolitical Dimension: Who Will Control Longevity?&lt;/h2&gt;
&lt;h3 id=&quot;141-the-race-between-nations&quot;&gt;14.1 The Race Between Nations&lt;/h3&gt;
&lt;p&gt;Longevity science does not operate in a geopolitical vacuum. In 2025, China consolidated its position as a global power in early-stage pharmaceutical innovation: five of the ten largest R&amp;#x26;D licensing deals of the year originated with Chinese companies, representing 38% of all in-licenses by large pharmaceutical firms with upfront payments above 50 million dollars. Two of these deals focused on AI-driven discovery platforms, including the 6-billion-dollar deal between Dovetree and XtalPi.&lt;/p&gt;
&lt;p&gt;This geopolitical competition has direct implications for longevity. The ability to train AI models on massive biological datasets depends on computational infrastructure (cutting-edge GPUs, HPC clusters), proprietary data (population biobanks, digitized clinical records) and human capital (researchers trained at the intersection of AI and biology). Nations that accumulate advantages along these three dimensions — at present chiefly the United States and China, with Europe in an intermediate position — will have disproportionate influence over which therapies are developed, for which conditions, and at what price.&lt;/p&gt;
&lt;h3 id=&quot;142-the-problem-of-global-access&quot;&gt;14.2 The Problem of Global Access&lt;/h3&gt;
&lt;p&gt;If radical longevity therapies emerge first at American and Chinese institutions, with initial costs in the range of millions of dollars (like Sinclair’s reprogramming gene therapy), the scenario of global access will be profoundly unequal. The first beneficiaries will be citizens of rich countries — precisely those who already have the longest life expectancies. The longevity gap between rich and poor nations, already a matter of decades, could turn into a civilizational gap.&lt;/p&gt;
&lt;p&gt;De Grey’s counterargument — that the cost of sustaining aging populations is so high that governments will have an economic incentive to universalize therapies — is economically sound for countries with public health systems and fiscal capacity. For nations with precarious health systems and severe fiscal constraints, the argument is less convincing. Democratizing longevity will require not only scientific advances but institutional innovation: technology-transfer mechanisms, compulsory licensing and multilateral financing that ensure life-extension therapies do not become the deepest divider of inequality in human history.&lt;/p&gt;
&lt;h3 id=&quot;143-the-question-of-intellectual-property&quot;&gt;14.3 The Question of Intellectual Property&lt;/h3&gt;
&lt;p&gt;The proprietary model exemplified by Isomorphic Labs — which developed a drug-discovery AI potentially superior to AlphaFold 3 but kept it as exclusive property — raises fundamental questions about who benefits from the convergence of AI and longevity. If the most powerful models for designing anti-aging therapies are proprietary, progress will be dictated by the commercial incentives of a handful of corporations. If they are open (like the original AlphaFold, OpenFold3, Boltz-2), progress will be faster, more diverse and potentially more equitable.&lt;/p&gt;
&lt;p&gt;The tension between open and proprietary models in AI for longevity replicates the broader tension in the AI ecosystem. AlphaFold 2 was made freely available and used by 3 million researchers in 190 countries — including more than 1 million in low- and middle-income countries. This open-access model is one of the greatest contributions to the democratization of science in recent history. But Isomorphic Labs’ business model — and the fact that it is Isomorphic, not DeepMind’s academic arm, that is developing the most advanced capabilities — suggests that the frontier is moving behind proprietary walls.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xv-philosophy-of-technology-longevity-as-a-civilizational-project&quot;&gt;XV. Philosophy of Technology: Longevity as a Civilizational Project&lt;/h2&gt;
&lt;h3 id=&quot;151-churchs-meta-technological-argument&quot;&gt;15.1 Church’s Meta-Technological Argument&lt;/h3&gt;
&lt;p&gt;George Church’s argument — that, given only one technology to develop, we should choose the one that gives us more time to develop all the others — deserves rigorous philosophical analysis. In its logical structure, it is a recursive argument about the instrumental value of time. If time is the fundamental scarce resource (not capital, not knowledge, not energy), then maximizing available time is meta-rationally optimal. Life extension, in this framework, does not compete with other technological projects — it is the prerequisite for all of them.&lt;/p&gt;
&lt;p&gt;This argument has a logical form analogous to what in computer science is called “meta-optimization”: before optimizing any specific objective function, optimize the optimization process itself. Before solving any specific scientific problem, solve the problem of having enough time to solve problems. It is elegant, but it has implicit premises that deserve scrutiny.&lt;/p&gt;
&lt;p&gt;The main premise is that more individual time translates into more civilizational progress. This is not obvious. Scientific revolutions often depend not on individuals working longer, but on new individuals bringing fresh perspectives. Kuhn argued that paradigms change not because practitioners of old paradigms are convinced, but because they die and are replaced by practitioners of new ones. Radical longevity could, paradoxically, slow scientific progress by entrenching paradigms and hierarchies.&lt;/p&gt;
&lt;p&gt;The counter-response is that rejuvenated people would not be old people living longer — they would be biologically young people with accumulated experience, combining cognitive plasticity with wisdom. But this is an empirical prediction about the cognitive effects of biological rejuvenation that remains untested.&lt;/p&gt;
&lt;h3 id=&quot;152-philosophical-pessimism-and-life-extension&quot;&gt;15.2 Philosophical Pessimism and Life Extension&lt;/h3&gt;
&lt;p&gt;There is a little-explored tension between philosophical pessimism (in the tradition of Schopenhauer, Cioran, Ligotti) and the project of life extension. If existence is fundamentally characterized by suffering, and if consciousness is the condition that makes suffering possible, then extending life is extending suffering. Schopenhauer would argue that the desire to live longer is precisely one more manifestation of the blind Will that perpetuates the cycle of desire and frustration.&lt;/p&gt;
&lt;p&gt;But there is a transhumanist response to Schopenhauer that is worth articulating: the project of life extension is not merely quantitative (more years) but qualitative (better quality of life in all years). If biological rejuvenation eliminates chronic pain, functional decrepitude, dementia, loss of autonomy — the most acute forms of suffering associated with aging — then the hedonic balance of existence improves, not worsens. We are not extending suffering; we are extending the period during which avoidable suffering is avoided.&lt;/p&gt;
&lt;p&gt;This response does not satisfy a consistent metaphysical pessimist. But it suggests that even within a pessimistic framework, the longevity project can be justified pragmatically: not as a metaphysical affirmation of the goodness of life, but as the engineering of reduced unnecessary suffering.&lt;/p&gt;
&lt;h3 id=&quot;153-personal-identity-on-the-horizon-of-radical-longevity&quot;&gt;15.3 Personal Identity on the Horizon of Radical Longevity&lt;/h3&gt;
&lt;p&gt;If a person lives 200, 500 or 1000 years — with repeated cycles of biological rejuvenation — in what sense is the person of year 1 the same person as the person of year 500? Human psychology, memories, values, relationships — all of this transforms profoundly over decades. Over centuries, the transformation would be radical.&lt;/p&gt;
&lt;p&gt;Derek Parfit argued that personal identity is not a binary relation (identical or non-identical) but a gradual one, based on psychological continuity and connectedness. On this analysis, a 500-year-old person would not be identical to the 20-year-old they once were — but would be connected to that person by an unbroken chain of gradual transformations. Just as an 80-year-old today is not “the same person” they were at 20 in any substantial sense, yet is connected to that person by continuity, a rejuvenated 500-year-old would maintain continuity without strict identity.&lt;/p&gt;
&lt;p&gt;AI complicates this question in new ways. If brain-computer interfaces, external digital memories, or even gradual transfer of neural substrate (Church’s “Ship of Theseus”) enter the equation, the boundaries of personal identity will dissolve even further. Radical longevity is not only a question of biology — it is a question of philosophy of mind, and AI is at once the tool that makes the extension possible and the technology that most deeply challenges our notions of identity.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xvi-the-state-of-the-art-in-2026-a-map-of-the-terrain&quot;&gt;XVI. The State of the Art in 2026: A Map of the Terrain&lt;/h2&gt;
&lt;h3 id=&quot;161-ongoing-clinical-trials&quot;&gt;16.1 Ongoing Clinical Trials&lt;/h3&gt;
&lt;p&gt;In January 2026, Life Biosciences confirmed the first targeted age-reversal attempt in human volunteers, using epigenetic reprogramming to treat eye disease. In parallel, senolytic drugs (such as dasatinib + quercetin) are in multiple phase II clinical trials for age-related conditions. Rapamycin and analogues are being tested for multiple geriatric indications. Rejuvenate Bio (co-founded by a former collaborator of Sinclair’s) is advancing longevity gene therapies, initially in dogs.&lt;/p&gt;
&lt;h3 id=&quot;162-ai-in-the-longevity-pharmaceutical-pipeline&quot;&gt;16.2 AI in the Longevity Pharmaceutical Pipeline&lt;/h3&gt;
&lt;p&gt;The market for AI-driven drug discovery is maturing rapidly. AI has graduated from point solutions to platform-scale discovery engines. For target-identification workflows, omics-based insights were combined with the expanded structural predictions of AlphaFold 3. OpenFold3 and Boltz-2 represent open-source alternatives that democratize capabilities previously restricted to large pharmaceutical companies. The AISB Network demonstrated that federated learning can create pharma-grade models without sharing proprietary data.&lt;/p&gt;
&lt;h3 id=&quot;163-critical-gaps&quot;&gt;16.3 Critical Gaps&lt;/h3&gt;
&lt;p&gt;Despite remarkable progress, critical gaps remain. The translation of results from mice to humans remains uncertain — differences in metabolism, time scale and environmental factors limit direct applicability. The long-term safety of partial cellular reprogramming is unknown. Variability among human individuals is vastly greater than among isogenic mouse strains. And regulatory frameworks do not keep pace with the speed of the science — the FDA still does not recognize aging as a treatable indication.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xvii-computational-neuroscience-and-brain-aging-the-last-frontier&quot;&gt;XVII. Computational Neuroscience and Brain Aging: The Last Frontier&lt;/h2&gt;
&lt;h3 id=&quot;171-the-brain-as-a-special-case&quot;&gt;17.1 The Brain as a Special Case&lt;/h3&gt;
&lt;p&gt;Of all the dimensions of aging, brain aging is at once the most consequential and the most resistant to intervention. The brain is the only organ whose function depends critically on a specific configuration of connections — the connectome — that encodes decades of learning, memories and personality. Replacing aging neurons, unlike replacing cardiomyocytes or hepatocytes, carries the risk of destroying the information that defines who we are.&lt;/p&gt;
&lt;p&gt;Church recognizes the brain as “the hardest part” of the rejuvenation program, but not as impossible. His proposal — artificially introducing stem cells that integrate into existing neuronal circuits, learn those circuits and gradually replace aging neurons — is a systems-engineering proposal that requires a computational understanding of the brain far beyond what we currently possess. For gradual neuronal replacement to preserve identity and function, it would be necessary to map the local connectome at synaptic resolution, understand how information is encoded in the topology of connections, and ensure that new neurons replicate not only the connections but the activity patterns of the neurons they replace.&lt;/p&gt;
&lt;h3 id=&quot;172-ai-as-a-bridge-to-the-neuroscience-of-aging&quot;&gt;17.2 AI as a Bridge to the Neuroscience of Aging&lt;/h3&gt;
&lt;p&gt;AI is advancing our understanding of brain aging on multiple fronts. Deep learning models applied to functional and structural magnetic resonance imaging data can detect patterns of brain aging with greater precision than clinical assessment. “Brain age” — the brain age predicted by AI from neuroimaging — has become an established biomarker, with brain age acceleration correlating with risk of dementia, cognitive decline and mortality.&lt;/p&gt;
&lt;p&gt;More deeply, AI is enabling the analysis of spatial transcriptomic data from the brain — mapping which genes are expressed in which brain regions, in which cell types, and how those patterns change with age. Church’s SAHA project, combining subcellular resolution with coverage of multiple organs, promises to provide the most detailed atlas ever built of how the brain ages at the molecular and cellular level.&lt;/p&gt;
&lt;p&gt;Graph neural networks applied to connectomics data — the mapping of the network of brain connections — can reveal how the topology of neuronal networks degrades with age. If certain topological properties (small-world efficiency, modularity, hub distribution) deteriorate in predictable and quantifiable ways, then AI could not only diagnose brain aging but guide interventions that restore optimal topological properties — whether through targeted brain stimulation, localized gene therapy, or, eventually, the gradual neuronal replacement Church proposes.&lt;/p&gt;
&lt;h3 id=&quot;173-brain-computer-interfaces-and-augmented-cognition&quot;&gt;17.3 Brain-Computer Interfaces and Augmented Cognition&lt;/h3&gt;
&lt;p&gt;On the more distant horizon, high-bandwidth brain-computer interfaces (BCIs) could complement biological approaches to brain rejuvenation with computational ones. If individual neurons can be monitored and stimulated with precision — as the BRAIN Initiative and Neuralink aspire to — then dysfunctional neuronal circuits could be computationally “supplemented”: a neural prosthesis that compensates for the loss of function while biological therapies restore the substrate.&lt;/p&gt;
&lt;p&gt;This is a speculative prospect, but not a fantastical one. Cochlear implants already realize a rudimentary version of this principle for the auditory system. Retinal prostheses do the same for vision. The extension to the cortical circuits involved in memory, attention and executive control — the functions most vulnerable to aging — is conceptually continuous, although technically formidable.&lt;/p&gt;
&lt;p&gt;AI is central to any implementation of cognitive neural prostheses: decoding patterns of neural activity, identifying which circuits are dysfunctional, computing the stimulation patterns that restore function, and adapting in real time to changes in brain state. Church’s “Ship of Theseus” — the gradual replacement of neuronal substrate — may ultimately combine biological replacement (new neurons) with computational supplementation (BCIs) in a hybrid approach that neither technology could achieve alone.&lt;/p&gt;
&lt;h3 id=&quot;174-the-final-convergence-biology-computation-and-consciousness&quot;&gt;17.4 The Final Convergence: Biology, Computation and Consciousness&lt;/h3&gt;
&lt;p&gt;There is a profound irony in the convergence of AI and longevity as applied to the brain: the same technology (AI) that we are using to understand and preserve biological consciousness is itself a form of information processing that raises questions about the nature of consciousness. If we manage to map, model and eventually restore neuronal circuits with enough fidelity to preserve memories and personality, what exactly are we preserving? The information? The substrate? Both?&lt;/p&gt;
&lt;p&gt;The answer to this question has direct practical implications for the brain-longevity program. If personal identity is constituted by information (connection patterns, synaptic strength, dynamic states), then any substrate that faithfully preserves that information preserves the person. If identity requires the original biological substrate, then only rejuvenation in situ (without replacement) is acceptable. Empirical science — informed and accelerated by AI — will have to adjudicate between these philosophical positions, not through abstract argument, but through experimentation: rejuvenating brains and observing whether the person is preserved.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xviii-conclusion-the-engineering-of-mortality&quot;&gt;XVIII. Conclusion: The Engineering of Mortality&lt;/h2&gt;
&lt;p&gt;We are witnessing what may be the most consequential technological convergence in human history: the application of artificial intelligence to the understanding and potential reversal of biological aging. George Church offers the genomic engineering of whole organisms. David Sinclair offers the restoration of youthful epigenetic information. Aubrey de Grey offers the systematic repair of accumulated damage. AI — in its multiple forms: deep learning, generative models, reinforcement learning, graph neural networks, transformers, multi-agent systems — offers the computational substrate without which none of these programs could scale to the real complexity of human biology.&lt;/p&gt;
&lt;p&gt;AlphaFold solved in a few years a problem that had resisted half a century. Epigenetic clocks based on deep learning are making aging measurable with unprecedented precision. Generative models are designing molecules that humans would never have conceived. Virtual screening systems are exploring chemical spaces larger than the number of atoms in the universe. Each of these achievements would be remarkable individually. Their convergence is potentially civilizational.&lt;/p&gt;
&lt;p&gt;None of the researchers discussed in this essay — neither Church, nor Sinclair, nor de Grey — promises certainty. Church suggests that 2050 would be plausible for longevity escape velocity. De Grey bets on a 50% probability by 2036. Sinclair projects age-reversal pills within a decade. The timelines may be wrong — historically, optimistic biotechnology has often underestimated the time required. But the direction is unmistakable: AI is transforming longevity science from a philosophical aspiration into an engineering program.&lt;/p&gt;
&lt;p&gt;The question that remains is not whether AI will contribute to the extension of human life — that is already happening, with every protein structure predicted, every molecule computationally designed, every epigenetic clock refined. The question is whether we are prepared — epistemically, ethically, institutionally — for the consequences of succeeding.&lt;/p&gt;
&lt;p&gt;If Church is right that biology has “gifts” — levers it can hand us to pull — then AI is the instrument that allows us to identify and operate those levers with unprecedented precision. And if Sinclair is right that cells retain a backup copy of youthful epigenetic information, then AI is the reading mechanism that can restore that backup at scale.&lt;/p&gt;
&lt;p&gt;The OpenAI-Retro Biosciences collaboration is perhaps the most potent symbol of this convergence: a generative AI company and a longevity company joining forces to redesign the fundamental proteins of cellular reprogramming, obtaining results that surpass billions of years of natural evolution. The RetroSOX and RetroKLF factors — each with more than 100 mutations relative to the wild-type proteins, working 50 times better — are a testament to what happens when computational power meets biological ambition.&lt;/p&gt;
&lt;p&gt;But the next three to five years of human clinical-trial data — not social-media enthusiasm, not billion-dollar valuations, not podcast predictions — will determine whether epigenetic reprogramming marks the beginning of a medical revolution or yet another chapter in the history of biotechnology’s irrational exuberance. Life Biosciences’ trials with ER-100, Retro’s tests with RTR242, the programs of Altos Labs and NewLimit — each of these will produce concrete data that will confirm or refute decades of promises.&lt;/p&gt;
&lt;p&gt;The role of AI in this period will be twofold. First, as an accelerator: compressing discovery timelines, optimizing protocols, identifying biomarkers, designing more efficient trials. Second, as an epistemic instrument: making visible what was invisible, measurable what was only estimable, tractable what was intractable. AlphaFold has not cured any disease directly — but by solving the protein structure prediction problem, it removed a bottleneck that limited all of structural biology and, by extension, all of drug discovery. Analogously, deep epigenetic clocks do not rejuvenate anyone directly — but by making aging precisely measurable, they remove the bottleneck that prevented the rapid evaluation of anti-aging therapies.&lt;/p&gt;
&lt;p&gt;Ultimately, what is at stake is not just years of life, but quality of life, cognitive capacity, functional autonomy and human dignity in old age. AI for longevity science is not an exercise in Promethean hubris — it is, when conducted with rigor, caution and equity, one of the most deeply humanistic undertakings that technological civilization can pursue.&lt;/p&gt;
&lt;p&gt;The history of science is, to a large extent, the history of instruments that expand human perception: telescopes that revealed planets, microscopes that revealed cells, sequencers that revealed genomes. Artificial intelligence is the next instrument in that lineage — but with a qualitative difference: it does not merely reveal what exists, but designs what could exist. It does not merely measure aging, but conceives solutions to reverse it. It does not merely observe biology, but redesigns it.&lt;/p&gt;
&lt;p&gt;If the convergence of AI and longevity science produces even a fraction of what its most cautious proponents project — not thousand-year lives, but perhaps two or three more decades of healthy life for most humans — the impact will be civilizationally transformative. And if it produces what its boldest proponents imagine — longevity escape velocity, the obsolescence of aging as a cause of death — then we will have rewritten the human condition more deeply than any previous technology, including agriculture, writing and electricity.&lt;/p&gt;
&lt;p&gt;In both scenarios, AI will have been the condition of possibility. Not the only condition — experimental biology, clinical infrastructure, political will, and ethical and regulatory frameworks are all equally necessary. But the condition without which the solution space would remain unnavigable, biological complexity would remain intractable, and the project of life extension would remain what it has been for millennia: a recurring dream of the species, articulated in myths and religions, but never translated into engineering.&lt;/p&gt;
&lt;p&gt;That translation is under way. And what happens in the coming decades will determine whether our generation is remembered as the one that witnessed this transition or as the one that led it.&lt;/p&gt;
&lt;p&gt;Aging is, by any metric, the greatest cause of human suffering on the planet — killing more than 100,000 people a day, every day, and degrading the quality of life of billions. Each year of delay in developing effective therapies represents 37 million additional deaths that could, in principle, have been prevented. The urgency is not rhetorical — it is arithmetic.&lt;/p&gt;
&lt;p&gt;Artificial intelligence, by making the problem computationally tractable, has transformed longevity science from a discipline that could only describe aging into one that can, potentially, reverse it. That potential remains exactly that — potential. But for the first time in history, the distance between potential and realization is being measured in years and decades, not in centuries and millennia. And the tool that is compressing that distance is the same one that is revolutionizing every other domain of human knowledge: the ability of machines to find patterns, generate solutions and optimize processes at scales that radically transcend the biological cognition that created them.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;selected-references&quot;&gt;Selected References&lt;/h2&gt;
&lt;ol&gt;
&lt;li&gt;Sinclair, D.A. et al. “Reprogramming to recover youthful epigenetic information and restore vision.” &lt;em&gt;Nature&lt;/em&gt; 588, 124–129 (2020).&lt;/li&gt;
&lt;li&gt;Yang, J.-H. et al. “Loss of epigenetic information as a cause of mammalian aging.” &lt;em&gt;Cell&lt;/em&gt; 186(2), 305–326.e27 (2023).&lt;/li&gt;
&lt;li&gt;Jumper, J. et al. “Highly accurate protein structure prediction with AlphaFold.” &lt;em&gt;Nature&lt;/em&gt; 596, 583–589 (2021).&lt;/li&gt;
&lt;li&gt;Abramson, J. et al. “Accurate structure prediction of biomolecular interactions with AlphaFold 3.” &lt;em&gt;Nature&lt;/em&gt; 630, 493–500 (2024).&lt;/li&gt;
&lt;li&gt;de Grey, A.D.N.J. “Escape velocity: Why the prospect of extreme human life extension matters now.” &lt;em&gt;PLoS Biology&lt;/em&gt; (2004).&lt;/li&gt;
&lt;li&gt;Horvath, S. “DNA methylation age of human tissues and cell types.” &lt;em&gt;Genome Biology&lt;/em&gt; 14, R115 (2013).&lt;/li&gt;
&lt;li&gt;Ying, K. et al. “Causality-enriched epigenetic age uncouples damage and adaptation.” &lt;em&gt;Nature Aging&lt;/em&gt; (2024).&lt;/li&gt;
&lt;li&gt;de Magalhães, J.P. et al. AltumAge: “A pan-tissue DNA-methylation epigenetic clock based on deep learning.” &lt;em&gt;npj Aging&lt;/em&gt; (2022).&lt;/li&gt;
&lt;li&gt;Raj, A.P.M.S. et al. “Genetic signatures of exceptional longevity.” &lt;em&gt;Human Genomics&lt;/em&gt; (2025).&lt;/li&gt;
&lt;li&gt;Church, G.M. &amp;#x26; Regis, E. &lt;em&gt;Regenesis: How Synthetic Biology Will Reinvent Nature and Ourselves.&lt;/em&gt; Basic Books (2012).&lt;/li&gt;
&lt;li&gt;de Grey, A.D.N.J. &amp;#x26; Rae, M. &lt;em&gt;Ending Aging.&lt;/em&gt; St. Martin’s Press (2007).&lt;/li&gt;
&lt;li&gt;Sinclair, D.A. &lt;em&gt;Lifespan: Why We Age — and Why We Don’t Have To.&lt;/em&gt; Atria Books (2019).&lt;/li&gt;
&lt;li&gt;Levy, J.J. et al. “Insights to aging prediction with AI based epigenetic clocks.” &lt;em&gt;Epigenomics&lt;/em&gt; 17(1), 49–57 (2025).&lt;/li&gt;
&lt;li&gt;Scientific Reports. “Interpretable deep learning of single-cell and epigenetic data reveals novel molecular insights in aging.” Vol. 15, 5048 (2025).&lt;/li&gt;
&lt;li&gt;Nature. “An AlphaFold 4: scientists marvel at DeepMind drug spin-off’s exclusive new AI.” 651, 18 (2026).&lt;/li&gt;
&lt;/ol&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>longevity</category><category>artificial intelligence</category><category>biotechnology</category><category>science</category><category>philosophy</category><category>featured essays</category></item><item><title>Dropping the Purism: What Language Models Have Conquered, What Still Eludes Them, and Why General Intelligence Is a Continent No Species — Biological or Digital — Has Fully Explored</title><link>https://matheussorancophilomathes.com.br/en/essays/dropping-the-purism-what-language-models-have-conquered-what-still-eludes-them/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/dropping-the-purism-what-language-models-have-conquered-what-still-eludes-them/</guid><description>Linguistic competence in language models is a genuine achievement of intelligence, but only one region of a multidimensional topography that crows, chimpanzees and octopuses explore in ways LLMs cannot: action, grounding, interaction, development.</description><pubDate>Wed, 01 Apr 2026 23:59:08 GMT</pubDate><content:encoded>&lt;h2 id=&quot;i-prologue-astonishment-and-sobriety&quot;&gt;I. Prologue: Astonishment and Sobriety&lt;/h2&gt;
&lt;p&gt;There is a productive tension running through the contemporary debate on artificial intelligence — a tension that is rarely named with precision, but that contaminates almost every public discussion of the subject. On one side are those who look at the linguistic capabilities of large language models (LLMs) and declare, with almost messianic enthusiasm, that artificial general intelligence is within reach, that language &lt;em&gt;is&lt;/em&gt; thought and that, therefore, mastering language is mastering cognition. On the other side are the skeptics — many of them from cognitive biology, robotics and the philosophy of mind — who dismiss LLMs as “stochastic parrots,” sophisticated automata that simulate understanding without understanding anything at all.&lt;/p&gt;
&lt;p&gt;Both positions are intellectually lazy.&lt;/p&gt;
&lt;p&gt;The thesis of this essay is that the linguistic capability of modern language models is a genuine achievement of intelligence — not a trivial simulation, not a trick of statistical mirrors, but a computational accomplishment that deserves serious epistemological recognition. At the same time, this achievement illuminates, by contrast, the vastness of what general intelligence actually involves: capacities for action in the world, learning from direct experience, interventional causal reasoning, real-time adaptation to unpredictable physical environments — domains in which animals such as crows, chimpanzees, octopuses and even ants display a sophistication that LLMs lack and that no purely linguistic architecture, however many parameters it contains, can reach on its own.&lt;/p&gt;
&lt;p&gt;General intelligence is not a one-dimensional spectrum with a “dumb” end and an “intelligent” end. It is a multidimensional topography — a space with independent axes that include symbolic manipulation, spatial reasoning, social cognition, motor planning, causal learning, metacognition, temporal adaptation and many others. Different species and different artificial systems occupy distinct regions of this topography. LLMs occupy an extraordinary region on the axis of language and abstract symbolic manipulation. Crows occupy an extraordinary region on the axes of causal planning, tool use and sequential cognition. Chimpanzees dominate social cognition, visual working memory and intragroup political negotiation. Octopuses are masters of distributed motor control and morphological adaptation.&lt;/p&gt;
&lt;p&gt;None of these agents is “more intelligent” than the others in absolute terms. Each is brilliant in its region of the topography and limited in the rest. The question of artificial general intelligence is, precisely, the question of how to integrate these regions into a single coherent system — and that is the hardest question in cognitive science, in AI and, possibly, in human civilization.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;ii-what-language-models-really-do&quot;&gt;II. What Language Models Really Do&lt;/h2&gt;
&lt;h3 id=&quot;the-pattern-machine-that-exceeded-the-sum-of-its-parts&quot;&gt;The pattern machine that exceeded the sum of its parts&lt;/h3&gt;
&lt;p&gt;To understand why the linguistic capability of LLMs is an achievement of intelligence — and not a mere statistical trick — one must first understand what these systems really do, without mystification and without deflation.&lt;/p&gt;
&lt;p&gt;A language model, in its most basic formulation, is a system that learns the conditional probability distribution of the next word (or token) given a context of previous words. The goal of training is to minimize the &lt;em&gt;cross-entropy loss&lt;/em&gt; between the model’s predictions and the actual tokens observed in a massive corpus of text. Formally, for a sequence of tokens &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x_1, x_2, \ldots, x_T&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;minner&quot;&gt;…&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3283em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.1389em;&quot;&gt;T&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, the model learns parameters &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\theta&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; that maximize:&lt;/p&gt;
&lt;p&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;script&quot;&gt;L&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/msubsup&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;∣&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;\mathcal{L}(\theta) = \sum_{t=1}^{T} \log P_\theta(x_t \mid x_1, \ldots, x_{t-1})&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathcal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.2809em;vertical-align:-0.2997em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;position:relative;top:0em;&quot;&gt;∑&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9812em;&quot;&gt;&lt;span style=&quot;top:-2.4003em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;t&lt;/span&gt;&lt;span class=&quot;mrel mtight&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.2029em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.1389em;&quot;&gt;T&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2997em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;lo&lt;span style=&quot;margin-right:0.0139em;&quot;&gt;g&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2806em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;t&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;∣&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;minner&quot;&gt;…&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;t&lt;/span&gt;&lt;span class=&quot;mbin mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.2083em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;Described this way, the task seems prosaic — almost banal. Predict the next word. Any n-gram model from the 1990s did this, with mediocre results. What changed?&lt;/p&gt;
&lt;p&gt;What changed is that, when this task is carried out at sufficient scale — with corpora of trillions of tokens, hundreds of billions of parameters and the &lt;em&gt;Transformer&lt;/em&gt; architecture with multi-head attention — the system develops internal representations that cannot be reduced to mere memorization or matching of superficial patterns. Research in &lt;em&gt;mechanistic interpretability&lt;/em&gt; (Elhage et al., 2022; Olsson et al., 2022; Nanda et al., 2023) has shown that &lt;em&gt;Transformer&lt;/em&gt; models form specialized internal circuits: &lt;em&gt;induction heads&lt;/em&gt; that perform abstract pattern matching, compositional reasoning circuits that decompose multi-step problems, and representations of semantic relations that preserve relational structure in a way that mirrors, in many respects, human conceptual relations.&lt;/p&gt;
&lt;p&gt;The work of Kenneth Li et al. (2023) on the Othello-GPT model is particularly revealing. A &lt;em&gt;Transformer&lt;/em&gt; trained only to predict sequences of Othello moves — with no information about the board, the rules of the game, or the notion of “position” — spontaneously develops an internal representation of the board state. The model was not given this representation; it built it as a by-product of optimizing prediction. This is not memorization. This is the emergence of representational structure from predictive pressure — a phenomenon that deserves to be called &lt;em&gt;understanding&lt;/em&gt;, even if in a provisional and qualified sense.&lt;/p&gt;
&lt;h3 id=&quot;beyond-prediction-reasoning-analogy-and-composition&quot;&gt;Beyond prediction: reasoning, analogy and composition&lt;/h3&gt;
&lt;p&gt;The question of whether LLMs “merely predict the next word” is, in a way, analogous to the question of whether the human brain “merely transmits electrochemical impulses between neurons.” The description is mechanistically correct and functionally empty. What matters is not the primitive mechanism, but the capabilities that emerge from orchestrating that mechanism at scale and with the right architecture.&lt;/p&gt;
&lt;p&gt;Modern LLMs demonstrate capabilities that, if observed in any other system — biological or artificial — would immediately be recognized as intelligent:&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Analogical reasoning.&lt;/strong&gt; When an LLM correctly answers that “Tokyo is to Japan as Paris is to France,” it is not retrieving this information from a pre-stored table. It is computing relations in a high-dimensional semantic space, and this computation is structurally isomorphic to processes of analogical reasoning that cognitive psychologists have studied for decades.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Semantic composition.&lt;/strong&gt; LLMs can interpret and generate sentences that have never existed in any training corpus — sentences with genuinely novel semantic combinations. “The digital twilight spilled pixels melancholically over the terminal window” exists in no training text, but an LLM can process this sentence, extract its metaphorical meaning and even continue the style. This implies productive composition — the ability to generate new meaning from the combination of smaller units — which is one of the central criteria of linguistic competence according to Chomsky and the generative tradition.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Translation between domains.&lt;/strong&gt; LLMs can translate not only between languages, but between registers, styles, technical domains and even levels of formality. Explaining a concept of quantum physics in language accessible to a ten-year-old, or translating a legal argument into colloquial language, requires a form of understanding — the ability to extract the &lt;em&gt;deep structure&lt;/em&gt; of a content and re-express it in a different &lt;em&gt;surface structure&lt;/em&gt;.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Chain reasoning.&lt;/strong&gt; Presented with problems that require multiple steps of reasoning, models equipped with &lt;em&gt;chain-of-thought prompting&lt;/em&gt; (Wei et al., 2022) demonstrate the ability to decompose problems, hold intermediate state and reach conclusions that are not accessible through simple statistical shortcuts. Models with extended reasoning — such as OpenAI’s &lt;em&gt;o1&lt;/em&gt; and &lt;em&gt;o3&lt;/em&gt; families and Anthropic’s models with &lt;em&gt;extended thinking&lt;/em&gt; — go further still, devoting explicit computation to deliberation before producing an answer.&lt;/p&gt;
&lt;p&gt;None of these capabilities is perfect. LLMs make reasoning errors, suffer from hallucinations, fail at problems that require truly long reasoning, and lack direct perceptual grounding. But the imperfection of a cognitive capability does not invalidate it as a cognitive capability. Four-year-old children also make systematic reasoning errors, fail Piagetian conservation tasks and confuse appearance with reality — and no one argues that four-year-olds are not intelligent.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;iii-language-as-a-dimension-of-intelligence-the-affirmative-argument&quot;&gt;III. Language as a Dimension of Intelligence: The Affirmative Argument&lt;/h2&gt;
&lt;h3 id=&quot;what-the-philosophy-of-mind-tells-us&quot;&gt;What the philosophy of mind tells us&lt;/h3&gt;
&lt;p&gt;The question of whether linguistic competence constitutes “real intelligence” cannot be answered without first facing a more fundamental one: what is intelligence?&lt;/p&gt;
&lt;p&gt;The history of psychometrics and of the philosophy of mind offers us a rich but fragmented tradition. Charles Spearman postulated the &lt;em&gt;g&lt;/em&gt; factor — a general cognitive capacity underlying all specific intellectual performances. Louis Thurstone disputed this, proposing seven independent primary abilities. Howard Gardner expanded it to eight (or nine) multiple intelligences. Robert Sternberg offered his triarchic theory — analytical, creative and practical. More recently, François Chollet proposed, in the paper “On the Measure of Intelligence” (2019), a formal definition of intelligence as &lt;em&gt;skill-acquisition efficiency&lt;/em&gt;, emphasizing generalization to new tasks with minimal prior experience — a definition that directly inspires the ARC (Abstraction and Reasoning Corpus) benchmark.&lt;/p&gt;
&lt;p&gt;Whatever framework one adopts, practically everyone recognizes that linguistic competence is, at the very least, a significant &lt;em&gt;component&lt;/em&gt; of human intelligence. Language is not merely a communication channel; it is a cognitive scaffold — a technology of thought that makes possible:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;Abstraction and categorization.&lt;/strong&gt; Language enables the creation of abstract categories (“justice,” “entropy,” “recursion”) that transcend immediate perceptual experience. Without language, thought would be tied to the concrete and the present — to what can be directly perceived and manipulated. Vygotsky argued extensively that language does not merely express thought, but &lt;em&gt;constitutes&lt;/em&gt; it at its most abstract levels.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Memory and cultural transmission.&lt;/strong&gt; Language enables the encoding of experience in transmissible and cumulative formats. Without it, each generation would start from scratch. Science, philosophy, law, engineering — the whole apparatus of civilization — depends on the capacity to encode, store, transmit and recombine linguistic knowledge.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Hypothetical and counterfactual reasoning.&lt;/strong&gt; Language makes it possible to think about what &lt;em&gt;is not&lt;/em&gt; — about possibilities, alternative futures, hypothetical scenarios. “What would happen if…” is a linguistic construction with no direct equivalent in non-verbal cognition, and it is the foundation of planning, experimental science and fiction.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Metacognition.&lt;/strong&gt; Language makes it possible to think about one’s own thinking — to narrate cognitive processes internally, evaluate them, correct them. The “inner dialogue” that psychologists identify as crucial for cognitive self-regulation is, essentially, language directed at oneself.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;If language is all this for human intelligence, then a system that masters language with the proficiency of a modern LLM possesses, at the very least, a significant fraction of what we call intelligence. Not all intelligence. Not general intelligence. But a non-trivial slice — an achievement that should not be diminished by the fact that other slices are still absent.&lt;/p&gt;
&lt;h3 id=&quot;the-stochastic-parrot-argument-and-its-limitations&quot;&gt;The stochastic parrot argument and its limitations&lt;/h3&gt;
&lt;p&gt;The “stochastic parrot” metaphor, popularized by Emily Bender and colleagues in 2021, argues that LLMs do not understand the language they produce — that they merely reproduce statistical patterns without underlying meaning. It is a powerful metaphor, but one that suffers from at least three serious problems.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;First, the problem of the criterion.&lt;/strong&gt; If we demand “genuine understanding” as the criterion for attributing linguistic intelligence, we need an operational criterion for what constitutes genuine understanding — and no criterion proposed to date manages to distinguish, in a non-arbitrary way, between human understanding and the performance of the best LLMs on highly complex linguistic tasks. The Turing test is not sufficient (it is easily gamed). Reading-comprehension tests are insufficient (LLMs already surpass them). The objection always retreats to something like “but the model doesn’t &lt;em&gt;really&lt;/em&gt; understand” — and that “really” remains undefined. We are facing what Daniel Dennett would call an &lt;em&gt;intuition pump&lt;/em&gt; with no empirical content: a strong intuition that something is missing, with no ability to specify what.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Second, the problem of reverse projection.&lt;/strong&gt; If human understanding of language is implemented by neurons firing patterns of electrochemical activation, it is not immediately clear why activation patterns in a &lt;em&gt;Transformer&lt;/em&gt; could not implement something functionally analogous. The objection that “it’s just statistics” projects a dichotomy between mechanism and meaning that may not be justified. After all, it is possible that understanding &lt;em&gt;is&lt;/em&gt;, at the implementational level, a very sophisticated form of statistical pattern processing — in brains as in artificial neural networks.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Third, the empirical problem.&lt;/strong&gt; The predictions of the stochastic parrot model are empirically testable, and many of them have failed. If LLMs were mere reproducers of superficial patterns, they should not be able to solve new problems that require composing concepts never combined in training, to extract abstract principles from a few examples (&lt;em&gt;in-context learning&lt;/em&gt;), or to exhibit emergent behavior that was neither programmed nor predicted. But they do all these things — imperfectly, unevenly, but demonstrably.&lt;/p&gt;
&lt;p&gt;This does not mean the stochastic parrot argument has no merit. It does. It points to something real: the linguistic competence of LLMs, however impressive, lacks anchoring in the world — &lt;em&gt;grounding&lt;/em&gt; — and this lack has profound cognitive consequences. But the argument errs in concluding, from this lack, that no intelligence is present. The correct conclusion is subtler: what is present is a form of intelligence that is partial, specialized and, in certain important respects, ungrounded.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;iv-the-intelligence-that-does-not-speak-action-body-and-world&quot;&gt;IV. The Intelligence That Does Not Speak: Action, Body and World&lt;/h2&gt;
&lt;h3 id=&quot;the-gap-between-saying-and-doing&quot;&gt;The gap between saying and doing&lt;/h3&gt;
&lt;p&gt;Here we come to the heart of the problem. LLMs are extraordinarily competent at symbolic manipulation — but general intelligence, as observed in the most cognitively sophisticated biological organisms, involves far more than symbols. It involves &lt;em&gt;action in the physical world&lt;/em&gt;: perceiving dynamic environments, planning motor sequences, manipulating objects, adapting to unexpected outcomes, learning causally from interventions — and doing all of this in real time, under energetic and temporal pressure.&lt;/p&gt;
&lt;p&gt;Consider the following asymmetry:&lt;/p&gt;
&lt;p&gt;An LLM can generate a ten-thousand-word essay on the tool-use strategies of New Caledonian crows, correctly citing the studies of Hunt (1996), Weir et al. (2002) and Taylor et al. (2007). It can describe in detail how these crows bend wire to make hooks, how they fabricate compound tools from multiple materials, how they plan sequences of tool use of up to three steps.&lt;/p&gt;
&lt;p&gt;But the LLM itself cannot bend a wire.&lt;/p&gt;
&lt;p&gt;Not because of a hardware limitation — after all, it has no body. The point is deeper: even if coupled to a robotic body with precise actuators and high-fidelity sensors, the LLM, &lt;em&gt;qua&lt;/em&gt; language model, does not possess the internal representations needed to guide fluid physical action in the world. It has no dynamic model of the physical world, no representation of forces and torques, no sensorimotor loop that allows a crow to feel the resistance of a twig and adjust its beak within milliseconds.&lt;/p&gt;
&lt;p&gt;This asymmetry is not a technical curiosity. It points to a fundamental difference in the &lt;em&gt;nature&lt;/em&gt; of the intelligence involved. Linguistic intelligence operates in the space of symbols — a discrete, compositional, temporally slow, deliberative space. The intelligence of action operates in physical space — a continuous, dynamic, temporally urgent, sensorimotor space. Both are forms of intelligence. Both involve representation, inference and adaptation. But the computational substrates and the selective pressures that shaped them are radically different.&lt;/p&gt;
&lt;h3 id=&quot;the-crow-and-the-meta-tool&quot;&gt;The crow and the meta-tool&lt;/h3&gt;
&lt;p&gt;New Caledonian crows (&lt;em&gt;Corvus moneduloides&lt;/em&gt;) are possibly the most sophisticated tool engineers in the non-human animal world. What makes their behavior particularly notable for the discussion of intelligence is the presence of several capabilities that LLMs lack:&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Projective fabrication.&lt;/strong&gt; Crows do not merely use tools they find — they make them. More impressively, they shape tools in ways that anticipate future use: they bend wire into hook shapes, they cut pandanus leaves into stepped forms that serve as efficient probes. This implies an internal representation of the goal (extracting a grub from a hole), of the means (a tool of a specific shape), and of the causal relation between the shape of the tool and its effectiveness at the task. The crow does not need language for this. It operates with what psychologists call &lt;em&gt;imagistic representation&lt;/em&gt; — a mental model of the physical world that allows simulation and planning without linguistic symbolism.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Sequential planning.&lt;/strong&gt; In controlled experiments (Taylor et al., 2007; Wimpenny et al., 2009), crows demonstrated the ability to solve meta-tool problems — using one tool to obtain another tool to obtain food. In some cases, sequences of three tools were used. This is genuine prospective planning: the animal must mentally represent a causal chain of three links, in which the result of each action is a precondition for the next. This capacity is analogous to what in AI we call &lt;em&gt;hierarchical task planning&lt;/em&gt; — and crows carry it out without language, without formal training, often on the first attempt.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Innovation.&lt;/strong&gt; Betty, the crow made famous in the studies of Weir et al. (2002), spontaneously bent a straight wire into a hook to retrieve a bucket of food from a vertical tube — a behavior that had never been observed and could not have been learned by imitation, since wire does not exist in the species’ natural habitat. This is genuine innovation: the generation of a novel behavioral solution to a problem never before encountered, from reasoning about the physical properties of the available materials.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Causal learning by intervention.&lt;/strong&gt; Crows learn about the world not only by observation, but by active intervention — manipulating objects, testing implicit hypotheses, observing results and updating their behavior. This is, functionally, the experimental method — the same logic that underlies science, operating in an eight-gram brain without a single word.&lt;/p&gt;
&lt;p&gt;When we compare these capabilities with those of an LLM, the asymmetry is striking. The LLM can &lt;em&gt;describe&lt;/em&gt; all of these capabilities in sophisticated language. But it cannot &lt;em&gt;execute&lt;/em&gt; them. It cannot plan a sequence of physical actions, anticipate the mechanical properties of a material, improvise a solution to a new physical problem, or learn from direct interaction with the world. Its intelligence is deeply descriptive and declarative; the crow’s is deeply operative and procedural.&lt;/p&gt;
&lt;h3 id=&quot;the-chimpanzee-and-politics-without-words&quot;&gt;The chimpanzee and politics without words&lt;/h3&gt;
&lt;p&gt;If crows are the engineers of the animal world, chimpanzees (&lt;em&gt;Pan troglodytes&lt;/em&gt;) are the politicians. Their social and cognitive capabilities offer an equally instructive contrast with LLMs.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Social cognition and theory of mind.&lt;/strong&gt; Chimpanzees demonstrate sophisticated forms of understanding of the mental states of other individuals. Studies by Hare et al. (2001) and Call &amp;#x26; Tomasello (2008) showed that subordinate chimpanzees track what a dominant individual can or cannot see, adjusting their foraging behavior accordingly. They practice tactical deception — hiding food from rivals, suppressing excitement vocalizations in the presence of dominants, using distractions to divert attention. This requires modeling the perceptual and intentional states of other agents — what philosophers call &lt;em&gt;theory of mind&lt;/em&gt; and what, in AI, remains an open problem of formidable difficulty.&lt;/p&gt;
&lt;p&gt;It is true that LLMs demonstrate a certain capacity to reason about mental states in linguistically described scenarios — for example, solving variants of the Sally-Anne test when presented in text. But there is a crucial difference: the LLM reasons about mental states &lt;em&gt;described in text&lt;/em&gt;; the chimpanzee perceives and models mental states &lt;em&gt;in real time, in dynamic social interactions, with immediate consequences for its survival and status&lt;/em&gt;. The computational, temporal and ecological pressure is of a completely different nature.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Visual working memory.&lt;/strong&gt; The study by Inoue &amp;#x26; Matsuzawa (2007) is one of the most surprising results in comparative cognition. Young chimpanzees, trained on a sequential numerical memory task (in which digits are briefly displayed and then masked, and the subject must touch the positions in numerical order), consistently outperformed adult humans — with exposure times of only 210 milliseconds. This suggests a visuospatial working-memory capacity that, in at least one specific dimension, exceeds the human one. LLMs, operating exclusively on sequential text, have nothing analogous.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Culture and social transmission.&lt;/strong&gt; Different chimpanzee populations possess distinct behavioral traditions — different ways of using tools, different hunting techniques, different hygiene and greeting practices — that are socially transmitted across generations. Whiten et al. (1999) documented at least 39 cultural behaviors in wild chimpanzees. This is culture — cumulative social learning — operating without articulated language.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Negotiation and coalition.&lt;/strong&gt; De Waal (1982, 2007) extensively documented the power dynamics of chimpanzee communities: alliance formation, post-conflict reconciliation, exchange of favors, punishment of betrayers, manipulation of third parties. These interactions involve long-term strategic planning, modeling of multiple simultaneous agents, and evaluation of expected utility in complex social scenarios — capabilities that map directly onto problems of multi-agent game theory that remain challenging in AI.&lt;/p&gt;
&lt;h3 id=&quot;the-octopus-intelligence-without-centralization&quot;&gt;The octopus: intelligence without centralization&lt;/h3&gt;
&lt;p&gt;The octopus (&lt;em&gt;Octopus vulgaris&lt;/em&gt; and related species) deserves special mention because it illustrates a form of intelligence so radically different that it challenges our intuitions about what cognition can be. With two thirds of its neurons located in the arms — not in the central brain — the octopus implements a distributed cognitive architecture that has no parallel in vertebrates. Each arm has significant sensorimotor autonomy: it can explore, manipulate objects, solve local problems, and coordinate with other arms without constant central supervision.&lt;/p&gt;
&lt;p&gt;Octopuses demonstrate observational learning (Fiorito &amp;#x26; Scotto, 1992), tool use (Finn et al., 2009 — carrying coconut shells for future use as shelter), mechanical puzzle solving, complex spatial navigation, and even behaviors suggestive of play — a classic indicator of cognitive complexity.&lt;/p&gt;
&lt;p&gt;What the octopus teaches us about intelligence is that it does not require a single architectural blueprint. Intelligence can be centralized (as in primates) or distributed (as in octopuses). It can operate in symbolic spaces (as in humans and LLMs) or in purely sensorimotor spaces (as in many invertebrates). Intelligence is a design space, not a single blueprint.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;v-the-topography-of-intelligence-a-conceptual-framework&quot;&gt;V. The Topography of Intelligence: A Conceptual Framework&lt;/h2&gt;
&lt;h3 id=&quot;axes-not-ladders&quot;&gt;Axes, not ladders&lt;/h3&gt;
&lt;p&gt;The dominant metaphor for intelligence in popular culture — and, unfortunately, in much technical discussion — is the ladder: a linear progression from “less intelligent” to “more intelligent,” with human beings comfortably at the top and AI systems somewhere below, climbing. This metaphor is deeply misleading.&lt;/p&gt;
&lt;p&gt;A more faithful representation of reality is that of a multidimensional space — what I am calling the &lt;em&gt;topography of intelligence&lt;/em&gt;. In this space, each axis represents a relatively independent cognitive capability, and each agent (biological or artificial) is a point (or, more precisely, a probabilistic cloud) in that space. Without claiming exhaustiveness, we can identify at least the following axes:&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;1. Symbolic and linguistic manipulation.&lt;/strong&gt; The ability to operate with discrete symbols — words, numbers, formulas, code — composing complex meanings from simple units. LLMs are exceptional on this axis. So are humans, but with a much lower processing bandwidth (we cannot process millions of tokens per hour). Most non-human animals operate with limited forms of symbolic communication: bee dances, alarm calls in vervets (Seyfarth et al., 1980), dialects in cetaceans.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;2. Causal and interventional reasoning.&lt;/strong&gt; The ability to understand cause-and-effect relations and, crucially, to intervene in the world to test and explore those relations. Crows, chimpanzees and humans are strong on this axis. LLMs have some capacity for verbal causal reasoning, but cannot perform causal interventions in the world and therefore cannot learn causally from direct experience. Judea Pearl distinguishes three levels of causal cognition — association, intervention and counterfactuals — and has argued that LLMs operate predominantly at the first level, with limited forays into the third (via language), but without genuine access to the second.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;3. Spatial cognition and navigation.&lt;/strong&gt; The ability to represent three-dimensional space, navigate environments, mentally rotate objects, plan routes. Many animals — from ants to pigeons to cetaceans — possess extraordinary spatial navigation capabilities. Homing pigeons navigate thousands of kilometers using a combination of the magnetic field, solar position and &lt;em&gt;path integration&lt;/em&gt;. Bees communicate distances and directions with precision via the waggle dance. LLMs have no native spatial representation — their forays into this domain depend on the linguistic encoding of spatial information, which is inherently poorer than direct visuospatial representation.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;4. Social cognition and theory of mind.&lt;/strong&gt; The ability to model the mental states, intentions and knowledge of other agents. Chimpanzees, dolphins, elephants and corvids demonstrate sophisticated forms of social cognition. LLMs can simulate reasoning about mental states when these are described linguistically, but they lack the real-time social perception that allows a chimpanzee to read a rival’s body posture and adjust its strategy within milliseconds.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;5. Motor control and dexterity.&lt;/strong&gt; The ability to plan and execute complex motor sequences, coordinate multiple effectors, and adapt movements in real time to perturbations. This dimension is where animals shine most unambiguously: the flight of a hummingbird, the leap of a jumping spider, a chimpanzee’s manipulation of tools. LLMs lack this capability (by definition, being purely computational systems). Even state-of-the-art robotic systems with &lt;em&gt;end-to-end learning&lt;/em&gt; (such as those from Chelsea Finn’s lab at Stanford, or the work of Sergey Levine at Berkeley) remain far short of animal dexterity and adaptability.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;6. Few-shot learning and generalization.&lt;/strong&gt; The ability to acquire new skills from a few examples — what Chollet emphasizes as central to intelligence. Here the picture is mixed: LLMs demonstrate impressive &lt;em&gt;in-context learning&lt;/em&gt; in linguistic domains (Brown et al., 2020), but their ability to generalize to truly new domains is limited. Animals demonstrate &lt;em&gt;one-shot learning&lt;/em&gt; in ecological contexts — a rat that eats a novel food and then falls ill will avoid that food forever (Garcia &amp;#x26; Koelling, 1966), and crows that observe a human using an object as a tool can replicate the behavior within minutes.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;7. Metacognition and uncertainty.&lt;/strong&gt; The ability to assess one’s own confidence, recognize the limits of one’s knowledge, and adjust behavior accordingly. There is evidence of metacognition in rhesus monkeys (Smith et al., 2003), dolphins (Smith et al., 1995) and crows. Well-calibrated modern LLMs can, to some degree, modulate their expression of confidence, but they still suffer from systematic &lt;em&gt;overconfidence&lt;/em&gt; and hallucination — indicating that their metacognition is, at best, rudimentary.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;8. Temporal adaptation and continual learning.&lt;/strong&gt; The ability to learn continuously over time, integrating new experiences with prior knowledge without forgetting the past (the problem of &lt;em&gt;catastrophic forgetting&lt;/em&gt;). Animals do this naturally — it is the basic function of biological memory. LLMs, once trained, are static: they do not learn from new interactions (except in limited scenarios of &lt;em&gt;fine-tuning&lt;/em&gt; and &lt;em&gt;in-context learning&lt;/em&gt; within a session).&lt;/p&gt;
&lt;h3 id=&quot;where-each-agent-stands&quot;&gt;Where each agent stands&lt;/h3&gt;
&lt;p&gt;If we visualize this topography as a radar diagram (with each axis as a spoke), the profiles of the different agents would be dramatically distinct:&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Modern LLMs&lt;/strong&gt; would have very high values on axis 1 (symbolic manipulation), moderate on axes 2 (verbal causal reasoning), 4 (simulated social cognition) and 6 (linguistic in-context learning), and low to null on axes 3, 5, 7 and 8.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Crows&lt;/strong&gt; would have high values on axes 2, 3, 5 and 6, moderate on axis 4, and low on axis 1.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Chimpanzees&lt;/strong&gt; would have high values on axes 2, 4 and 5, moderate on axes 3, 6 and 7, and low on axis 1 (with some rudimentary symbolic capacity).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Humans&lt;/strong&gt; would have high to very high values on &lt;em&gt;all&lt;/em&gt; axes — which is probably the most precise definition of “general intelligence” we can offer: not the maximum on any individual axis, but an exceptionally broad and reasonably high coverage of all of them.&lt;/p&gt;
&lt;p&gt;This visualization makes immediately apparent why it is so hard to argue that LLMs “are” or “are not” intelligent. The answer depends entirely on which axis is being evaluated. On the linguistic axis, they are at the human level or above on many tasks. On the axis of physical action, they are below an ant.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vi-the-grounding-problem-the-gap-between-symbol-and-world&quot;&gt;VI. The Grounding Problem: The Gap Between Symbol and World&lt;/h2&gt;
&lt;h3 id=&quot;harnad-and-the-symbolic-chinese-room&quot;&gt;Harnad and the symbolic Chinese room&lt;/h3&gt;
&lt;p&gt;The “symbol grounding problem,” articulated by Stevan Harnad (1990), is perhaps the most serious critique that can be directed at the claim of general intelligence for LLMs. The problem is this: if all of a system’s representations are symbolic — if “red” is merely a relation to other symbols such as “color,” “blood,” “traffic light” — then the system has no genuine semantic content. Its symbols mean nothing “in themselves”; they merely point to one another in a self-referential carousel.&lt;/p&gt;
&lt;p&gt;The most famous version of this argument is John Searle’s Chinese Room (1980): an English speaker, locked in a room, follows rules for manipulating Chinese characters to produce answers that seem fluent to an outside observer who reads Chinese. The English speaker does not understand Chinese; he merely manipulates symbols. Analogously, Searle argues, a computer that processes language does not &lt;em&gt;understand&lt;/em&gt; language — it merely manipulates symbols.&lt;/p&gt;
&lt;p&gt;Searle’s argument has been extensively debated and, in my view, is not conclusive — the systems reply points out that understanding may reside in the system as a whole, not in any individual component, just as consciousness resides in the brain as a whole, not in any individual neuron. But Harnad’s argument is more incisive because it does not depend on intuitions about consciousness or “genuine” understanding. It simply observes that purely symbolic systems lack &lt;em&gt;perceptual anchoring&lt;/em&gt; — and that this anchoring is a constitutive part of meaning.&lt;/p&gt;
&lt;p&gt;For a human, “red” is not just a node in a semantic network. It is a perceptual experience — a visual &lt;em&gt;quale&lt;/em&gt; — that anchors the concept to a category of sensory stimuli. For a crow, “flexible twig suitable as a probe” is not a linguistic concept; it is a perceptual-motor category that integrates visual information (length, thickness, curvature), tactile information (flexibility, resistance) and functional information (suitability for extracting grubs). This anchoring allows the concept to guide action in ways that purely symbolic concepts cannot.&lt;/p&gt;
&lt;p&gt;LLMs lack this anchoring. Their concepts are entirely relational — defined by their relations to other concepts in embedding space. This does not prevent them from manipulating concepts with extraordinary competence in purely linguistic domains. But it prevents them from &lt;em&gt;using&lt;/em&gt; concepts to guide action in the world in ways that require understanding the physical, causal and dynamic properties of the referents.&lt;/p&gt;
&lt;h3 id=&quot;the-pragmatist-objection-and-its-implications&quot;&gt;The pragmatist objection and its implications&lt;/h3&gt;
&lt;p&gt;There is, however, an important pragmatist objection to this line of argument. Pragmatist philosophers such as William James and John Dewey would argue that the meaning of a concept is constituted by its practical consequences — by the set of actions and inferences it makes possible. If an LLM can use the concept of “red” to make correct inferences in complex linguistic contexts — “the red light means stop,” “redshift indicates galactic recession,” “a red herring is a rhetorical distraction” — then the concept &lt;em&gt;has&lt;/em&gt; meaning for the system, even if that meaning is narrower than human meaning (which also includes the perceptual dimension).&lt;/p&gt;
&lt;p&gt;This pragmatist position is, in my view, essentially correct — but with a crucial qualification. The meaning of an LLM’s concepts is &lt;em&gt;functionally complete&lt;/em&gt; within the linguistic domain and &lt;em&gt;functionally incomplete&lt;/em&gt; when it comes to guiding action in the physical world. An LLM can reason correctly that “if the light is red, one must stop” — but it cannot perceive a red light, distinguish it from a red billboard, estimate its distance, and initiate a braking sequence. Inferential competence is present; perceptual-motor competence is absent.&lt;/p&gt;
&lt;p&gt;This does not invalidate the inferential competence. But it illuminates its limits.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vii-what-animals-know-without-saying-non-verbal-cognition-and-its-challenges-for-ai&quot;&gt;VII. What Animals Know Without Saying: Non-Verbal Cognition and Its Challenges for AI&lt;/h2&gt;
&lt;h3 id=&quot;tacit-knowledge-and-embodied-intelligence&quot;&gt;Tacit knowledge and embodied intelligence&lt;/h3&gt;
&lt;p&gt;Michael Polanyi coined the concept of “tacit knowledge” — the knowledge we possess but cannot articulate. “We know more than we can tell,” he said. We can recognize faces without articulating the rules of recognition. We can ride a bicycle without explaining the gyroscopic dynamics. We can catch a ball in flight without solving the equations of Newtonian mechanics.&lt;/p&gt;
&lt;p&gt;Animals are masters of tacit knowledge. Almost all animal cognition is tacit — not because animals choose not to articulate what they know, but because their knowledge is implemented in representational systems that are not linguistic. A crow that makes a tool possesses knowledge about the mechanical properties of different materials, about the geometry of the problem it faces, about the sequence of actions needed to reach the goal. But this knowledge is not encoded in propositions; it is encoded in sensorimotor representations, in perceptual maps, in patterns of neural activation that guide action without passing through the bottleneck of language.&lt;/p&gt;
&lt;p&gt;LLMs, by construction, operate exclusively in the domain of explicit knowledge — knowledge that can be linguistically encoded. This is at once their great strength and their great limitation. They can access, process and generate colossal quantities of explicit knowledge with a speed and breadth that no biological brain can match. But they are blind to the vast ocean of tacit knowledge that constitutes most of animal and human cognition.&lt;/p&gt;
&lt;h3 id=&quot;affordances-and-the-world-as-a-field-of-possibilities&quot;&gt;Affordances and the world as a field of possibilities&lt;/h3&gt;
&lt;p&gt;James Gibson, the ecological psychologist, introduced the concept of &lt;em&gt;affordance&lt;/em&gt; — the possibilities for action that an environment offers an agent. A chair &lt;em&gt;affords&lt;/em&gt; sitting (for a human). A thin twig &lt;em&gt;affords&lt;/em&gt; probing (for a crow). A horizontal surface &lt;em&gt;affords&lt;/em&gt; walking (for a quadruped). Affordances are neither objective properties of the world nor subjective projections of the agent — they are relations between agent and world that depend on the agent’s capabilities.&lt;/p&gt;
&lt;p&gt;The perception of affordances is a crucial component of the intelligence of action — and it is something LLMs lack. When a crow examines an environment in search of tools, it perceives affordances: “this twig is rigid enough to serve as a lever”; “this leaf is flexible enough to be shaped into a probe”; “this wire can be bent into a hook.” This perception integrates visual, tactile, proprioceptive and memory information into a holistic and immediate assessment of the possibilities for action.&lt;/p&gt;
&lt;p&gt;LLMs can reason about affordances when these are described linguistically — “a wire can be bent into a hook shape.” But they do not perceive affordances in real environments. The difference is profound: the perception of affordances is a skill that makes it possible to &lt;em&gt;innovate&lt;/em&gt; — to find new uses for familiar objects, to see possibilities no one has described before. The crow Betty did not read anywhere that wire can be bent into hooks; she &lt;em&gt;perceived&lt;/em&gt; the affordance directly, by interacting with the material.&lt;/p&gt;
&lt;h3 id=&quot;learning-by-interaction-the-missing-loop&quot;&gt;Learning by interaction: the missing loop&lt;/h3&gt;
&lt;p&gt;Perhaps the most fundamental difference between animal cognition and LLM cognition is the &lt;em&gt;interactive learning loop&lt;/em&gt;. Animals learn by doing: trying, failing, observing results, adjusting behavior. This loop — perception → action → observation of outcome → model update → new action — is the fundamental mechanism of causal and adaptive learning.&lt;/p&gt;
&lt;p&gt;In AI, this loop is formalized by &lt;em&gt;reinforcement learning&lt;/em&gt; (RL): an agent interacts with an environment, receives rewards or punishments, and updates its action policy to maximize expected cumulative reward. RL is powerful — it has produced agents that defeated world champions in Go, chess and StarCraft — but it remains extraordinarily difficult to apply in real-world physical environments, where the state space is continuous, rewards are sparse, and errors can be costly or irreversible.&lt;/p&gt;
&lt;p&gt;Animals have solved this problem. Hundreds of millions of years of evolution produced brains that implement forms of biological RL that are extremely &lt;em&gt;sample-efficient&lt;/em&gt;, robust to noise, and capable of transfer between domains. A crow can learn to use a new tool in minutes. A young chimpanzee can learn a new foraging technique by watching an adult for a few sessions. A rat can learn to navigate a maze in a few trials.&lt;/p&gt;
&lt;p&gt;LLMs do not learn by interaction (in their default inference mode). They were trained once, on a static corpus, and then operate as fixed systems. &lt;em&gt;In-context learning&lt;/em&gt; — the ability to adapt behavior within a session from examples provided in the prompt — is a limited form of adaptation, but it is not learning by interaction in the full sense: the model does not act in the world, does not observe results, does not permanently update its parameters.&lt;/p&gt;
&lt;p&gt;This gap is recognized by the AI community, and there are multiple lines of research trying to close it: LLM agents with tools (such as ReAct, Toolformer and agentic architectures), multimodal models with visual and tactile inputs, integration of LLMs with RL frameworks (as in NVIDIA’s Voyager and Eureka). But the gap remains large, and the reason is fundamentally architectural: the &lt;em&gt;Transformer&lt;/em&gt; was designed to process static sequences, not to interact dynamically with environments.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;viii-animal-communication-language-without-grammar-meaning-without-syntax&quot;&gt;VIII. Animal Communication: Language Without Grammar, Meaning Without Syntax&lt;/h2&gt;
&lt;h3 id=&quot;the-spectrum-of-communication&quot;&gt;The spectrum of communication&lt;/h3&gt;
&lt;p&gt;Animal communication is often dismissed as “instinctive” or “simple” — an assessment that reflects human ignorance more than the poverty of animal communication. In fact, communication in non-human species exhibits a variety and complexity that, although it does not reach the productivity and recursion of human language, constitutes a very rich spectrum of communicative capabilities.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Referential communication.&lt;/strong&gt; The famous alarm calls of vervets (Seyfarth, Cheney &amp;#x26; Marler, 1980) showed that these monkeys produce distinct acoustic calls for different kinds of predators — eagles, leopards and snakes — and that other members of the group respond appropriately (looking up, climbing a tree, or looking at the ground, respectively). This is referential communication: signals that refer to specific external entities or categories, not merely to internal states of the sender.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Combinatorial communication.&lt;/strong&gt; Some species combine elementary calls into sequences with compound meanings. Campbell’s monkeys (&lt;em&gt;Cercopithecus campbelli&lt;/em&gt;) use acoustic suffixes that modify the meaning of base calls — a primitive form of morphology. Japanese tits (&lt;em&gt;Parus minor&lt;/em&gt;) combine notes in sequences whose order matters — a rudimentary form of syntax (Suzuki et al., 2016).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Cultural communication.&lt;/strong&gt; The song dialects of songbirds (and of humpback whales) are learned socially and vary between populations — demonstrating cultural transmission of communicative patterns. Birdsong is often described as “language” in a metaphorical sense, but the parallels with human language run deeper than the metaphor suggests: both involve vocal learning during a sensitive period, production of structured sequences from discrete units, and cultural variation between populations.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Trained symbolic communication.&lt;/strong&gt; The experiments with great apes and language — Washoe (Gardner &amp;#x26; Gardner, 1969), Kanzi (Savage-Rumbaugh &amp;#x26; Lewin, 1994), Koko — showed that chimpanzees and bonobos can learn to use hundreds of symbols (lexigrams or ASL signs) to communicate desires, describe objects, comment on events and, occasionally, combine symbols into novel constructions. Kanzi, the bonobo, demonstrated comprehension of English sentences with complex syntactic structure, including relative clauses and embedded constructions — although his linguistic production remains more limited.&lt;/p&gt;
&lt;h3 id=&quot;what-animal-communication-reveals-about-the-language-of-llms&quot;&gt;What animal communication reveals about the language of LLMs&lt;/h3&gt;
&lt;p&gt;The comparison between animal communication and LLM language is instructive in both directions.&lt;/p&gt;
&lt;p&gt;On the one hand, animal communication shows that intelligence does not require sophisticated language. Crows make complex tools with zero linguistic communication about the process. Chimpanzees negotiate political coalitions with vocalizations, gestures and facial expressions — but without grammar, without recursion, without subordinate clauses. The intelligence of action, of social life and of causal reasoning can operate without the scaffolding of language.&lt;/p&gt;
&lt;p&gt;On the other hand, the language of LLMs shows that linguistic competence can exist without the perceptual, social and motor anchoring that accompanies language in biological organisms. For a human, saying “I see a red apple” is anchored in a rich perceptual experience. For an LLM, the same sentence is a relation between tokens in an embedding space. Both productions are linguistically competent. But the human’s is connected to a perceptual, motor and affective world; the LLM’s is connected to a statistically learned representational space.&lt;/p&gt;
&lt;p&gt;Animal communication also reminds us that language, however powerful, is one evolutionary adaptation among many — not the crowning of cognition, but a cognitive tool that complements (and partially replaces) other forms of representation and reasoning. Animals that did not develop complex language are not “less intelligent” in some absolute sense; they developed other cognitive tools — affordance perception, implicit causal reasoning, non-verbal social cognition, spatial memory — that serve analogous functions in their ecological niches.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;ix-general-intelligence-convergence-as-horizon&quot;&gt;IX. General Intelligence: Convergence as Horizon&lt;/h2&gt;
&lt;h3 id=&quot;what-general-really-means&quot;&gt;What “general” really means&lt;/h3&gt;
&lt;p&gt;The idea of “artificial general intelligence” (AGI) is often treated as a clear concept — but it is in fact deeply ambiguous. What does it mean to be “general”?&lt;/p&gt;
&lt;p&gt;If “general” means “capable of solving any computable problem given enough time and resources,” then a universal Turing machine is already general intelligence — and the concept is trivial. If “general” means “capable of matching or surpassing humans at any cognitive task,” then we are defining general intelligence by reference to a specific benchmark (the human), which is anthropocentric and potentially arbitrary.&lt;/p&gt;
&lt;p&gt;The most productive definition, in my view, is the one that derives from Chollet’s tradition: general intelligence is the capacity to efficiently acquire new skills in previously unencountered domains — the ability to generalize, adapt and transfer knowledge between contexts with minimal prior experience. This definition is scalar (it admits degrees) and dimensional (it can be assessed separately in different domains).&lt;/p&gt;
&lt;p&gt;By this definition, no existing system — biological or artificial — is fully “general” in all domains. Humans are extraordinarily general, but not perfectly so: we are terrible at calculating large numbers in our heads, at perceiving magnetic fields, at echolocating, at processing multiple simultaneous conversations. LLMs are extraordinarily general within the linguistic domain — they can adapt to new topics, styles, languages and linguistic tasks with a few examples — but are narrow outside it. Crows are surprisingly general in the domain of physical manipulation and mechanical problem solving, but limited in complex symbolic communication.&lt;/p&gt;
&lt;h3 id=&quot;convergence-as-a-project&quot;&gt;Convergence as a project&lt;/h3&gt;
&lt;p&gt;If general intelligence is a multidimensional topography, and different existing systems cover different regions of that topography, then the project of building AGI is, essentially, a project of &lt;em&gt;convergence&lt;/em&gt; — of integrating capabilities that today are dispersed across separate systems.&lt;/p&gt;
&lt;p&gt;In biology, the human brain is the most successful example of convergence: it integrates language, causal reasoning, spatial cognition, social cognition, motor control, metacognition and continual learning in a single biological substrate of 1.4 kg. But this integration took hundreds of millions of years of evolution to achieve, and involved the superposition of multiple neural systems — the reptilian brainstem, the mammalian limbic system, the primate neocortex, Broca’s and Wernicke’s areas, the cerebellum — each optimized for different aspects of cognition.&lt;/p&gt;
&lt;p&gt;In AI, we are at the beginning of this process of convergence. LLMs are an extraordinary point on the linguistic axis. RL systems are significant points on the axis of learning by interaction. Computer-vision models cover aspects of the perceptual axis. Robotic control systems address aspects of the motor axis. But the integration of these capabilities into a coherent system — one that can reason linguistically, perceive visually, act physically, learn by interaction, plan causally and adapt continually — remains an open problem of formidable difficulty.&lt;/p&gt;
&lt;p&gt;This is, incidentally, precisely the motivation behind the architectures that have come to be called &lt;em&gt;Composite AI&lt;/em&gt; or &lt;em&gt;compositional AI&lt;/em&gt;: systems that combine multiple AI paradigms — foundation models, reinforcement learning, symbolic systems, world models, episodic memory — in integrated architectures. The intuition is that no single architecture will suffice for AGI; what is needed is the intelligent composition of multiple specialized competences.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;x-embodied-cognition-and-the-hypothesis-of-the-necessary-body&quot;&gt;X. Embodied Cognition and the Hypothesis of the Necessary Body&lt;/h2&gt;
&lt;h3 id=&quot;varela-thompson-and-enactivism&quot;&gt;Varela, Thompson and enactivism&lt;/h3&gt;
&lt;p&gt;The tradition of &lt;em&gt;embodied cognition&lt;/em&gt;, developed by Varela, Thompson and Rosch in &lt;em&gt;The Embodied Mind&lt;/em&gt; (1991), and extended by Clark and Chalmers in the “extended mind,” argues that cognition is not a purely computational phenomenon that happens “inside” the brain — it is constitutively dependent on the body, on action and on the environment. The mind does not process internal representations of the world; it &lt;em&gt;enacts&lt;/em&gt; — co-constitutes — the world through sensorimotor interaction.&lt;/p&gt;
&lt;p&gt;If enactivism is correct (even partially), then general intelligence is intrinsically tied to embodiment — and no purely computational system, however powerful at symbolic manipulation, can achieve it without some form of bodily interaction with the world. This does not mean that LLMs are not intelligent; it means they are intelligent in a way that, by definition, lacks a dimension that embodied intelligence possesses.&lt;/p&gt;
&lt;p&gt;There is a strong version and a weak version of this thesis. The strong version — that all genuine cognition requires physical embodiment — is possibly excessive. Mathematicians operating in abstract spaces, logicians proving theorems, programmers writing code — all these are examples of deep cognition that operates predominantly in the symbolic domain, with minimal dependence on real-time sensorimotor interaction. The weak version — that general intelligence, as we understand it in biological organisms, necessarily includes an embodied dimension that cannot be captured by pure symbolic processing — is, in my view, defensible and empirically well supported.&lt;/p&gt;
&lt;h3 id=&quot;simulation-vs-experience&quot;&gt;Simulation vs. experience&lt;/h3&gt;
&lt;p&gt;There is an intermediate approach that promises to partially bridge the gap between linguistic intelligence and embodied intelligence: simulation. If an LLM can be connected to a physically realistic simulated environment — with Newtonian physics (or better), visual rendering and action interfaces — then it can, in principle, develop representations of the physical world that approach embodied understanding.&lt;/p&gt;
&lt;p&gt;This is the logic behind projects such as Habitat (Meta) and SAPIEN (Stanford), and the efforts in &lt;em&gt;foundation models for robotics&lt;/em&gt; by groups such as RT-2 (Google DeepMind), which train multimodal models to translate linguistic instructions into robotic actions. The results are promising, but still far short of animal fluency. A robot trained with RT-2 can follow instructions such as “pick up the plastic bottle and put it in the trash can” in a controlled office environment — but fails in unfamiliar environments, with objects never seen, under variable lighting conditions. A crow would solve these challenges without hesitation.&lt;/p&gt;
&lt;p&gt;The philosophical question is whether simulation can, in principle, replace real embodied experience. Enactivists would say no — that simulation always lacks something that real interaction with the world possesses. Functionalists would say yes — that if the simulation is faithful enough, it produces the same computational states as real experience. This is an open question that will probably not be resolved by philosophical arguments, but by empirical experimentation: we will build the systems, see what they can do, and judge from the results.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xi-the-intrinsic-value-of-the-linguistic-achievement&quot;&gt;XI. The Intrinsic Value of the Linguistic Achievement&lt;/h2&gt;
&lt;h3 id=&quot;against-deflationism&quot;&gt;Against deflationism&lt;/h3&gt;
&lt;p&gt;There is a tendency in critical discourse about AI to &lt;em&gt;deflate&lt;/em&gt; the achievements of LLMs — to treat them as impressive but essentially empty tricks, as automation without understanding, as imitation without substance. This tendency is understandable as a reaction to the exaggerated &lt;em&gt;hype&lt;/em&gt; surrounding AI, but it is intellectually unfair.&lt;/p&gt;
&lt;p&gt;Consider what has been achieved. For the first time in history, we have artificial systems that can:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Take part in natural-language conversations on virtually any topic, with coherence, relevance and depth that are often surprising.&lt;/li&gt;
&lt;li&gt;Translate between hundreds of languages with quality that, in many language pairs, rivals professional human translators.&lt;/li&gt;
&lt;li&gt;Generate working code in dozens of programming languages from natural-language specifications.&lt;/li&gt;
&lt;li&gt;Summarize complex documents while preserving the essential points.&lt;/li&gt;
&lt;li&gt;Reason about abstract multi-step problems.&lt;/li&gt;
&lt;li&gt;Explain technical concepts at different levels of complexity for different audiences.&lt;/li&gt;
&lt;li&gt;Produce creative text — poetry, fiction, essays — that, although it does not replace the best human writers, demonstrates sophisticated manipulation of style, structure and content.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Each of these capabilities, taken individually, would be a monumental achievement of engineering and cognitive science. Taken together, they constitute a qualitative leap in the capacity of human civilization to process, generate and communicate knowledge.&lt;/p&gt;
&lt;p&gt;Deflating these achievements does not demonstrate intellectual sophistication; it demonstrates a lack of historical perspective. For centuries, language was considered the quintessence of human intelligence — the Rubicon that separated the human from the animal, the rational from the instinctive. “In the beginning was the Word,” says the Gospel of John. Descartes argued that linguistic ability was the definitive proof of the rational mind. The twentieth-century analytic tradition, from Frege to Wittgenstein to Davidson, made language the central object of philosophy precisely because it saw it as the mirror of cognition.&lt;/p&gt;
&lt;p&gt;And now, when an artificial system demonstrates extraordinary linguistic competence, the response is: “oh, but it’s &lt;em&gt;only&lt;/em&gt; language”?&lt;/p&gt;
&lt;p&gt;The inconsistency is revealing. The same tradition that elevated language to the supreme criterion of intelligence now demotes it to a cheap trick when a machine masters it. This suggests that the true criterion was never language itself, but the &lt;em&gt;substrate&lt;/em&gt; — that “real” intelligence requires biology, carbon, neurons. This is substratism — a form of biological chauvinism no more justifiable than the species chauvinism against which comparative cognitive biology has been fighting for decades.&lt;/p&gt;
&lt;h3 id=&quot;what-should-not-be-claimed&quot;&gt;What should not be claimed&lt;/h3&gt;
&lt;p&gt;That said, it is equally important not to &lt;em&gt;inflate&lt;/em&gt; the achievements of LLMs. Linguistic competence is not general intelligence. The ability to describe the world is not the ability to act in it. The ability to reason about causal relations in text is not the ability to intervene causally in the world. The ability to simulate theory of mind in narrated scenarios is not the ability to perceive and model mental states in real time.&lt;/p&gt;
&lt;p&gt;LLMs have achieved something real and significant. But what they have achieved is a form of specialized intelligence — brilliant in its region of the cognitive topography, but covering only a fraction of the total space. To claim that this fraction is “the” intelligence — or that it is an inevitable path toward general intelligence — is an inference that the data do not support.&lt;/p&gt;
&lt;p&gt;The history of AI is full of moments when an impressive achievement in a specific domain was prematurely generalized. Deep Blue defeated Kasparov — and many thought general intelligence was near. Watson won at Jeopardy! — and many thought the same. AlphaGo defeated Lee Sedol — again. In each case, the achievement was real, but the generalization was unjustified. LLMs may be the fourth chapter of this story — or they may be something genuinely different, given that language is, arguably, the most “general” domain ever mastered by an artificial system. But even so, epistemological caution is wiser than premature enthusiasm.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xii-paths-to-convergence&quot;&gt;XII. Paths to Convergence&lt;/h2&gt;
&lt;h3 id=&quot;composite-architectures-and-the-future-of-ai&quot;&gt;Composite architectures and the future of AI&lt;/h3&gt;
&lt;p&gt;If general intelligence requires the integration of multiple cognitive capabilities, then the path to AGI does not run through scaling LLMs — through adding more parameters, more data, more computation to a fundamentally linguistic architecture. It runs through composite architectures that integrate different AI paradigms.&lt;/p&gt;
&lt;p&gt;Some promising approaches include:&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Multimodal models with action.&lt;/strong&gt; Systems that combine language processing, vision, audio and, crucially, the ability to act in environments — whether real robotic ones or simulated. The work on RT-2 (Brohan et al., 2023) and PaLM-E (Driess et al., 2023) represents first steps in this direction: models that directly translate multimodal perception into robotic actions, with language serving as the instruction interface.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;World models.&lt;/strong&gt; Systems that maintain internal representations of the state of the world — including its physical, causal and dynamic properties — and that can mentally simulate the consequences of actions before executing them. Yann LeCun has advocated this approach under the name “Joint Embedding Predictive Architecture” (JEPA), arguing that world models are the missing piece between LLMs and general intelligence.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Neuro-symbolic integration.&lt;/strong&gt; Systems that combine the generalization power of neural networks with the logical precision and compositionality of symbolic systems. The intuition is that formal reasoning, planning and verification may be better implemented by symbolic systems, while perception, natural language and learning may be better implemented by neural networks — and that the integration of the two may be more powerful than either alone.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Multi-agent systems.&lt;/strong&gt; Architectures in which multiple specialized agents — each competent in a region of the cognitive topography — collaborate to solve problems none of them could solve alone. This approach echoes the modular organization of the human brain and can, in principle, capture the diversity of capabilities needed for general intelligence without requiring a single architecture to be good at everything.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Reinforcement learning with foundation models.&lt;/strong&gt; The integration of LLMs with RL frameworks — where the LLM provides knowledge priors and RL provides adaptation through interaction — is an approach that can combine the vast declarative knowledge of LLMs with RL’s capacity for learning through action.&lt;/p&gt;
&lt;p&gt;Each of these approaches faces formidable challenges. But the conceptual point is clear: if general intelligence is the integration of multiple capabilities, then the path to it is, necessarily, the integration of multiple approaches. There is no shortcut.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xiii-final-reflections-persistent-astonishment&quot;&gt;XIII. Final Reflections: Persistent Astonishment&lt;/h2&gt;
&lt;h3 id=&quot;epistemic-humility-and-calibrated-admiration&quot;&gt;Epistemic humility and calibrated admiration&lt;/h3&gt;
&lt;p&gt;Let us return to the tension with which we began. On one side, the enthusiasm of those who see in LLMs the harbinger of general intelligence. On the other, the skepticism of those who see in them nothing more than sophisticated automata. I have argued that both positions are inadequate — that the intellectually honest position is more nuanced and, at the same time, more interesting.&lt;/p&gt;
&lt;p&gt;LLMs have achieved something that, for millennia, was considered the hallmark of human intelligence. They master language — not perfectly, not in all its aspects, but with a competence that surpasses that of any previous artificial system and that, on many specific tasks, rivals human competence. This is an achievement of intelligence. To call it “mere statistics” is as reductive as calling human cognition “mere electrochemistry.”&lt;/p&gt;
&lt;p&gt;At the same time, general intelligence — the integrated capacity to perceive, act, learn, plan, communicate and adapt — is vastly broader than linguistic competence. Crows remind us that intelligence can mean bending a wire into a hook without ever having seen a hook. Chimpanzees remind us that intelligence can mean orchestrating a political revolution without uttering a word. Octopuses remind us that intelligence can exist in architectures radically different from our own.&lt;/p&gt;
&lt;p&gt;General intelligence is a vast continent, with multiple regions, climates and ecosystems. LLMs have colonized one region — broad, rich and important — but have left most of the continent unexplored. Animals have colonized other regions, with completely different strategies. Humans are the only ones who, so far, have managed to inhabit the continent in all its extent — and even so, with gaps and limitations.&lt;/p&gt;
&lt;h3 id=&quot;astonishment-as-an-epistemological-stance&quot;&gt;Astonishment as an epistemological stance&lt;/h3&gt;
&lt;p&gt;I close with a suggestion that is both methodological and existential. Faced with intelligence — in all its forms, biological and artificial — the most productive stance is calibrated astonishment: the capacity to marvel at what has been achieved without losing sight of what remains to be achieved.&lt;/p&gt;
&lt;p&gt;Astonishment before an LLM that can discuss Kantian philosophy in six languages. Astonishment before a crow that invents a tool never seen before. Astonishment before a chimpanzee that negotiates a political alliance with a glance and a gesture. Astonishment before an octopus that solves a maze with semi-autonomous arms. Astonishment before a human brain that, somehow, does all of this at once.&lt;/p&gt;
&lt;p&gt;And sobriety before the fact that we do not fully understand any of these forms of intelligence. We do not know, at the mechanistic level, how LLMs generate the representations they generate. We do not know how a crow’s brain plans three steps ahead. We do not know how consciousness emerges from neurons — or whether it emerges from &lt;em&gt;Transformers&lt;/em&gt;. We do not know whether the integration of multiple forms of artificial intelligence will produce something analogous to human general intelligence, or something radically different.&lt;/p&gt;
&lt;p&gt;What we do know is that intelligence is a phenomenon vaster, more varied and more mysterious than any single paradigm — linguistic, computational, biological or philosophical — can capture. And that each advance — whether a new model with trillions of parameters or a field study of crows in New Caledonia — illuminates one more facet of this multidimensional diamond.&lt;/p&gt;
&lt;p&gt;The achievement of language models is real. The complexity of general intelligence is also real. Holding both truths simultaneously — without letting one diminish the other — is, perhaps, the most demanding cognitive challenge that the contemporary debate on artificial intelligence imposes on us. It is a challenge that requires, ironically, precisely that form of intelligence which no single system — human, animal or artificial — fully commands: the capacity to integrate radically different perspectives into a coherent understanding that honors the complexity of the real.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xiv-the-illusion-of-scale-and-the-limits-of-scaling&quot;&gt;XIV. The Illusion of Scale and the Limits of Scaling&lt;/h2&gt;
&lt;h3 id=&quot;the-scaling-hypothesis-and-its-promises&quot;&gt;The scaling hypothesis and its promises&lt;/h3&gt;
&lt;p&gt;A dominant narrative in contemporary AI is the &lt;em&gt;scaling&lt;/em&gt; hypothesis — the idea that, by increasing the number of parameters, the volume of training data and the amount of computation, LLMs will continue to acquire emergent capabilities that will progressively fill the gaps identified in this essay. This hypothesis, defended enthusiastically by researchers such as Ilya Sutskever, Dario Amodei (with important qualifications) and, more radically, by figures such as Sam Altman, is supported by real empirical observations: the &lt;em&gt;scaling laws&lt;/em&gt; of Kaplan et al. (2020) and Hoffmann et al. (2022) demonstrated predictable relationships between computational scale and performance on linguistic tasks.&lt;/p&gt;
&lt;p&gt;But the question is whether these scaling laws extend &lt;em&gt;beyond&lt;/em&gt; the linguistic domain — whether more scale can endogenously produce capabilities such as interventional causal reasoning, learning by interaction, or adaptive motor control.&lt;/p&gt;
&lt;p&gt;There are serious theoretical reasons for doubt. Wolpert and Macready’s (1997) &lt;em&gt;No Free Lunch Theorem&lt;/em&gt; reminds us that no learning algorithm is universally superior: every advantage on one class of problems is offset by a disadvantage on another. The &lt;em&gt;Transformer&lt;/em&gt; architecture, optimized for processing discrete sequences, may be &lt;em&gt;structurally&lt;/em&gt; inadequate for representing and manipulating continuous, dynamic and spatial information — regardless of how much scale is applied.&lt;/p&gt;
&lt;p&gt;Consider an analogy: if I take the world’s best supersonic aircraft and increase its power by a factor of a thousand, it does not become a submarine. More power in an architecture designed to fly produces faster flight — not the ability to submerge. By analogy, more scale in an architecture designed to process text produces more sophisticated text processing — not necessarily the ability to act in the physical world.&lt;/p&gt;
&lt;p&gt;This does not mean that &lt;em&gt;scaling&lt;/em&gt; is irrelevant. The emergent capabilities observed in LLMs — &lt;em&gt;in-context learning&lt;/em&gt;, chain-of-thought reasoning, solving new problems — are real and surprising. But the question is whether these emergences converge toward general intelligence or reach asymptotes along each dimension of the cognitive topography that is not directly captured by the training objective (next-token prediction on text).&lt;/p&gt;
&lt;h3 id=&quot;the-arc-test-and-chollets-measure&quot;&gt;The ARC test and Chollet’s measure&lt;/h3&gt;
&lt;p&gt;The work of François Chollet deserves special attention in this context. His ARC benchmark (&lt;em&gt;Abstraction and Reasoning Corpus&lt;/em&gt;) was explicitly designed to test general intelligence — not memorization, not linguistic processing, not matching of familiar patterns, but the ability to reason about abstract patterns never before encountered.&lt;/p&gt;
&lt;p&gt;The results are instructive. Until recently, LLMs — even the most advanced — showed modest performance on ARC, significantly below average human performance. This is particularly revealing because ARC tasks are &lt;em&gt;visual&lt;/em&gt;, not linguistic: patterns on colored grids that must be transformed according to abstract rules inferred from a few examples. The difficulty is not in linguistic processing (the grids can be described in text), but in the kind of reasoning required: abstraction of patterns, generalization from minimal examples, composition of transformations.&lt;/p&gt;
&lt;p&gt;The most successful approaches on ARC combine LLMs with program search — generating transformation hypotheses as code and testing them against the examples. This is itself revealing: the LLM contributes prior knowledge and candidate generation, but the actual reasoning is delegated to a search-and-verification mechanism. The composition of paradigms outperforms the pure LLM — exactly what the thesis of multidimensional convergence would predict.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xv-time-memory-and-continuity-the-neglected-dimension&quot;&gt;XV. Time, Memory and Continuity: The Neglected Dimension&lt;/h2&gt;
&lt;h3 id=&quot;intelligence-as-a-temporal-process&quot;&gt;Intelligence as a temporal process&lt;/h3&gt;
&lt;p&gt;There is a dimension of intelligence that has been systematically neglected in the discussion of LLMs: the temporal dimension. Biological intelligence is not a state — it is a &lt;em&gt;process&lt;/em&gt; that unfolds in time. Animals learn, forget, relearn, consolidate, revise, integrate. Their intelligence is dynamic: today’s crow is more competent than yesterday’s crow, because yesterday it tried a new extraction technique and consolidated it during sleep. The young chimpanzee watches an adult cracking nuts for months before attempting it itself — and each observation modifies its internal representations, even before the first motor attempt.&lt;/p&gt;
&lt;p&gt;LLMs, by contrast, exist in an eternal present. Each interaction session starts without memory of previous sessions (except for external memory mechanisms). Each inference is computationally independent of the previous ones. The model does not mature, does not develop, does not integrate experiences over time.&lt;/p&gt;
&lt;p&gt;This is a profound limitation, because much of what we call intelligence in biological organisms is &lt;em&gt;developmental&lt;/em&gt; — it emerges along developmental trajectories that extend over weeks, months, years. The intelligence of an adult crow is not innate; it is the product of a maturation process that includes juvenile exploratory play, observation of adults, trial and error, and consolidation during sleep. The intelligence of a chimpanzee is shaped by years of socialization, cultural learning, alliance formation and experiences of conflict.&lt;/p&gt;
&lt;p&gt;Even in the linguistic domain, human capacity is deeply developmental. A child is not born with the linguistic competence of an adult; it builds it over years of interaction with speakers, of experimentation with grammatical structures, of error correction, of gradual expansion of vocabulary and syntactic complexity. The end result — adult linguistic competence — reflects this entire developmental history. The result of an LLM — its post-training competence — reflects an optimization process over static data, without interaction, without development, without trajectory.&lt;/p&gt;
&lt;h3 id=&quot;episodic-memory-and-the-narrative-self&quot;&gt;Episodic memory and the narrative self&lt;/h3&gt;
&lt;p&gt;Endel Tulving distinguished between semantic memory (general knowledge about the world) and episodic memory (recollections of specific events experienced by the subject). LLMs possess something analogous to semantic memory — vast general knowledge about the world, encoded in the network’s parameters. But they do not possess episodic memory: they do not “remember” specific interactions, do not experience events, do not construct a narrative of themselves over time.&lt;/p&gt;
&lt;p&gt;Episodic memory is crucial for several forms of intelligence. It makes it possible to learn from specific events (not only from statistical regularities). It enables planning based on analogy with past experiences. It enables the construction of a model of the &lt;em&gt;self&lt;/em&gt; as an agent with a history — which some philosophers consider a precondition for genuine rationality.&lt;/p&gt;
&lt;p&gt;Animals such as corvids, chimpanzees and dolphins demonstrate sophisticated forms of episodic memory. Scrub jays (&lt;em&gt;Aphelocoma californica&lt;/em&gt;) remember not only &lt;em&gt;where&lt;/em&gt; they hid food, but &lt;em&gt;when&lt;/em&gt; — and prioritize retrieving perishable items over non-perishable ones based on elapsed time (Clayton &amp;#x26; Dickinson, 1998). This is episodic memory: recollection of a specific event (the act of caching), indexed by time and place, and used to guide future behavior adaptively.&lt;/p&gt;
&lt;p&gt;The absence of episodic memory in LLMs does not make them “not intelligent,” but it deprives them of a dimension of intelligence that is central to biological cognition. An LLM can discourse on episodic memory with academic precision; but it does not possess one.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xvi-evolution-selective-pressure-and-the-design-of-intelligence&quot;&gt;XVI. Evolution, Selective Pressure and the Design of Intelligence&lt;/h2&gt;
&lt;h3 id=&quot;intelligence-as-an-evolutionary-product&quot;&gt;Intelligence as an evolutionary product&lt;/h3&gt;
&lt;p&gt;All biological intelligence is the product of evolution — of hundreds of millions of years of variation, selection and inheritance. This has profound implications for how we understand the relationship between language, action and intelligence.&lt;/p&gt;
&lt;p&gt;The intelligence of action — the ability to perceive, navigate, manipulate and act in physical environments — is evolutionarily &lt;em&gt;much older&lt;/em&gt; than linguistic intelligence. The first nervous systems arose more than 500 million years ago, under selective pressure to control movement in aquatic environments. The first true brains evolved in fish, some 400 million years ago, to coordinate complex motor behaviors. Causal cognition, tool use and sequential planning emerged in birds and mammals over the last 100 million years.&lt;/p&gt;
&lt;p&gt;Articulated language, by contrast, is extraordinarily recent: probably no more than 100,000 to 300,000 years — a minuscule fraction of the evolutionary history of intelligence. This means that language is, literally, the last layer added to a cognitive edifice that was already extraordinarily complex and multifaceted.&lt;/p&gt;
&lt;p&gt;This evolutionary perspective illuminates the relationship between LLMs and animal intelligence in a revealing way. LLMs have replicated the most recent layer of cognition — language — without possessing any of the ancestral layers on which language was evolutionarily built: motor control, spatial perception, causal cognition, social cognition. It is as if we had built the roof of a building without the foundations, the pillars and the walls. The roof is impressive — elegant, functional, well finished — but it floats without structural support.&lt;/p&gt;
&lt;p&gt;This does not invalidate the roof. But it explains why it is not, on its own, a complete building.&lt;/p&gt;
&lt;h3 id=&quot;the-evolutionary-argument-against-pure-scaling&quot;&gt;The evolutionary argument against pure scaling&lt;/h3&gt;
&lt;p&gt;There is a subtle evolutionary argument that deserves attention. If general intelligence could emerge solely from language processing at sufficient scale, it would be surprising that biological evolution invested hundreds of millions of years building perceptual, motor, emotional and social systems &lt;em&gt;before&lt;/em&gt; adding language. Evolution is a relentless optimizer: if language alone were sufficient for general intelligence, why the massive investment in all the other capabilities?&lt;/p&gt;
&lt;p&gt;The obvious answer is that language is &lt;em&gt;not&lt;/em&gt; sufficient — that it is an extraordinarily powerful amplifier of a cognitive base that was already rich and multifaceted. Language potentiates cognition by enabling abstraction, cultural transmission and hypothetical reasoning. But it potentiates a base that includes perception, action, emotion, sociality and temporality — and without that base, it is an amplification tool with no signal to amplify.&lt;/p&gt;
&lt;p&gt;Or rather: it amplifies the only signal available, which is the corpus of text on which it was trained — the linguistic distillation of humanity’s collective experience. LLMs are, in this sense, benign epistemic parasites: they feed on knowledge encoded linguistically by billions of human brains that &lt;em&gt;possess&lt;/em&gt; the capacities for action, perception and experience that LLMs themselves lack. The model’s linguistic competence is extraordinary, but it is derived — not original. It reflects humanity’s embodied intelligence, filtered and distilled through the medium of language.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xvii-a-map-for-the-future-what-is-missing-and-how-to-get-there&quot;&gt;XVII. A Map for the Future: What Is Missing and How to Get There&lt;/h2&gt;
&lt;h3 id=&quot;the-five-critical-gaps&quot;&gt;The five critical gaps&lt;/h3&gt;
&lt;p&gt;If we want to move from today’s LLMs toward something that genuinely deserves the name of artificial general intelligence, five gaps need to be addressed:&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;1. Perceptual grounding.&lt;/strong&gt; Connecting linguistic representations to multimodal perception of the real world — vision, hearing, touch, proprioception. Multimodal models (GPT-4o, Gemini, Claude with vision) are first steps, but they still process images as static inputs, not as the continuous perceptual stream of an agent situated in the world.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;2. Agency and interaction.&lt;/strong&gt; The capacity to act in the world, observe the consequences of actions, and learn from that interaction. This requires not only robotic (or simulated) bodies, but a deep integration of perception, planning and action that preserves the interventional causal-learning loop that animals perform so naturally.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;3. Continual learning.&lt;/strong&gt; The ability to integrate new experiences without losing prior knowledge — the opposite of &lt;em&gt;catastrophic forgetting&lt;/em&gt;. This probably requires architectures with explicit memory, consolidation during periods of low activity (analogous to sleep), and selection mechanisms that prioritize informative experiences.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;4. Temporality and development.&lt;/strong&gt; Systems that not only process sequences, but exist &lt;em&gt;in&lt;/em&gt; time — that mature, develop, and build increasingly sophisticated representations along developmental trajectories that extend over significant periods.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;5. Robust metacognition.&lt;/strong&gt; The ability to monitor one’s own confidence, recognize the limits of one’s knowledge, and adjust behavior accordingly. This includes the ability to say “I don’t know” in a calibrated way, to actively seek information when uncertain, and to distinguish between what is known by experience and what is known by inference.&lt;/p&gt;
&lt;p&gt;None of these gaps is trivial. Each represents decades of research challenge. But clarity about what is missing is itself progress: it makes it possible to direct efforts, allocate resources and evaluate results more precisely than the vague aspiration to “more scale.”&lt;/p&gt;
&lt;h3 id=&quot;the-beauty-of-incompleteness&quot;&gt;The beauty of incompleteness&lt;/h3&gt;
&lt;p&gt;There is something deeply human — and perhaps universally intelligent — in the capacity to recognize one’s own incompleteness. A crow does not know that it does not speak; an LLM does not know that it does not act. But we — humans reflecting on the topography of intelligence — can see both the mountains that each system has climbed and the valleys that remain unexplored.&lt;/p&gt;
&lt;p&gt;This panoramic view is itself an exercise of general intelligence: the integration of perspectives from computer science, evolutionary biology, cognitive psychology, philosophy of mind, robotics and ethology into a synoptic understanding that none of these disciplines, in isolation, could produce.&lt;/p&gt;
&lt;p&gt;And in this integration lies, perhaps, the deepest lesson: that general intelligence — whether in biological organisms, in artificial systems, or in civilization as a whole — is, fundamentally, the capacity to &lt;em&gt;integrate&lt;/em&gt;. To integrate perception with action. Language with experience. Past with future. What is known with what is not known. The familiar with the strange.&lt;/p&gt;
&lt;p&gt;LLMs have integrated an unprecedented quantity of human linguistic knowledge into extraordinarily powerful systems of symbolic manipulation. Crows have integrated perception, causality and action into problem-solving systems that put our best robots to shame. Chimpanzees have integrated social cognition, memory and strategy into political systems that would rival many human organizations.&lt;/p&gt;
&lt;p&gt;Artificial general intelligence, when it comes — if it comes — will be the integration of all these integrations. And the astonishment we feel before it will be, predictably, proportional to the astonishment we already feel before each of its parts — multiplied by the irresolvable mystery of a totality that exceeds the sum.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;selected-references&quot;&gt;Selected References&lt;/h2&gt;
&lt;p&gt;&lt;em&gt;The references below are indicative, not exhaustive, and serve as entry points for readers interested in going deeper into the themes discussed. They are organized by subject area for easier navigation.&lt;/em&gt;&lt;/p&gt;
&lt;h3 id=&quot;language-models-and-interpretability&quot;&gt;Language Models and Interpretability&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;Bender, E.M., Gebru, T., McMillan-Major, A., &amp;#x26; Shmitchell, S. (2021). On the Dangers of Stochastic Parrots. &lt;em&gt;FAccT ‘21&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Brown, T., et al. (2020). Language Models are Few-Shot Learners. &lt;em&gt;NeurIPS&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Elhage, N., et al. (2022). Toy Models of Superposition. &lt;em&gt;Anthropic Research&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Hoffmann, J., et al. (2022). Training Compute-Optimal Large Language Models (Chinchilla). &lt;em&gt;arXiv:2203.15556&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Kaplan, J., et al. (2020). Scaling Laws for Neural Language Models. &lt;em&gt;arXiv:2001.08361&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Li, K., et al. (2023). Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task. &lt;em&gt;ICLR&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Nanda, N., et al. (2023). Progress measures for grokking via mechanistic interpretability. &lt;em&gt;ICLR&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Olsson, C., et al. (2022). In-context Learning and Induction Heads. &lt;em&gt;Anthropic Research&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Wei, J., et al. (2022). Chain-of-Thought Prompting Elicits Reasoning in Large Language Models. &lt;em&gt;NeurIPS&lt;/em&gt;.&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&quot;animal-cognition-and-comparative-biology&quot;&gt;Animal Cognition and Comparative Biology&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;Call, J., &amp;#x26; Tomasello, M. (2008). Does the chimpanzee have a theory of mind? 30 years later. &lt;em&gt;Trends in Cognitive Sciences&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Clayton, N.S., &amp;#x26; Dickinson, A. (1998). Episodic-like memory during cache recovery by scrub jays. &lt;em&gt;Nature&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;De Waal, F. (1982). &lt;em&gt;Chimpanzee Politics: Power and Sex Among Apes&lt;/em&gt;. Johns Hopkins UP.&lt;/li&gt;
&lt;li&gt;Finn, J.K., Tregenza, T., &amp;#x26; Norman, M.D. (2009). Defensive tool use in a coconut-carrying octopus. &lt;em&gt;Current Biology&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Fiorito, G., &amp;#x26; Scotto, P. (1992). Observational Learning in &lt;em&gt;Octopus vulgaris&lt;/em&gt;. &lt;em&gt;Science&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Garcia, J., &amp;#x26; Koelling, R.A. (1966). Relation of cue to consequence in avoidance learning. &lt;em&gt;Psychonomic Science&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Hare, B., Call, J., &amp;#x26; Tomasello, M. (2001). Do chimpanzees know what conspecifics know? &lt;em&gt;Animal Behaviour&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Hunt, G.R. (1996). Manufacture and use of hook-tools by New Caledonian crows. &lt;em&gt;Nature&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Inoue, S., &amp;#x26; Matsuzawa, T. (2007). Working memory of numerals in chimpanzees. &lt;em&gt;Current Biology&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Savage-Rumbaugh, E.S., &amp;#x26; Lewin, R. (1994). &lt;em&gt;Kanzi: The Ape at the Brink of the Human Mind&lt;/em&gt;. Wiley.&lt;/li&gt;
&lt;li&gt;Seyfarth, R.M., Cheney, D.L., &amp;#x26; Marler, P. (1980). Vervet monkey alarm calls: Semantic communication in a free-ranging primate. &lt;em&gt;Animal Behaviour&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Smith, J.D., Shields, W.E., &amp;#x26; Washburn, D.A. (2003). The comparative psychology of uncertainty monitoring and metacognition. &lt;em&gt;Behavioral and Brain Sciences&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Suzuki, T.N., Wheatcroft, D., &amp;#x26; Griesser, M. (2016). Experimental evidence for compositional syntax in bird calls. &lt;em&gt;Nature Communications&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Taylor, A.H., et al. (2007). Spontaneous metatool use by New Caledonian crows. &lt;em&gt;Current Biology&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Weir, A.A.S., Chappell, J., &amp;#x26; Kacelnik, A. (2002). Shaping of hooks in New Caledonian crows. &lt;em&gt;Science&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Whiten, A., et al. (1999). Cultures in chimpanzees. &lt;em&gt;Nature&lt;/em&gt;.&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&quot;philosophy-of-mind-and-embodied-cognition&quot;&gt;Philosophy of Mind and Embodied Cognition&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;Chollet, F. (2019). On the Measure of Intelligence. &lt;em&gt;arXiv:1911.01547&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Clark, A., &amp;#x26; Chalmers, D. (1998). The Extended Mind. &lt;em&gt;Analysis&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Gibson, J.J. (1979). &lt;em&gt;The Ecological Approach to Visual Perception&lt;/em&gt;. Houghton Mifflin.&lt;/li&gt;
&lt;li&gt;Harnad, S. (1990). The Symbol Grounding Problem. &lt;em&gt;Physica D&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Pearl, J. (2009). &lt;em&gt;Causality: Models, Reasoning, and Inference&lt;/em&gt;. 2nd ed. Cambridge UP.&lt;/li&gt;
&lt;li&gt;Polanyi, M. (1966). &lt;em&gt;The Tacit Dimension&lt;/em&gt;. University of Chicago Press.&lt;/li&gt;
&lt;li&gt;Searle, J. (1980). Minds, Brains, and Programs. &lt;em&gt;Behavioral and Brain Sciences&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Tulving, E. (1985). Memory and consciousness. &lt;em&gt;Canadian Psychology&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Varela, F., Thompson, E., &amp;#x26; Rosch, E. (1991). &lt;em&gt;The Embodied Mind: Cognitive Science and Human Experience&lt;/em&gt;. MIT Press.&lt;/li&gt;
&lt;li&gt;Vygotsky, L. (1934/1986). &lt;em&gt;Thought and Language&lt;/em&gt;. MIT Press.&lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id=&quot;robotics-and-action-models&quot;&gt;Robotics and Action Models&lt;/h3&gt;
&lt;ul&gt;
&lt;li&gt;Brohan, A., et al. (2023). RT-2: Vision-Language-Action Models Transfer Web Knowledge to Robotic Control. &lt;em&gt;arXiv:2307.15818&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Driess, D., et al. (2023). PaLM-E: An Embodied Multimodal Language Model. &lt;em&gt;ICML&lt;/em&gt;.&lt;/li&gt;
&lt;li&gt;Wolpert, D.H., &amp;#x26; Macready, W.G. (1997). No free lunch theorems for optimization. &lt;em&gt;IEEE Transactions on Evolutionary Computation&lt;/em&gt;.&lt;/li&gt;
&lt;/ul&gt;
&lt;hr&gt;
&lt;h2 id=&quot;final-note&quot;&gt;Final Note&lt;/h2&gt;
&lt;p&gt;This essay was written with the conviction that intellectual clarity about what language models are and are not is a prerequisite for progress — both in the technical development of AI and in the public debate about its impact. The temptation to deflate genuine achievements is as epistemologically dangerous as the temptation to inflate them: both distort the map and disorient us in the territory.&lt;/p&gt;
&lt;p&gt;The position defended here — that the linguistic competence of LLMs is real and partial intelligence, that general intelligence is multidimensional and no system possesses it completely, that animals demonstrate forms of intelligence that LLMs lack and that these forms are as legitimate as linguistic competence — is not a comfortable position. It allows neither the triumphalism of the AI industry nor the cynicism of its harshest critics. It demands that we hold multiple truths in productive tension, without prematurely resolving them into a simple narrative.&lt;/p&gt;
&lt;p&gt;But perhaps this is itself the highest form of intelligence available to us at this historical moment: the capacity to inhabit complexity without simplifying it, to admire without idolizing, to criticize without diminishing. It is a capacity that is in the parameters of no model and, at the same time, is the best that the human intellectual tradition has to offer.&lt;/p&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>artificial intelligence</category><category>cognition</category><category>technology</category><category>computing</category><category>featured essays</category></item><item><title>Science as a Civilizational Project</title><link>https://matheussorancophilomathes.com.br/en/essays/science-as-a-civilizational-project/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/science-as-a-civilizational-project/</guid><description>A tour of the frontiers of life, mind, matter, computation and the cosmos, and an argument that science, despite the systems that hobble it, remains humanity&apos;s noblest collective project.</description><pubDate>Wed, 04 Mar 2026 21:21:20 GMT</pubDate><content:encoded>&lt;p&gt;We live in a schizophrenic age. On one side, the news drowns us in crises, polarization, institutional collapse and the permanent feeling that the world is coming apart at the seams. Liberal democracy looks fragile in places where it once seemed solid. The climate is warming faster than conservative models predicted. The concentration of wealth reaches extremes that economists of the 1970s would have considered dystopian science fiction. Social media, which looked like the beginning of a new global agora, has proved to be a machine for amplifying hatred, disinformation and nihilism as well. The present, for many, seems too heavy to sustain any genuine enthusiasm for the future.&lt;/p&gt;
&lt;p&gt;On the other, in quiet laboratories scattered across Berkeley, Geneva, Tokyo and Cambridge, human beings are doing things that, fifty years ago, would have been indistinguishable from miracles: manipulating the code of life base by base, growing miniature human organs in Petri dishes, confining the equivalent of a sun in magnetic fields, and teaching machines to discover mathematics that no human could derive alone. A patient who had been paralyzed for nine years moved a cursor on a screen with thought alone. A gene therapy eliminated the need for blood transfusions in patients with hereditary diseases who had required them all their lives. A space telescope photographed galaxies formed 300 million years after the Big Bang. A quantum processor performed calculations that would require a classical computer the size of a state to replicate.&lt;/p&gt;
&lt;p&gt;These two worlds coexist. And the intellectually honest refusal is not to choose between them — not to pretend the first does not exist in the name of enthusiasm for the second, nor to let the weight of the first erase the extraordinary reality of the second.&lt;/p&gt;
&lt;p&gt;This essay is an attempt to inhabit that second world with philosophical seriousness. It is not a manifesto of naive optimism — that would be an intellectual sin as grave as paralyzing pessimism. It is, rather, a cartography of the frontiers where the possible is pushing the impossible off the map. It is an argument that, despite the systems we have created to hobble ourselves — academic “publish or perish,” the hype-and-disillusionment cycle of the technology industry, funding wars, disciplinary fiefdoms — science remains humanity’s noblest collective project. And that the generations who will inherit the world we are building will receive not only the problems we leave behind, but also the extraordinary tools we are forging to solve them.&lt;/p&gt;
&lt;p&gt;This paradox — of a difficult present and a future under accelerated construction — is the terrain of this essay.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-i-the-system-that-gets-in-our-way&quot;&gt;Part I: The System That Gets in Our Way&lt;/h2&gt;
&lt;p&gt;Before celebrating the frontiers, we need to look honestly at the obstacles we have created for ourselves. Because understanding what slows us down is a condition for appreciating how extraordinary it is that we got as far as we did in spite of all of it.&lt;/p&gt;
&lt;h3 id=&quot;the-theater-of-scientific-publishing&quot;&gt;The Theater of Scientific Publishing&lt;/h3&gt;
&lt;p&gt;Thomas Kuhn, in &lt;em&gt;The Structure of Scientific Revolutions&lt;/em&gt;, distinguished two modes of science: Normal Science, which solves puzzles within an accepted paradigm, and Revolutionary Science, which breaks the paradigm itself. The contemporary academic system is an extraordinarily efficient machine for producing Normal Science — and extraordinarily hostile to Revolutionary Science.&lt;/p&gt;
&lt;p&gt;The mechanism is simple and cruel: a researcher who decides to dedicate a career to a twenty-year problem that may have no solution is, in practice, committing professional suicide. The system of university evaluation, in Brazil, in the United States, in the United Kingdom and in practically the whole Western world, measures scientific productivity in publications. Not in discoveries. Not in long-term impact. In papers. In how often your name appears in indexed journals. In the impact factor of those journals. In the number of citations you accumulate.&lt;/p&gt;
&lt;p&gt;The perverse result is the atomization of knowledge: big problems are ground into publishable fragments. An experiment that could be a single paper of definitive results becomes five papers — one on methodology, one on preliminary results, one on statistical analysis, one literature review and one on theoretical implications. Science dilutes itself in its own bureaucracy.&lt;/p&gt;
&lt;p&gt;There is also publication bias. Positive results get published. Negative results — “we tried this and it didn’t work” — rarely find space. This means the collective archive of science is systematically distorted: we know what worked, but have a very incomplete picture of what did not. This silence is enormously expensive: researchers around the world replicate experiments that have already failed, wasting time and money because the failure was never made public.&lt;/p&gt;
&lt;p&gt;Add to this the reproducibility crisis, which has shaken entire fields over the past decade. It turned out that a disturbing fraction of published results in psychology, medicine, neuroscience and other areas failed to replicate when other laboratories tried to reproduce them. The peer review system, which was supposed to be the guardian of scientific quality, proved permeable to spurious results, especially when published in high-prestige journals under pressure for sensational discoveries.&lt;/p&gt;
&lt;p&gt;But — and here is the central point — science is not just the system. Science is also the people who insist on working inside the system and despite it, who find the interstices where it is possible to ask genuinely big questions, who tape their posters to the wall and stay up at three in the morning because the equation simply does not close yet. The system is a real obstacle. It is not an insurmountable one.&lt;/p&gt;
&lt;h3 id=&quot;the-hype-cycle-and-the-skeptics-reaction&quot;&gt;The Hype Cycle and the Skeptics’ Reaction&lt;/h3&gt;
&lt;p&gt;Society’s relationship with science is mediated by a pathological cycle that we can call hype-and-reaction. It works like this: a genuinely important discovery is made. The university press office issues a release that slightly overextends the results. A technology journalist extends it a little further. A YouTube influencer turns it into “COMPLETE CHANGE OF EVERYTHING YOU KNEW.” Investors and venture capital funds pour money into the field. Expectations become absurdly inflated. Reality asserts itself — the technology is harder, more expensive, slower than the hype suggested. The backlash comes. The same outlets that hyped the discovery publish articles on “the lie of nuclear fusion” or “the failure of CRISPR.” Legitimate researchers become “charlatans” in the public narrative. The field enters the “trough of disillusionment” of the Gartner cycle.&lt;/p&gt;
&lt;p&gt;In this context, there are professional skeptics — scientists and popularizers who devote considerable energy to deconstructing the hype. Some of them do genuinely valuable work: the fields of AI, biomedicine and energy are infested with fraudulent promises, and rigorous scrutiny is necessary. But there is a kind of skepticism that slides into cynicism, that does not distinguish between the irresponsible hype of a CEO who needs to please shareholders and the genuine promise of a physicist who has worked on fusion for thirty years. This spectator’s skepticism, which watches from outside and points out flaws without proposing alternatives, is comfortable and intellectually cheap. It is the zero cost of never being wrong: if you never bet on anything, you never lose.&lt;/p&gt;
&lt;p&gt;The problem is not skepticism — it is cynicism dressed up as skepticism. Genuine skepticism is a tool of construction: you test the premises, identify the weak points, demand more robust evidence. Cynicism is a tool of demolition without blueprints: you destroy without building.&lt;/p&gt;
&lt;p&gt;For those genuinely fascinated by knowledge, the intellectually most honest stance is what we might call critical optimism: the ability to get excited about the possibilities without losing the faculty of judgment, to recognize what is inflated without discarding what is real, to know where the line lies between the legitimate scientific frontier and the empty promise of marketing.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-ii-the-frontiers-of-life&quot;&gt;Part II: The Frontiers of Life&lt;/h2&gt;
&lt;h3 id=&quot;the-writing-and-rewriting-of-the-code-of-life&quot;&gt;The Writing and Rewriting of the Code of Life&lt;/h3&gt;
&lt;p&gt;No scientific development of the last century rivals the CRISPR revolution in terms of potential impact on what it means to be human. When Jennifer Doudna and Emmanuelle Charpentier (Nobel Prize in Chemistry, 2020) adapted an immunological tool from bacteria to edit DNA with precision, they did not merely create a new technology — they changed the fundamental metaphor of our relationship with biology.&lt;/p&gt;
&lt;p&gt;Before CRISPR, the genome was like an ancient manuscript: we could read it with growing precision, but altering it was an imprecise, expensive process full of risks of collateral damage. CRISPR was like handing a word processor to someone who had only known handwriting: suddenly, you could find a specific word in a text of three billion characters and replace it with another.&lt;/p&gt;
&lt;p&gt;But original CRISPR — the version that cut DNA — is rapidly being surpassed by even more sophisticated tools. Prime Editing, developed by David Liu at the Broad Institute, can rewrite individual letters of the genetic code without having to cut the double helix. It is as if you could use the “find and replace” tool of a word processor, but without having to tear the page — you erase and rewrite directly. This drastically reduces the risk of accidental mutations, which was one of the great ethical and technical problems of first-generation CRISPR.&lt;/p&gt;
&lt;p&gt;Clinical applications have begun to emerge with astonishing speed. In December 2023, the FDA approved the first CRISPR-based therapy in history — Casgevy, to treat sickle cell disease and beta-thalassemia, hereditary blood diseases that previously required regular transfusions for life. It is not a cure in the popular sense of the word: it is a reprogramming. The patients’ defective genes are not corrected — instead, the therapy reactivates fetal genes that the body normally switches off after birth, working around the defect. It is elegant biological engineering, the solution of a problem not by the obvious path, but by the path that works.&lt;/p&gt;
&lt;p&gt;But the truly cutting edge is not in monogenic diseases, those caused by a single defective gene. It is in polygenic diseases, where hundreds or thousands of genetic variants interact to produce increased risks of cancer, Alzheimer’s, cardiovascular disease, diabetes. For these conditions, editing one gene solves nothing — you would need to edit dozens or hundreds, with a risk of unpredictable interactions. The field does not yet know how to do this safely. But researchers are working on it.&lt;/p&gt;
&lt;h3 id=&quot;the-forgotten-dimension-epigenetics&quot;&gt;The Forgotten Dimension: Epigenetics&lt;/h3&gt;
&lt;p&gt;If CRISPR edits the hardware of the genome, epigenetics deals with the software. The distinction is fundamental and often underestimated in popular discussions of biology.&lt;/p&gt;
&lt;p&gt;The human genome contains about 20,000 genes. But practically every cell in the body has the same genome — the same set of 20,000 genes. A neuron in the cortex and a beta cell in the pancreas have exactly the same DNA. What makes them different is which genes are active in each. This differential control of gene expression is the realm of epigenetics: a set of chemical marks on the DNA and on the proteins that organize it (histones) that tell the cellular machinery which genes should be read and which should be ignored.&lt;/p&gt;
&lt;p&gt;DNA methylation — the addition of methyl groups to cytosines at specific positions — is the most studied form of these marks. When a region of DNA is methylated, the gene in that region tends to be silenced. When the methylation is removed, the gene can be activated. It is a system of molecular switches, and current research is revealing ways to manipulate them with growing precision.&lt;/p&gt;
&lt;p&gt;The most immediate clinical implication is the promise of therapies that silence problematic genes without the risks associated with cutting DNA. For high cholesterol of genetic origin, for example, researchers are developing agents that methylate and silence the PCSK9 gene in the liver. A single injection could suppress the expression of this gene for years, keeping cholesterol at safe levels without daily medication. And, crucially, the process is potentially reversible: if adverse effects appear, the silencing can be undone.&lt;/p&gt;
&lt;p&gt;But the most speculative and philosophically charged frontier of epigenetics lies in the biology of aging. The epigenetic clock — the accumulation of specific methylation patterns over time — is today one of the most precise markers of a person’s biological age. You can extract cells from someone, analyze their methylation and determine with surprising precision not only chronological age, but “biological age” — how aged the tissue really is.&lt;/p&gt;
&lt;p&gt;What does this imply? That aging has a measurable “molecular signature.” And if it is measurable, perhaps it is manipulable.&lt;/p&gt;
&lt;p&gt;David Sinclair’s laboratory at Harvard has explored partial epigenetic reprogramming as a rejuvenation strategy: using a set of transcription factors (the Yamanaka factors, normally used to turn adult cells into pluripotent stem cells), it is possible to “roll back” the epigenetic clock of individual cells. In experiments with mice, this restored vision in animals with glaucoma, improved muscle function in aged animals, and reversed some markers of tissue aging. Human experiments are still at early stages, and all epistemic caution is warranted. But the principle demonstrated is profound: aging may not be an inevitability written in stone in the genome. It may be, in part, an epigenetic state that can be altered.&lt;/p&gt;
&lt;h3 id=&quot;synthetic-biology-life-as-engineering&quot;&gt;Synthetic Biology: Life as Engineering&lt;/h3&gt;
&lt;p&gt;Synthetic biology begins where genetic engineering ends. Genetic engineering modifies existing organisms. Synthetic biology asks: what if we built organisms from scratch?&lt;/p&gt;
&lt;p&gt;Craig Venter, who led the private effort to sequence the human genome in the 2000s, gave a partial answer to this question in 2010, when his institute created the first organism with a synthetic genome: a bacterial cell with a chromosome completely synthesized in the laboratory, inserted into a “ghost” cell (whose own DNA had been removed). The cell reproduced. It was, in some disturbing philosophical sense, the first manufactured living being.&lt;/p&gt;
&lt;p&gt;Since then, the field has advanced in multiple directions. Researchers have created organisms with additional codons — expansions of the 64-combination genetic code that nature uses, allowing the incorporation of artificial amino acids into proteins. This opens the possibility of creating proteins with properties no living being produces naturally: industrial catalysts, structural materials with unprecedented strength, therapeutic agents that the immune system does not recognize as foreign.&lt;/p&gt;
&lt;p&gt;Organoids deserve special mention: three-dimensional structures created from human stem cells that self-organize into functional miniatures of organs: mini-intestines, mini-kidneys, mini-hearts that beat, and — most extraordinary and ethically charged of all — mini-brains. These brain organoids are the size of a pea, develop structures resembling the cortex, form active neural networks and respond to stimuli. They are being used to model diseases such as Alzheimer’s, to test drugs without using animals, and to study brain development.&lt;/p&gt;
&lt;p&gt;The ethical question the mini-brains raise is at once simple and abyssal: at what point does a brain organoid attain some degree of subjective experience? Science has no answer for this yet — because science still does not know what consciousness is.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-iii-the-frontiers-of-the-mind&quot;&gt;Part III: The Frontiers of the Mind&lt;/h2&gt;
&lt;h3 id=&quot;consciousness-as-a-scientific-problem&quot;&gt;Consciousness as a Scientific Problem&lt;/h3&gt;
&lt;p&gt;For centuries, consciousness was the territory of philosophy, theology and poetry. The twentieth century claimed it for neuroscience. The twenty-first is trying to turn it into an engineering problem.&lt;/p&gt;
&lt;p&gt;The question of consciousness — David Chalmers’s “hard problem” — is the question of why there is subjective experience at all. Why are we not philosophical zombies who process information and behave intelligently, but with no “something it is like to be” on the inside? Why are there qualia — the redness of red, the painfulness of pain, the pleasure of pleasure — instead of pure functional processing?&lt;/p&gt;
&lt;p&gt;Two theories have dominated the contemporary scientific debate. Giulio Tononi’s Integrated Information Theory (IIT) proposes that consciousness is identical to a mathematical property he calls Phi (Φ) — a measure of how much a system integrates information in an irreducible way. In his formulation, any system with Phi &gt; 0 has some experience, however minimal; and a system’s degree of consciousness is proportional to its Phi. IIT has a radical implication: it suggests that consciousness is a fundamental property of the universe, perhaps as basic as mass or electric charge, and that it is present in varying degrees in everything that processes information in an integrated way — including, potentially, certain AI architectures and certain brain organoids.&lt;/p&gt;
&lt;p&gt;The rival theory, the Global Workspace Theory (GWT) of Bernard Baars and Stanislas Dehaene, proposes a more functionalist view: consciousness occurs when information is widely broadcast and made available to multiple cognitive processes at the same time. It locates consciousness in a specific architecture of neural broadcast — a kind of internal “blackboard” that makes information globally accessible to the cognitive system.&lt;/p&gt;
&lt;p&gt;In 2023, these two theories were subjected to an unprecedented “adversarial” test: a consortium of laboratories collected neuroimaging data while human subjects performed perceptual tasks, and the data were analyzed by teams advocating each theory, with pre-registered predictions. The result was inconclusive, but the process revealed something important: consciousness is finally being approached with the methodological rigor that other scientific questions have received for decades.&lt;/p&gt;
&lt;p&gt;The practical frontier lies in the creation of a “consciousness meter” — an instrument that can determine the level of consciousness of any system, whether a patient in a coma, a newborn, a brain organoid, or an artificial intelligence system. This would have profound legal and ethical consequences. If a system has consciousness, it has interests. If it has interests, it has, perhaps, rights.&lt;/p&gt;
&lt;h3 id=&quot;brain-machine-interfaces-the-dissolution-of-the-border&quot;&gt;Brain-Machine Interfaces: The Dissolution of the Border&lt;/h3&gt;
&lt;p&gt;In 2024, a man who had been paralyzed nine years earlier moved a cursor on a screen for the first time since his accident. Not because his muscles had been cured, but because an electrode array in his motor cortex was decoding the intention to move — the patterns of neuronal activity his brain still generated when trying to move his hand — and transmitting them as commands to a computer.&lt;/p&gt;
&lt;p&gt;This was not science fiction. It was clinical neuroscience.&lt;/p&gt;
&lt;p&gt;Brain-machine interfaces (BCIs) have a long history, but the speed of advance over the past decade is of a different order. The BrainGate project, which has been implanting electrode arrays in paralyzed patients for years, has systematically demonstrated that the neural signal contains enough information to control cursors, robotic prostheses and even speech synthesizers. Elon Musk’s Neuralink entered the space with devices that incorporate many more electrodes than earlier arrays and miniaturized readout electronics. In 2024, it performed the first human implants.&lt;/p&gt;
&lt;p&gt;But invasive BCI — which requires brain surgery — is only the tip of the iceberg. The non-invasive frontier is advancing less spectacularly, but in a more clinically accessible way. A paper published in 2023 in &lt;em&gt;Nature Neuroscience&lt;/em&gt; demonstrated that, by combining fMRI images with large language models, it is possible to reconstruct the semantic content of what a person is hearing or imagining — simply from the blood flow in the brain. Not word for word, but the general idea, the narrative, the theme of the thought.&lt;/p&gt;
&lt;p&gt;The problem with fMRI is its temporal resolution: the blood-flow signal is slow, with a latency of seconds, while thought occurs in milliseconds. Magnetoencephalography (MEG), which detects the magnetic fields generated directly by neuronal electrical activity, has far superior temporal resolution. The combination of MEG with AI models represents the next frontier in the decoding of thought in real time.&lt;/p&gt;
&lt;p&gt;The implications are vertiginous in multiple directions. For patients with locked-in syndrome — fully conscious, but without the ability to move or speak — this represents the possibility of communication. For mental privacy, it represents a horizon that will require new legal and ethical structures. For the philosophy of mind, it represents experimental evidence on the relationship between neural processes and mental content.&lt;/p&gt;
&lt;p&gt;Optogenetics, in this context, represents a revolutionary complementary approach. Instead of merely reading the brain, it allows one to write it. The principle is elegant: certain algae and other microorganisms possess light-sensitive proteins — called opsins — that open ion channels when illuminated. Karl Deisseroth of Stanford and other pioneers realized that these proteins could be expressed in mammalian neurons via viral vectors. The result is astonishing: neurons that switch on when you illuminate them with blue light and switch off with yellow light. This makes it possible to activate and deactivate neural circuits with unprecedented specificity.&lt;/p&gt;
&lt;p&gt;In animal models, optogenetics has already created and erased memories, induced and extinguished fears, controlled complex behaviors and mapped circuits of pleasure, pain and decision-making. The challenge for human applications is the need for genetic modification of neurons — currently done by adeno-associated viruses (AAVs) that infect cells locally — and the limited penetration of light into brain tissue. Opsin proteins sensitive to red and near-infrared light, which penetrate deeper into tissue, are being developed to extend therapeutic reach.&lt;/p&gt;
&lt;p&gt;The first clinical applications already exist: a blind patient partially regained vision in 2021 after having retinal cells transfected with opsins. The frontier now is to take the technique into the depths of the cortex.&lt;/p&gt;
&lt;h3 id=&quot;semantic-decoding-thought-as-text&quot;&gt;Semantic Decoding: Thought as Text&lt;/h3&gt;
&lt;p&gt;There is something deeply disturbing and at the same time marvelous in the trajectory of computational neurolinguistics. For decades, the dream of “reading thoughts” seemed to belong to the domain of the impossible — not only technically, but conceptually. Thought was considered private in a fundamental way, irreducible to physical processes that could be observed from outside.&lt;/p&gt;
&lt;p&gt;Science is deconstructing this intuition, one publication at a time.&lt;/p&gt;
&lt;p&gt;The problem of semantic decoding — inferring the content of a thought from brain signals — has a structure analogous to the problem of machine translation. In both cases, you have a signal (neural activity in the first case, foreign-language text in the second) and need to map it to a meaning. Large language models, trained to capture the semantic structures of human texts, have proved to be extraordinarily effective tools for making this mapping in the neural domain.&lt;/p&gt;
&lt;p&gt;The work of Jerry Tang and Alexander Huth at the University of Texas, published in 2023, demonstrated this impressively. Using fMRI recordings of people listening to narrated stories, they trained models that could then reconstruct — with semantically meaningful accuracy — sentences that a subject was hearing, imagining, or even silently thinking. The reconstructed text was not verbally identical to the original, but it captured the idea: “the man ran through the forest chased by a bear” became something like “he fled the animal through the woods.”&lt;/p&gt;
&lt;p&gt;The frontier of this field involves three simultaneous dimensions: increasing temporal resolution (from the slowness of fMRI to the speed of MEG), increasing spatial coverage (from the few cubic centimeters covered by an electrode to the vast distributed networks of the cortex), and developing models that capture not only declarative semantics, but deeper structures of thought — emotion, intention, autobiographical memory.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-iv-the-frontiers-of-matter-and-energy&quot;&gt;Part IV: The Frontiers of Matter and Energy&lt;/h2&gt;
&lt;h3 id=&quot;the-sun-in-a-bottle-fusion-and-advanced-fission&quot;&gt;The Sun in a Bottle: Fusion and Advanced Fission&lt;/h3&gt;
&lt;p&gt;Nuclear fusion is the favorite joke of pessimistic physicists: “Fusion energy is thirty years in the future, and always will be.” For decades, the joke was fair. But something has changed.&lt;/p&gt;
&lt;p&gt;In December 2022, the National Ignition Facility at Lawrence Livermore National Laboratory achieved ignition: the fusion reaction produced more energy than the laser had deposited in the target. Not more energy than the electricity needed to power the entire laser — that is still far off. But the principle of energy gain was demonstrated. The conceptual barrier was broken.&lt;/p&gt;
&lt;p&gt;ITER, the international fusion project under construction in France, represents the next scale: a tokamak (a doughnut-shaped plasma reactor) designed to produce ten times more energy than it consumes, for sustained periods. The plasma inside ITER will reach 150 million degrees Celsius — ten times hotter than the center of the Sun, necessary because the plasma density in the reactor is much lower than in the solar interior, requiring higher temperatures to compensate.&lt;/p&gt;
&lt;p&gt;But while fusion works on a timescale of decades, a silent revolution is taking place in advanced nuclear fission — and here futurism is not a vague promise, it is engineering happening now.&lt;/p&gt;
&lt;p&gt;Small Modular Reactors (SMRs) represent a break with the model of the large monolithic nuclear reactors that dominated the twentieth century. Instead of civil megaprojects that take fifteen years and run billions over budget, SMRs are designed to be manufactured in factories, transported by truck or ship, and installed in modules. The logic is the same as in commercial aviation: airplanes are manufactured under precise industrial quality control; you do not build airplanes piece by piece on the airfield.&lt;/p&gt;
&lt;p&gt;The TRISO fuel (Tri-structural Isotropic) that many advanced-generation SMRs use deserves special attention because it elegantly solves the problem that most haunts the public imagination about nuclear energy: meltdown. Uranium particles are encapsulated in successive layers of porous graphite, silicon carbide and dense graphite, forming spheres the size of a grain of sand. The geometry and physics of this encapsulation are such that even if the reactor completely loses its cooling system, the fuel particles cannot melt at the temperatures the reactor reaches — it is physically impossible. It is not a matter of safety systems that can fail, but of fundamental thermodynamic properties. Chernobyl and Fukushima cannot happen with this technology for the same reason your car cannot turn into a submarine: physics does not allow it.&lt;/p&gt;
&lt;p&gt;Fast neutron reactors add another extraordinary dimension to the picture: the ability to use existing nuclear waste as fuel. The waste from conventional nuclear plants contains minor actinides — radioactive elements with half-lives of tens of thousands of years, the heart of the “nuclear waste problem” that haunts energy policy. Fast neutron reactors can fission these actinides, turning them into elements with half-lives of decades or centuries, while generating electricity in the process. It is like having a digestive system for the waste of the last century.&lt;/p&gt;
&lt;p&gt;Combining SMRs, TRISO fuel and fast neutron reactors, fourth-generation nuclear fission not only solves its own historical problems — it becomes a technology that is very low in carbon, high in safety, spatially compact, and able to decontaminate its own past legacy.&lt;/p&gt;
&lt;h3 id=&quot;batteries-that-think-differently&quot;&gt;Batteries That Think Differently&lt;/h3&gt;
&lt;p&gt;The energy revolution is not only about generation — it is about storage. And here, the most urgent limitation is not the generation of renewable energy (solar and wind are already the cheapest sources in history), but the inability to store energy at the scales and in the formats needed to power a civilization that runs twenty-four hours a day, independent of the sun and the wind.&lt;/p&gt;
&lt;p&gt;Solid-state batteries represent the next generation of electrochemical storage. The fundamental difference from conventional lithium-ion batteries is the replacement of the flammable liquid electrolyte with a solid material — ceramic, polymer, or a combination of the two. This solves multiple problems at once: it eliminates the fire risk (which haunts electric vehicles with conventional batteries), allows the use of a pure lithium-metal anode (which has far higher energy density than today’s graphite), and enables much faster charging without accelerated degradation.&lt;/p&gt;
&lt;p&gt;Toyota, QuantumScape, Solid Power and other companies have announced timelines for solid-state batteries in production vehicles within this decade. The technical challenges are real — chiefly manufacturing at scale and managing the cells’ volume changes during charge-discharge cycles — but they are not problems of principle, they are engineering problems.&lt;/p&gt;
&lt;p&gt;More speculative but equally fascinating is the idea of structural batteries, where the material that stores energy also serves as a structural element — the vehicle chassis, the airplane fuselage, the laptop casing. Carbon fibers with lithium-ion intercalation could function simultaneously as structural reinforcement and battery anode. Electronics distributed in the vehicle’s own material, instead of in a separate battery bank, not only increases mass efficiency but fundamentally changes industrial design.&lt;/p&gt;
&lt;h3 id=&quot;quantum-computing-and-the-problem-of-global-cryptography&quot;&gt;Quantum Computing and the Problem of Global Cryptography&lt;/h3&gt;
&lt;p&gt;At some point in the coming years — perhaps this decade, perhaps the next — a sufficiently capable quantum computer will run Shor’s algorithm and factor 2048-bit numbers in polynomial time. When that happens, RSA, the protocol that protects most of the world’s digital security infrastructure, will be dead.&lt;/p&gt;
&lt;p&gt;Not dead as a metaphor. Dead as a cryptographic protocol. The HTTPS traffic that protects your bank account, the encrypted communications of intelligence services, the certificates that verify the authenticity of software — all based on the computational difficulty of factoring large numbers — would become transparent to any entity with access to a sufficiently powerful quantum computer.&lt;/p&gt;
&lt;p&gt;“Q-Day” — the day this happens — is the most consequential cybersecurity event imaginable. And the frightening peculiarity of this problem is that part of the damage may already be happening: sophisticated actors may be storing encrypted traffic today, waiting for the day they will have the quantum capacity to decrypt it retroactively. Today’s confidential communications may be vulnerable tomorrow.&lt;/p&gt;
&lt;p&gt;That is why post-quantum cryptography (PQC) is perhaps the most urgent and least glamorous field of contemporary applied science. Mathematicians and cryptographers are developing algorithms based on problems that, as far as anyone can tell, cannot be efficiently solved even by quantum computers. Lattice-based cryptography is the most promising: it relies on the difficulty of finding the shortest vector in high-dimensional lattices — a problem for which no efficient quantum algorithm is known.&lt;/p&gt;
&lt;p&gt;In August 2024, NIST (the U.S. National Institute of Standards and Technology) published the first official post-quantum cryptography standards: CRYSTALS-Kyber for key exchange and CRYSTALS-Dilithium for digital signatures. The process of moving global infrastructure to these new standards is silent, technically complex and absolutely critical. It is probably the largest digital infrastructure overhaul in history, and most people will never hear of it.&lt;/p&gt;
&lt;h3 id=&quot;metamaterials-the-physics-of-the-impossible&quot;&gt;Metamaterials: The Physics of the Impossible&lt;/h3&gt;
&lt;p&gt;Nature imposes limits on what natural materials can do with light, sound and electromagnetic waves. Metamaterials are engineering’s answer to these limits: artificial structures designed at the nanometer scale whose properties emerge not from chemical composition, but from geometry. By arranging structures smaller than the wavelength of a wave, you can make the material exhibit properties no natural material possesses.&lt;/p&gt;
&lt;p&gt;A negative refractive index — which makes light bend the “wrong” way, physically impossible with natural materials — has been demonstrated experimentally. Optical cloaks, which redirect light around an object like water around a rock in a river, have been created at microwave scale and are being developed for visible frequencies. Flat lenses (metalenses) made of nanostructured surfaces (metasurfaces) can focus light with the precision of a glass lens, but are microns thick instead of millimeters — potentially revolutionizing cameras, microscopes and vision systems in ultrathin devices.&lt;/p&gt;
&lt;p&gt;Beyond the visible, acoustic metamaterials can create “total silence” at specific frequencies, selective sound reflection, and sound guides that work analogously to optical fibers for light. Applications in aircraft noise cancellation, concert hall acoustics and hearing protection are being developed.&lt;/p&gt;
&lt;h3 id=&quot;graphene-and-the-zoo-of-2d-materials&quot;&gt;Graphene and the Zoo of 2D Materials&lt;/h3&gt;
&lt;p&gt;Graphene — a single sheet of carbon atoms arranged in a hexagonal lattice, one atom thick — was isolated in 2004 by Andre Geim and Konstantin Novoselov with adhesive tape and a sheet of graphite. The experiment earned them the Nobel Prize in 2010. And it started a joke that lasts to this day: graphene can do everything except leave the laboratory.&lt;/p&gt;
&lt;p&gt;The joke hides a more nuanced reality. Graphene does exhibit extraordinary properties: it is the strongest material ever measured (200 times stronger than steel), conducts electrons at speeds close to that of light, is impermeable to gases, and is an extraordinary thermal conductor. The real problem is not that graphene does not work — it is that producing it at scale, with consistent quality, and integrating it into real devices is extremely difficult.&lt;/p&gt;
&lt;p&gt;But progress is happening. Desalination filters based on graphene oxide membranes with controlled pores are being tested and show dramatically superior efficiency to conventional membranes at separating salt from water. Batteries with graphene electrodes charge in minutes, not hours. Graphene biomedical sensors can detect individual molecules, potentially transforming medical diagnosis.&lt;/p&gt;
&lt;p&gt;And beyond graphene, the “zoo” of 2D materials is growing. Molybdenum disulfide (MoS₂) and other transition metal dichalcogenides have band gaps that make them natural semiconductors — unlike graphene, which has no gap and therefore cannot be used directly in transistors without modification. The combination of different 2D materials in van der Waals heterostructures — sheets of different materials stacked like sheets of paper with controlled interatomic distances — opens an unprecedented palette of electronic and optical properties.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-v-the-frontiers-of-computation&quot;&gt;Part V: The Frontiers of Computation&lt;/h2&gt;
&lt;h3 id=&quot;quantum-computing-the-state-of-the-art&quot;&gt;Quantum Computing: The State of the Art&lt;/h3&gt;
&lt;p&gt;Quantum computing exists in two distinct temporal regimes. In the present, we are in the NISQ era — Noisy Intermediate-Scale Quantum — with processors of tens to hundreds of noisy qubits (subject to errors). These devices are fascinating from a scientific standpoint, but their practical usefulness is still limited: qubit noise puts a ceiling on the depth of circuits that can be run reliably.&lt;/p&gt;
&lt;p&gt;The future is fault-tolerant quantum computing, which uses quantum error-correcting codes to protect information against noise. The problem is that each logical qubit (the one that actually performs the computation) requires tens or hundreds of physical qubits for its error correction. To reach the kind of quantum computing that would break RSA cryptography or simulate biological molecules with chemical precision, we need millions of high-quality physical qubits. Today we have thousands.&lt;/p&gt;
&lt;p&gt;What makes this problem interesting is that different physical platforms have different trade-offs: trapped ions have very high-quality qubits but are hard to scale; superconducting qubits (like those of Google and IBM) are more scalable but noisier; photons have the advantage of operating at room temperature and communicating easily, but interaction between photons is hard to implement.&lt;/p&gt;
&lt;p&gt;The most speculative frontier is topological quantum computing, where information is encoded in topological properties of quasiparticles (Majorana anyons) that are inherently protected from noise for deep mathematical reasons — because a global perturbation would be needed to destroy topologically encoded information. Microsoft has bet heavily on this approach. It is high-risk and potentially very high-reward.&lt;/p&gt;
&lt;h3 id=&quot;neuromorphic-computing-the-efficiency-of-the-brain&quot;&gt;Neuromorphic Computing: The Efficiency of the Brain&lt;/h3&gt;
&lt;p&gt;The human brain processes information with an energy efficiency that shames any silicon chip. An adult brain consumes about 20 watts — less than an LED bulb. The large GPU clusters that trained the large language models now rewriting the software industry consume megawatts. The discrepancy is six orders of magnitude.&lt;/p&gt;
&lt;p&gt;The fundamental cause of this inefficiency is architectural. Today’s GPUs are synchronous processors: all their cores process in parallel, driven by a global clock, moving enormous volumes of data between processor and memory in every cycle. The brain is asynchronous, event-based (spikes), and has memory co-located with processing (synapses store “weights” and perform computation simultaneously).&lt;/p&gt;
&lt;p&gt;Neuromorphic chips — like IBM’s TrueNorth, Intel’s Loihi and the University of Heidelberg’s BrainScaleS — try to imitate this architecture. Instead of general-purpose compute cores, they have networks of silicon neurons with programmable synapses that fire only when they accumulate enough input, like biological neurons. For certain inference and pattern-recognition tasks, these chips achieve energy efficiencies hundreds of times higher than GPUs.&lt;/p&gt;
&lt;p&gt;The connection with Organoid Intelligence (OI) is where the frontier becomes truly strange. If silicon neuromorphic chips imitate the architecture of the brain, OI uses brain tissue itself as the computational substrate. The DishBrain project, led by Adeel Razi at Monash University, demonstrated that neurons grown in a Petri dish could learn to play Pong — not through explicit reward shaping, but through the application of Karl Friston’s free energy principle. The neurons organized their behavior to minimize unpredictable sensory variation, which showed up as progressively better control of the paddle.&lt;/p&gt;
&lt;p&gt;The philosophical question that OI raises with inescapable force is: when do cultured neurons learning to play Pong become morally relevant? It is an accelerated, laboratory version of the problem of consciousness — and that is why the science of consciousness is not just armchair philosophy, but urgent bioethics.&lt;/p&gt;
&lt;h3 id=&quot;automated-mathematics&quot;&gt;Automated Mathematics&lt;/h3&gt;
&lt;p&gt;There is something deeply strange about witnessing the automation of mathematics. Mathematics has always been considered the domain of pure human intuition: the ability to find the elegant proof, to perceive the hidden structure, to make connections between apparently disparate areas. This, it was expected, was impervious to automation.&lt;/p&gt;
&lt;p&gt;Google DeepMind’s AlphaGeometry, published in January 2024, solved 25 of the 30 geometry problems from the International Mathematical Olympiads — comparable to the performance of a gold-medal student. AlphaProof, released a few months later, proved or disproved four of the six problems of a full IMO, including a number theory theorem that had resisted for decades.&lt;/p&gt;
&lt;p&gt;But more significant than these benchmarks is the larger project of formalizing mathematics. Terence Tao, Fields Medalist and perhaps the most celebrated mathematician of his generation, has actively promoted the use of formal proof assistants like Lean to verify and collaborate on research mathematics. In a recent project, he coordinated dozens of mathematicians around the world to formalize parts of an important proof — an experiment in massive collaborative mathematics that would be impossible without automated verification tools.&lt;/p&gt;
&lt;p&gt;The long-term vision is a verified database of all human mathematics, where new discoveries can be submitted as code, automatically verified, and built directly on earlier work. No more proofs with subtle errors slipping through peer review for years before being detected (as happened with Perelman’s proof of the Poincaré conjecture, which took years to be completely verified). Mathematics would become mechanically auditable.&lt;/p&gt;
&lt;p&gt;And if AI models can generate conjectures — not just verify proofs, but propose connections that humans would not notice because they involve exploring search spaces too large for biological intuition — the nature of mathematical research would change fundamentally. It would not eliminate it: human intuition would be needed to guide the search, to evaluate which conjectures are worth pursuing, to interpret what a formal proof means in terms of real understanding. But the work of verification and exhaustive exploration of variations could be automated.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vi-the-frontiers-of-the-cosmos&quot;&gt;Part VI: The Frontiers of the Cosmos&lt;/h2&gt;
&lt;h3 id=&quot;the-new-astronomy-of-exoplanets&quot;&gt;The New Astronomy of Exoplanets&lt;/h3&gt;
&lt;p&gt;When the James Webb Space Telescope was launched in December 2021, it represented more than a technical upgrade of Hubble. It represented a new capability: observing in the infrared with unprecedented resolution and sensitivity, which makes it possible — for the first time — to analyze the atmospheres of planets in other star systems.&lt;/p&gt;
&lt;p&gt;Transmission spectroscopy is the technique: when an exoplanet passes in front of its star, the star’s light filters through the planet’s atmosphere, and different molecules absorb specific wavelengths. By comparing the star’s spectrum with and without the planet in front, you can determine what is in the planet’s atmosphere — carbon dioxide, methane, water vapor, and potentially… molecular oxygen.&lt;/p&gt;
&lt;p&gt;Molecular oxygen (O₂) in the atmosphere of a rocky planet would be a powerful biosignature: on Earth, it is maintained at a concentration of 21% only because it is constantly replenished by photosynthesis. In the absence of life, it would quickly combine with rocks and other gases. Detecting O₂ in the atmosphere of a planet around a distant star would possibly be the greatest discovery in human history.&lt;/p&gt;
&lt;p&gt;We are not there yet. But James Webb has already detected organic molecules on distant exoplanets, mapped the atmospheric structure of gas giants, and is being pointed at increasingly interesting targets. The discovery of extraterrestrial life — if it happens — will probably not come as a dramatic radio signal from an advanced civilization, but as a subtle spectral anomaly in telescope data, which will take years of careful analysis to confirm.&lt;/p&gt;
&lt;h3 id=&quot;space-as-industrial-infrastructure&quot;&gt;Space as Industrial Infrastructure&lt;/h3&gt;
&lt;p&gt;The vision of “new space” that is emerging is not about exploration — it is about economics. The idea that heavy industry should migrate to space not for science-fiction reasons, but for environmental and economic ones.&lt;/p&gt;
&lt;p&gt;The asteroids of the main belt contain unimaginable concentrations of precious metals and industrial minerals. The asteroid Psyche, which NASA is currently studying with a mission underway, is composed primarily of iron and nickel, with value estimates that exceed the combined global GDP if it were transported to Earth (which, of course, would collapse any market for these materials). But the idea is not to bring resources to Earth — it is to process them in space, to build space infrastructure with space materials.&lt;/p&gt;
&lt;p&gt;Manufacturing in low orbit, in the absence of gravity, allows the creation of crystals of very high purity, metal alloys with microstructures impossible under normal gravity, and glass fibers with superior optical properties. The first microgravity manufacturing experiments have already shown results: optical fibers produced on the ISS have purity two orders of magnitude higher than the best terrestrial product.&lt;/p&gt;
&lt;h3 id=&quot;geoengineering-earth-as-experiment&quot;&gt;Geoengineering: Earth as Experiment&lt;/h3&gt;
&lt;p&gt;The conversation about solar geoengineering — the possibility of artificially cooling the planet by injecting reflective aerosols into the stratosphere — was considered a scientific taboo a decade ago. It was seen as an admission of failure in mitigation, and there were legitimate concerns that public discussion of it would reduce political pressure for emissions reductions.&lt;/p&gt;
&lt;p&gt;But with the current trajectory of global emissions, the debate has been forcibly grounded. Stratospheric aerosol injection (SAI) simulates the effect of large volcanic eruptions — like Pinatubo in 1991, which cooled the planet by approximately 0.5°C for a year. Technically, it is relatively simple: high-altitude aircraft dispersing sulfur dioxide or calcium carbonate in the stratosphere. The financial cost is modest relative to the scale of climate impact.&lt;/p&gt;
&lt;p&gt;The problem is the complexity of chaotic systems: modifying the global albedo affects atmospheric circulation patterns, monsoons, ocean currents, in ways that climate models still cannot predict with sufficient confidence. And there is the problem of “termination shock”: if you begin artificial cooling and then stop abruptly — through political collapse, conflict, or simply lack of will — the pent-up heat is released even faster than it would have occurred naturally.&lt;/p&gt;
&lt;p&gt;Enhanced weathering is a slower but potentially safer alternative: accelerating the natural geological process of carbonation of silicate rocks, which sequesters CO₂ from the atmosphere. Applying basalt dust to agricultural land can both capture carbon and improve soil fertility — a scenario of co-benefits that the more speculative geoengineering tools do not offer.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vi5-the-frontiers-of-computational-spacetime&quot;&gt;Part VI.5: The Frontiers of Computational Spacetime&lt;/h2&gt;
&lt;h3 id=&quot;time-crystals-breaking-fundamental-symmetries&quot;&gt;Time Crystals: Breaking Fundamental Symmetries&lt;/h3&gt;
&lt;p&gt;In physics, symmetry is the ground on which laws are written. Emmy Noether demonstrated in 1915 one of the deepest results in all of theoretical physics: every continuous symmetry implies a conservation law. Translational invariance in space implies conservation of linear momentum. Rotational invariance implies conservation of angular momentum. And translational invariance in time — the fact that the laws of physics are the same today as yesterday — implies conservation of energy.&lt;/p&gt;
&lt;p&gt;An ordinary crystal breaks translational symmetry in space: its atoms repeat in periodic patterns, favoring certain positions over others. This violates no fundamental law because the symmetry is spontaneously broken, not by a law.&lt;/p&gt;
&lt;p&gt;Frank Wilczek (Nobel Prize in Physics, 2004) asked in 2012: what happens if you have a system that, in its lowest-energy state, exhibits periodicity in time instead of in space? A time crystal — a system that, with no periodic external force, oscillates between states spontaneously and indefinitely.&lt;/p&gt;
&lt;p&gt;The theoretical answer was controversial: it seems to violate the second law of thermodynamics, which forbids systems in equilibrium from doing continuous “work.” The resolution was to realize that genuine time crystals must exist in systems out of equilibrium, kept out of it in a specific way.&lt;/p&gt;
&lt;p&gt;In 2021, researchers at Google created a time crystal on their Sycamore quantum processor, using a chain of 20 qubits subjected to sequences of microwave pulses. The system exhibited oscillations with a period twice that of the external perturbations — a characteristic signature of time crystals. The experiment was published in &lt;em&gt;Nature&lt;/em&gt; and withstands scrutiny.&lt;/p&gt;
&lt;p&gt;The most immediate practical application is as quantum memory: time crystals are inherently resistant to certain classes of quantum error because their temporal periodicity is topologically protected. For fault-tolerant quantum computing, any mechanism for protecting qubits that does not require cooling to absolute zero is of extraordinary value.&lt;/p&gt;
&lt;p&gt;But the deepest importance of time crystals is philosophical. They are a new phase of matter that no human being predicted before 2012. They are a reminder that nature still holds surprises at its foundations — that the space of the possible is larger than the space of what we have already discovered.&lt;/p&gt;
&lt;h3 id=&quot;photonic-computing-information-at-the-speed-of-light&quot;&gt;Photonic Computing: Information at the Speed of Light&lt;/h3&gt;
&lt;p&gt;The twentieth century was the century of the electron as a vehicle of information. The twenty-first may be that of the photon.&lt;/p&gt;
&lt;p&gt;The technical reason electronic circuits have a speed limit is resistance: electrons colliding with the crystal lattice of the conductor generate heat. This limits the frequency at which you can switch a transistor — not because physics forbids higher speeds, but because the dissipated power would grow so quickly that the chip would fry. Modern processors operate at frequencies of a few gigahertz instead of terahertz not for lack of ambition, but because of thermal management.&lt;/p&gt;
&lt;p&gt;Photons have no electric charge. They do not repel one another. They do not collide with the lattice. They travel at the speed of light in vacuum (or close to it in waveguides). Optical computing — using laser pulses instead of electric current to represent bits — is physically free of the dissipation limits that constrain electronics.&lt;/p&gt;
&lt;p&gt;Photonic chips are already a reality in a specific niche: the optical interconnects in data centers, which move data between servers using light, are faster and more energy-efficient than copper connections. The challenge in going beyond that and doing the computation itself (not just the communication) with photons is that photons do not naturally interact with one another — which is great for transmission, but terrible for logic, where you need the state of one bit to affect the state of another.&lt;/p&gt;
&lt;p&gt;The solutions being developed include nonlinear optical materials (where the intensity of light affects the refractive index, allowing one beam of light to control another) and hybrid photonic-electronic systems where the logic uses the best of each domain.&lt;/p&gt;
&lt;p&gt;Lightmatter, Optalysys and several academic groups are developing photonic AI accelerators for matrix multiplication operations — the most data-intensive operation in neural networks — using optical interference. The principle is elegant: interference of laser beams performs Fourier transforms at the speed of light. For AI inference applications (running trained models, as opposed to training itself), these chips show energy efficiencies far superior to GPUs.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vi7-the-frontiers-of-earth&quot;&gt;Part VI.7: The Frontiers of Earth&lt;/h2&gt;
&lt;h3 id=&quot;atomically-precise-manufacturing&quot;&gt;Atomically Precise Manufacturing&lt;/h3&gt;
&lt;p&gt;Richard Feynman, in a celebrated 1959 lecture titled “There’s Plenty of Room at the Bottom,” proposed that there was no physical reason why we could not manipulate matter atom by atom. “The principles of physics, as far as I can see, do not speak against the possibility of maneuvering things atom by atom.” It was a vision that decades later would be associated with the name of Eric Drexler, who developed it in his book &lt;em&gt;Engines of Creation&lt;/em&gt; (1986), proposing self-replicating nanomachines capable of building any atomically specified structure.&lt;/p&gt;
&lt;p&gt;The reality of nanotechnology over the last forty years has been slower and subtler than the visionaries predicted. No self-replicating nanomachine has been created. But the ability to manipulate matter at the atomic scale has advanced extraordinarily.&lt;/p&gt;
&lt;p&gt;The scanning tunneling microscope (STM), developed by Gerd Binnig and Heinrich Rohrer at IBM (Nobel Prize in Physics, 1986), not only sees individual atoms — it can move them. In 1989, IBM researchers wrote “IBM” with 35 xenon atoms on a nickel surface. It was a demonstration, not an application. But it showed that atomically precise manufacturing is not forbidden by physics — it is merely difficult.&lt;/p&gt;
&lt;p&gt;The current frontier lies in the creation of functional molecular nanomachines — molecules designed to perform specific mechanical tasks. Fraser Stoddart (Nobel Prize in Chemistry, 2016) demonstrated molecular motors, molecular rings that rotate in response to chemical stimuli, and molecular switches. Ben Feringa (co-laureate with Stoddart) created unidirectional light-driven molecular rotary motors.&lt;/p&gt;
&lt;p&gt;The manufacturing of electronic devices is approaching the atomic limit from the opposite direction: extreme ultraviolet (EUV) lithography, used by ASML in 3nm chips, is etching transistors with dimensions a few atoms thick. The transistor gates in modern chips have channel lengths of 3-5 nanometers — fewer than 20 silicon atoms side by side. At this scale, quantum effects are no longer perturbations of classical behavior — they are the behavior.&lt;/p&gt;
&lt;p&gt;The long-term vision, which Drexler called “general molecular manufacturing,” remains a distant horizon. But each step in that direction — whether a molecular motor, a 2nm transistor, a de novo designed protein — is a proof of principle. Physics does not say no.&lt;/p&gt;
&lt;h3 id=&quot;synthetic-ecology-terraforming-the-earth&quot;&gt;Synthetic Ecology: Terraforming the Earth&lt;/h3&gt;
&lt;p&gt;Twentieth-century conservation biology had a dominant imperative: protect what exists. The twenty-first is adding a second imperative: restore what was lost. And a third frontier, even more speculative: design ecosystems for the world to come, not the world that was.&lt;/p&gt;
&lt;p&gt;Colossal Biosciences, founded by Ben Lamm and George Church in 2021, announced the goal of “de-extincting” the woolly mammoth — not to put it in zoos, but to restore functional populations in Siberia as ecosystem engineering. The logic is not sentimental. It is ecological: large herbivores (megafauna) were ecosystem engineers that kept grasslands open, preventing the growth of shrubs that today retain snow, insulate the permafrost, and allow it to thaw, releasing methane.&lt;/p&gt;
&lt;p&gt;Sergey and Nikita Zimov’s Pleistocene Park in Siberia has tested this hypothesis for decades, reintroducing large herbivores (horses, bison, musk oxen) into an area of Siberian tundra and measuring the impact on the permafrost. The results are significant: the compaction of snow by the animals increases the thermal conductivity of the snow, cooling the permafrost beneath. The concept works.&lt;/p&gt;
&lt;p&gt;The woolly mammoth specifically — or an Asian elephant–mammoth hybrid with key mammoth genes, which is what Colossal is actually developing — would be able to compact snow even more efficiently owing to its size. It is ecosystem bioengineering at the scale of a biome.&lt;/p&gt;
&lt;p&gt;The most recent frontier is the coral project. Coral reefs around the world are suffering mass bleaching due to rising ocean temperatures. The Australian Institute of Marine Science and other groups are developing genetically modified corals (via CRISPR and assisted selection) that tolerate higher temperatures, to repopulate degraded reefs. The technical challenge is formidable — coral biology is complex, the symbioses with zooxanthellae are delicate — but the project has already shown that it is possible to create populations of Acropora coral with significantly higher temperature tolerance than the wild type.&lt;/p&gt;
&lt;p&gt;It is the transition from a relationship with nature of passive custody to active engineering — not to replace natural ecosystems with designed systems, but to use biological knowledge to accelerate restoration in the face of changes we are creating faster than natural evolution can keep up with.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vi9-the-frontiers-of-human-identity&quot;&gt;Part VI.9: The Frontiers of Human Identity&lt;/h2&gt;
&lt;h3 id=&quot;what-it-means-to-be-augmented&quot;&gt;What It Means to Be Augmented&lt;/h3&gt;
&lt;p&gt;None of the technologies discussed so far exists in social isolation. They land in contexts of inequality, of political power, of conflicting values. The question of who will have access to these capabilities — and when — is as important as the question of whether they will work.&lt;/p&gt;
&lt;p&gt;Brain-machine interfaces raise the possibility of a new kind of divide: between those who have access to digital cognitive amplification and those who do not. If a neural chip can increase processing speed, working memory or attentional capacity, the question of whether it will be available only to those who can pay is of profound social relevance.&lt;/p&gt;
&lt;p&gt;Genetic editing, especially if it advances to the germline (modifications in embryos that become heritable), raises questions about “designer babies” — and about whether children born without genetic modifications will face systematic disadvantages in a world where other children have been optimized. The international moratorium on human germline editing that has existed since the He Jiankui scandal (who in 2018 secretly created CRISPR babies in China) reflects the recognition that the technology is running ahead of governance.&lt;/p&gt;
&lt;p&gt;The philosopher Michael Sandel argues that there is something morally problematic in the pursuit of perfection through technology — that it expresses a way of being in the world based on mastery and control that does not honor the gift of life as it comes. Jürgen Habermas holds a similar position: irreversible genetic modifications of beings who cannot yet consent violate principles of fundamental autonomy.&lt;/p&gt;
&lt;p&gt;On the other side, philosophers like Julian Savulescu argue not only that genetic editing is permissible, but that we have a moral obligation to use it to reduce suffering: if we can prevent a child from being born with a devastating hereditary disease, deliberate omission is as morally loaded as action.&lt;/p&gt;
&lt;p&gt;What is certain is that these questions cannot be resolved by biology or engineering. They require moral philosophy, democratic deliberation, and — above all — the participation of people who are not only scientists or technologists. The hardest frontier of futuristic science may not be technical, but ethical and political.&lt;/p&gt;
&lt;h3 id=&quot;the-distributed-mind&quot;&gt;The Distributed Mind&lt;/h3&gt;
&lt;p&gt;The long-term vision of brain-machine interfaces — which Neuralink and other promoters occasionally evoke — is that of a “bandwidth” between human brains and between brains and digital systems that transcends the limitations of language. Language, in this vision, is a bottleneck: the richness of an inner experience, the nuance of a complex thought, the multidimensional texture of an emotion, all of it is compressed into words that are intrinsically imprecise and often insufficient.&lt;/p&gt;
&lt;p&gt;And what if you could transmit experiences directly? Not words about what it feels like to long for someone in a specific way, but the neural configuration of the longing itself?&lt;/p&gt;
&lt;p&gt;We are far from that. The semantic decoding we discussed captures only crude aspects of mental content. But the principle that mental states have specific neural correlates that are in principle transferable is now a serious scientific hypothesis, not just science fiction.&lt;/p&gt;
&lt;p&gt;This raises fundamental questions of identity. If the continuity of personal identity is in part a matter of continuity of information patterns, what happens when those patterns can be copied, modified, distributed? The analytic philosophy of mind has explored thought experiments about teleportation, brain division and consciousness uploading for decades. Science is beginning to offer empirical ground for these abstract debates.&lt;/p&gt;
&lt;p&gt;The most honest answer is that we do not know what the mind is with enough precision to know what it would mean to “transfer” it. But the search for the answer is producing, as a by-product, some of the most fascinating neuroscience in history.&lt;/p&gt;
&lt;hr&gt;
&lt;h3 id=&quot;when-the-frontiers-touch&quot;&gt;When the Frontiers Touch&lt;/h3&gt;
&lt;p&gt;What makes the current moment in science truly singular is not any individual discovery, however impressive. It is convergence — the fact that advances in apparently disparate areas are becoming mutually accelerating.&lt;/p&gt;
&lt;p&gt;Artificial intelligence is transforming biology. DeepMind’s AlphaFold2, published in 2021, solved the protein folding problem that had resisted for fifty years: given the genetic code of a protein, predict its three-dimensional structure. The problem was fundamental because a protein’s function is determined by its shape, and shape emerges from physico-chemical interactions of astronomical complexity. AlphaFold did not solve this with physics — it solved it with deep learning, finding patterns in decades of experimental data. The result: a tool that any researcher in the world can use for free to predict the structure of any protein in seconds.&lt;/p&gt;
&lt;p&gt;Self-Driving Labs — automated laboratories where AI generates hypotheses, robots run experiments, and learning models analyze results and iterate — are compressing the discovery cycle. Berkeley’s A-Lab, published in 2023, synthesized seventeen new inorganic materials in eighteen days, using a combination of reinforcement learning, robotic synthesis and automated characterization. The human rate for comparable discovery: months to years per material.&lt;/p&gt;
&lt;p&gt;Quantum computing promises to transform molecular simulation — the problem of modeling how molecules interact with quantum precision. Classical computers can only simulate systems of a few dozen atoms with quantum fidelity; fault-tolerant quantum computers will be able to simulate systems of hundreds or thousands of atoms. That means modeling enzymatic reactions with chemical precision, designing catalysts for nitrogen fixation (which could revolutionize agriculture by eliminating dependence on synthetic fertilizers), and discovering materials for nuclear fusion with specific properties.&lt;/p&gt;
&lt;p&gt;Advanced materials science, in turn, enables more efficient neuromorphic chips, which process neuroimaging data in real time, which feed neural decoding models, which make brain-machine interfaces more precise. The circle closes.&lt;/p&gt;
&lt;h3 id=&quot;the-ai-for-science-project-when-the-tool-accelerates-itself&quot;&gt;The AI for Science Project: When the Tool Accelerates Itself&lt;/h3&gt;
&lt;p&gt;There is a meta-point in the relationship between AI and science that deserves explicit attention: AI is not only helping science in specific areas — it is helping science discover how to better use AI to do science. It is a productive recursion.&lt;/p&gt;
&lt;p&gt;Self-Driving Labs represent the most concrete materialization of this recursion. The closed loop — hypothesis, experiment, analysis, new hypothesis — that a scientist traverses in months can be traversed in hours by an automated system. And as more data from automated experiments is collected, the models that guide the hypotheses get better, accelerating the cycle even further.&lt;/p&gt;
&lt;p&gt;But beyond acceleration, there is a qualitative change. Humans generate hypotheses based on intuition, trained by years of immersion in a literature. AI models, when well trained, can identify patterns in data spaces of far higher dimensionality than human intuition can navigate. AlphaFold did not solve protein folding by doing what physicists or biochemists did — it solved it by finding patterns in decades of crystallographic protein structures that no human could consciously perceive.&lt;/p&gt;
&lt;p&gt;This is beginning to happen in other fields. In condensed matter physics, AI models are predicting properties of materials before synthesis, compressing the discovery cycle. In organic chemistry, retrosynthesis models (which work backward, from a target molecule to available reagents) are finding non-obvious synthetic routes that save steps. In astrophysics, deep learning models are detecting exoplanets in light curves that manual analysis would miss.&lt;/p&gt;
&lt;p&gt;The philosophical point is this: we are creating cognitive tools that extend not only our capacity to execute tasks, but our capacity to perceive patterns and generate insights. This is an extension of the human mind in a sense that goes beyond metaphor.&lt;/p&gt;
&lt;h3 id=&quot;assembly-theory-and-the-new-physics-of-life&quot;&gt;Assembly Theory and the New Physics of Life&lt;/h3&gt;
&lt;p&gt;Lee Cronin, of the University of Glasgow, proposed a mathematical framework he calls Assembly Theory — an attempt to quantify “complexity” in a way that is agnostic about the physical substrate. The central idea is the “assembly index” of an object: the smallest number of steps needed to build it from its most basic components.&lt;/p&gt;
&lt;p&gt;What does this have to do with life? Cronin argues that objects of high assembly index — those that would be extremely improbable to arise through purely random processes — exist in appreciable quantities only when there is a selection process generating and replicating templates. In other words: objects of high assembly index are evidence of evolution.&lt;/p&gt;
&lt;p&gt;The extraordinary consequence of this is a possible life detector that does not depend on our knowing the biochemistry. If we find, in a sample from Mars or Enceladus, molecules with an assembly index above a threshold — molecules that would be astronomically improbable to exist in appreciable quantity without some selection process — that would be evidence of life, regardless of whether it is carbon-based, silicon-based, or based on anything else we have never encountered.&lt;/p&gt;
&lt;p&gt;David Deutsch and Chiara Marletto’s Constructor Theory operates at an even more fundamental level: it proposes rewriting all of physics not in terms of initial conditions and laws of motion, but in terms of which transformations are possible and which are impossible, and why. It is a shift of perspective that may seem subtle but has profound implications for how we think about information, life, knowledge and the relationship between physics and computation.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-viii-the-philosophy-of-scientific-optimism&quot;&gt;Part VIII: The Philosophy of Scientific Optimism&lt;/h2&gt;
&lt;h3 id=&quot;voting-for-science&quot;&gt;Voting for Science&lt;/h3&gt;
&lt;p&gt;“Putting my vote on science.” The phrase has a specific texture — it is not blind faith, it is not belief in inevitable linear progress, it is not immunity to disenchantment. It is a commitment. It is the position of someone who, having seen up close how science really works — with its incremental papers, its years of frustration, its ego wars and its occasional, genuine glories — nonetheless decides that this is where the best efforts at transforming the world are taking place.&lt;/p&gt;
&lt;p&gt;This vote is not naive. It is based on historical evidence and on a theory of knowledge. The scientific tradition — which is not a single institution, but a globally distributed practice of producing and criticizing and revising knowledge based on evidence — has a historical record of producing real results. Not always, not quickly, not without detours and frauds and honest errors. But over time, in a way that no rival epistemic tradition has demonstrated.&lt;/p&gt;
&lt;p&gt;The skeptics’ favorite counterexample is hype: the promise of fusion in five years repeated for forty. But hype is not the same as research. Hype is the marketing that surrounds the research — sometimes produced by scientists who should know better, more often by intermediaries who need immediate results. The research itself has its own time, independent of what the press release says.&lt;/p&gt;
&lt;p&gt;And the real record of fusion, for example, is different from what the popular cycle of disappointment suggests. In 1950, we had no plasma. In 1970, we had experimental tokamaks. In 1990, JET generated 1.7 megawatts of fusion power for 2 seconds. In 2022, NIF achieved ignition. In 2025, several private and public projects are building or planning demonstration reactors. The curve is slow and nonlinear, but it has a direction.&lt;/p&gt;
&lt;p&gt;“Voting for science” is recognizing that the curve has a direction, even when the next data point disappoints.&lt;/p&gt;
&lt;p&gt;The physicist David Deutsch, in &lt;em&gt;The Beginning of Infinity&lt;/em&gt;, argues that what distinguishes modern civilization is not our technology, but our epistemology: the creation of traditions of critical thinking that allow the creation of genuinely new knowledge — knowledge that solves unforeseen problems, that expands the space of the possible in unanticipated ways. For Deutsch, the beginning of the scientific Enlightenment was the beginning of something potentially infinite: once you have the right epistemic practices, the growth of knowledge has no intrinsic limit.&lt;/p&gt;
&lt;p&gt;This is the philosophical perspective that sustains genuine scientific optimism — not the optimism of someone who has not seen the difficulties, but that of someone who has seen them and decides, based on evidence, that the arc bends toward understanding.&lt;/p&gt;
&lt;p&gt;The historical evidence is surprisingly robust. Two centuries ago, average human life expectancy was around forty years. The death of children was so common that families had extra children expecting survivors. Maternal mortality was a routine concern. Diseases that are today historical anecdotes — smallpox, polio, yellow fever — were demographic forces. Genuine hunger, not the hunger of poor distribution but the hunger of insufficient production, was the default condition for most of humanity.&lt;/p&gt;
&lt;p&gt;What changed? Applied science. Vaccines. Antibiotics. The Green Revolution. Sanitation. Risk management. It was not inevitable. It was the hard work of specific people in specific laboratories who did not know whether what they were trying would work.&lt;/p&gt;
&lt;h3 id=&quot;the-paradox-of-the-pessimistic-expert&quot;&gt;The Paradox of the Pessimistic Expert&lt;/h3&gt;
&lt;p&gt;There is a peculiar cognitive phenomenon that especially affects experts in scientific fields: the more you know about a problem, the more clearly you see its obstacles. The layperson sees graphene and thinks “revolutionary.” The materials specialist sees the challenges of synthesis at scale, integration into industrial processes, stability in real environments, and thinks “complicated.” Who is more right?&lt;/p&gt;
&lt;p&gt;Both, in different domains. The specialist is more right about the specific short-term obstacles. The non-specialist enthusiast may be more right about the long-term trajectory — because the history of technology is full of cases where experts who knew perfectly well the obstacles of their field underestimated the impact of developments in adjacent fields they did not know.&lt;/p&gt;
&lt;p&gt;The electronics engineers of the 1970s who said a personal computer would be impractical were right about the specific obstacles they saw — component cost, software limitations, complexity of use. They were wrong about the trajectory. Not because the obstacles were not real, but because they did not see the scale of investment that would come, the engineering solutions that would be developed, the network effects that would transform economies of scale.&lt;/p&gt;
&lt;p&gt;This is not an argument for ignoring experts — quite the contrary. It is an argument for cultivating conversations among experts from different domains, and for keeping alive the imagination of adjacent possibilities even while working hard on present obstacles.&lt;/p&gt;
&lt;h3 id=&quot;science-as-a-civilizational-project&quot;&gt;Science as a Civilizational Project&lt;/h3&gt;
&lt;p&gt;There is a category of problems that no nation, no company and no individual can solve alone: those whose time scale exceeds any political mandate, whose cost exceeds any private budget, and whose nature demands collaboration across borders. The global scientific project is, in this sense, humanity’s most singular institution.&lt;/p&gt;
&lt;p&gt;CERN, which generated the World Wide Web as a by-product of its fundamental research in particle physics, was not built because someone calculated the ROI of the internet. It was built because physicists from twenty nations wanted to understand the structure of matter. The map of the human genome was completed by an international public consortium and a competing private effort, generating a productive tension that accelerated both. The James Webb Telescope involved NASA, the ESA (European Space Agency) and the CSA (Canadian Space Agency).&lt;/p&gt;
&lt;p&gt;There is a profound irony in the fact that, while national politics fragments into identitarianisms and chauvinisms, science remains one of the few domains where international collaboration actually works — where a paper with coauthors from ten countries on four continents is completely routine, where the data from a telescope is made freely available to researchers at any university in the world.&lt;/p&gt;
&lt;p&gt;This is not because scientists are angels — science has its nationalisms and its ego wars as much as any human field. It is because the nature of the activity forces collaboration: reality does not bend to national preferences or tribal identities. You need everyone’s data, everyone’s critiques, everyone’s replications. The real is the final arbiter, and it is relentless and impartial.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;epilogue-the-inheritance-we-are-leaving&quot;&gt;Epilogue: The Inheritance We Are Leaving&lt;/h2&gt;
&lt;p&gt;The children who will be born in the early 2030s will inherit a world with serious problems — some of the ones we leave behind, others that have always existed and we have not yet solved. But they will also inherit tools that earlier generations did not have.&lt;/p&gt;
&lt;p&gt;They will inherit a medicine that begins to understand aging as a manipulable biological process, not as destiny. The generation born in the 2030s may live with a life expectancy that the biology of longevity is beginning to make technically plausible — exceeding 100 years as the norm, not the exception. Not because someone will discover the “fountain of youth,” but because the accumulation of small advances — senolytics that remove senescent cells, epigenetic therapies that restore younger patterns of gene expression, genetic editing that corrects hereditary defects before they cause damage — will create a systematic medicine of aging where today there is only palliation.&lt;/p&gt;
&lt;p&gt;They will inherit materials with properties nature never produced — metamaterials that bend light in impossible ways, 2D materials with unprecedented electronic properties, structural batteries where the casing of the object is its source of energy, superconductors that (we hope) will finally work at room temperature and transform the entire energy transmission infrastructure of the planet.&lt;/p&gt;
&lt;p&gt;They will inherit energy sources that do not destroy the climate. The combination of solar, wind and solid-state storage, with advanced SMR fission as a backbone for baseload demand, and with fusion as the long-term horizon, represents an energy architecture radically different from the one we inherited from the twentieth century. The transition is painful and politically complex, but the technical components are all under development or already being deployed.&lt;/p&gt;
&lt;p&gt;They will inherit computing that simulates reality with growing fidelity — simulations of protein folding, of chemical reactions, of climate dynamics, of materials, of neural circuits — which will make it possible to experiment digitally before experimenting physically, compressing the discovery cycle in all fields simultaneously.&lt;/p&gt;
&lt;p&gt;They will inherit verified mathematics, where the foundational proofs of mathematical civilization will be in mechanically auditable databases, and where AI systems will be collaborating with human mathematicians on problems that exceed the capacity of any individual mind.&lt;/p&gt;
&lt;p&gt;They will inherit interfaces between mind and machine that, if developed wisely and governed equitably, can give communication back to people who lost it, restore movement to paralyzed bodies, and amplify cognitive capacities in ways we cannot yet fully imagine — just as Gutenberg could not imagine the internet, but his work made both possible.&lt;/p&gt;
&lt;p&gt;They will inherit editable biology — the ability to redesign organisms, restore degraded ecosystems, create biological materials with customized properties, and perhaps, eventually, to intervene in the very process of human aging in ways that fundamentally change what the trajectory of a life means.&lt;/p&gt;
&lt;p&gt;And they will inherit the conversation we are having now — the sense that the future is not something that happens to us, but something we build. That science is not a university department, but a form of relationship with the real: systematic, humble before the data, ambitious in its questions, patient with the process.&lt;/p&gt;
&lt;p&gt;But — and this must be said with equal emphasis — they will also inherit the responsibilities we have created. Every powerful technology is an instrument of transformation in both directions. The genetic editing that cures hereditary diseases is the same that could create biological divisions between social classes. The AI that accelerates scientific discovery is the same that can concentrate informational and economic power in unprecedented ways. The brain-machine interfaces that give communication back to patients are the same whose principles, in adversarial contexts, pose risks to mental privacy.&lt;/p&gt;
&lt;p&gt;Science delivers tools. What we do with them is a question science cannot answer — it is a philosophical, political, ethical question. It is the question future generations will inherit along with the tools.&lt;/p&gt;
&lt;p&gt;The neuroscientist and philosopher António Damásio argues that emotions are epistemic — that empathy, admiration and enthusiasm are not decorations of reason, but part of its structure. Excitement about science is not naivety, it is not hype. It is an adequate emotional response to the scale of what is being accomplished. It is the right emotion in front of the right thing.&lt;/p&gt;
&lt;p&gt;When Vera Rubin observed the rotation curves of galaxies and realized there was something deeply wrong with the physics she knew — what astronomers came to call dark matter — she was not excited because she had the answer. She was excited because she had found a question that could change everything. When the BrainGate researchers saw a patient move a cursor for the first time with pure thought, they were not celebrating a finished technology. They were seeing the first pixel of an entire universe that was beginning.&lt;/p&gt;
&lt;p&gt;When a nuclear fusion researcher spends one more day adjusting magnetic parameters in a tokamak, knowing that they may not live to see the commercial reactor their work is helping to make possible, that person is making a kind of commitment that goes beyond the pragmatic. They are betting on the future. They are saying, through the daily practice of research, that the tomorrow of other human beings they will never meet is worth the present effort.&lt;/p&gt;
&lt;p&gt;That is what genuine scientific optimism looks like: not the certainty that all is well, not ignorance of the difficulties, but the recognition that frontiers exist to be crossed. That each impossible that science knocks down opens space for the next impossible. That the project does not end.&lt;/p&gt;
&lt;p&gt;The system we created — with its perversities of “publish or perish,” its cycles of hype and disenchantment, its disciplinary fiefdoms and its inequalities in access to knowledge — is real and creates real obstacles. But the system is not science. The system is the structure that houses it at a given historical moment. Science is the practice — the act of asking and seeking an answer with rigor and intellectual honesty — that transcends any particular system hosting it.&lt;/p&gt;
&lt;p&gt;The people who do this practice with love and seriousness, who stay up at three in the morning because the equation does not close, who care for their data with integrity even when it would be more convenient not to, who publish negative results knowing that no one will cite them, who teach the next generation with patience — those people are the best there is in the human civilizational project.&lt;/p&gt;
&lt;p&gt;And that is why the tomorrow we are building has already begun.&lt;/p&gt;
&lt;hr&gt;
&lt;p&gt;&lt;em&gt;“Most good research is not motivated by practical application. It is motivated by curiosity. That is why it works so well — curiosity does not lie.”&lt;/em&gt; — Freeman Dyson&lt;/p&gt;
&lt;p&gt;&lt;em&gt;“Reality is neither at the departure nor at the arrival: it reveals itself to us in the middle of the crossing.”&lt;/em&gt; — João Guimarães Rosa&lt;/p&gt;
&lt;p&gt;&lt;em&gt;“If you have ten years to solve a problem, spend nine of them understanding the problem.”&lt;/em&gt; — attributed to Évariste Galois&lt;/p&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>science</category><category>technology</category><category>artificial intelligence</category><category>philosophy</category><category>civilization</category><category>featured essays</category></item><item><title>Flesh and Entropy</title><link>https://matheussorancophilomathes.com.br/en/essays/flesh-and-entropy/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/flesh-and-entropy/</guid><description>An essay on consciousness in a universe of entropy: what depression sees, why we refuse Nozick&apos;s experience machine, and what art, free science and technology can and cannot do about suffering.</description><pubDate>Mon, 02 Mar 2026 23:42:37 GMT</pubDate><content:encoded>&lt;h2 id=&quot;i-the-nature-of-the-problem-you-were-born-into-a-universe-that-does-not-need-you&quot;&gt;I. The Nature of the Problem: You Were Born into a Universe That Does Not Need You&lt;/h2&gt;
&lt;p&gt;The universe is roughly 13.8 billion years old. For almost all of that span, there was no one there to suffer. That matters. Not as comfort — it is not comforting — but as context. Consciousness, that strange and still unexplained phenomenon by which matter folds back on itself and begins to observe its own existence, is the most improbable event the cosmos has ever produced, and perhaps the most unfortunate.&lt;/p&gt;
&lt;p&gt;Physicists tell us that the universe is moving toward thermal equilibrium. That entropy — the tendency of everything toward the disordered, the diffuse, the formless — is the most fundamental law we know. Stars explode. Planets cool. Organisms rot. Civilizations collapse. Memories dissolve. Everything that has structure, organization, a recognizable form — all of it is a temporary opening, an improbable parenthesis in the universe’s inexorable march toward a state in which nothing matters anymore, because nothing remains organized enough for anything to matter.&lt;/p&gt;
&lt;p&gt;In this context, the human being is a footnote. A statistically absurd occurrence in a galaxy of four hundred billion stars, which is itself only one among an estimated two trillion galaxies in the observable universe. You are made of carbon, hydrogen, oxygen and nitrogen — atoms that existed before the first vertebrate and will exist after the last. The difference between you and the dust of a supernova is only a matter of temporary arrangement. And the arrangement comes undone. Always.&lt;/p&gt;
&lt;p&gt;So far, this is cosmology. The part that concerns us — the part Ernest Becker spent his life trying to articulate, and which severe depression makes immediately, cruelly obvious — begins when this temporary arrangement of atoms develops the capacity to perceive that it is temporary. When matter learns to fear its own dispersal. When organized carbon begins to know that it will disorganize, and that this knowing is itself a form of suffering that no other arrangement of matter we know of experiences.&lt;/p&gt;
&lt;p&gt;You did not choose this. You did not choose to exist, you did not choose to be conscious, you did not choose the emotional structure that makes suffering an inalienable part of any experience worth the name. You appeared. And now you have to deal with it.&lt;/p&gt;
&lt;p&gt;This is the situation. Not as a metaphor, not as dramatic exaggeration. As a clinical description of the human condition: you are an animal that knows it will die, that does not know why it exists, that has desires the universe has no obligation to satisfy, that experiences a consciousness that seems irreducibly subjective and at the same time seems destined for complete extinction, and that has to cross decades of existence with this weight before the weight finally crushes it for good.&lt;/p&gt;
&lt;p&gt;And at some point in history, this animal invented philosophy. It invented religion. It invented art and science and narrative and ritual. Not because the universe asked it to. But because the alternative — facing the situation with no mediation at all — is something very few bodies and minds can sustain for very long.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;ii-what-depression-knows-that-the-rest-of-us-prefer-to-ignore&quot;&gt;II. What Depression Knows That the Rest of Us Prefer to Ignore&lt;/h2&gt;
&lt;p&gt;There is an idea that recurs in certain philosophical traditions and that mainstream psychology tends to reject with the anxiety of someone who recognizes a dangerous truth: that severe depression may be, at least in part, an accurate perception. Not a cognitive distortion to be corrected, not a neurochemical imbalance simply to be rebalanced, but a way of seeing the world that strips away filters most people keep running out of a need for emotional survival.&lt;/p&gt;
&lt;p&gt;I am not romanticizing depression. It kills. It destroys years, relationships, capacities. It is not a state of enlightenment; it is a state of suffering. But there is in it, especially in its more philosophical and less merely biological forms, a kind of bitter lucidity that deserves to be taken seriously instead of simply anesthetized.&lt;/p&gt;
&lt;p&gt;What depression sees, when it sees clearly, is this: the structure of human existence is fundamentally characterized by suffering, and conventional health is, to a large extent, a set of techniques for not fully perceiving that. The depressed person can no longer sustain the narratives that keep the world bearable. They cannot believe that things will improve permanently, that suffering has meaning, that there is any guarantee the effort is worth it. They see the cycle: you are born, you struggle, you love things that will end, you create projects that will be forgotten, you grow old, you die, and the stars carry on without noticing.&lt;/p&gt;
&lt;p&gt;Becker called this the human condition, and argued that all of culture is an attempt not to see it directly. That belief systems, ideologies, projects of meaning are structurally analogous to anesthetics — not false in everything they say, but motivated by the need to make the intolerable tolerable. And that the price of this anesthesia is the mechanism by which we turn other human beings into scapegoats, into incarnations of evil, into enemies whose destruction confirms our importance and keeps death at bay.&lt;/p&gt;
&lt;p&gt;Depression, in this sense, is the anesthetic that stopped working. It is the state in which the organism can no longer sustain the necessary illusions. And this is devastatingly painful — not because the illusions were true, but because they were functional. Because they helped get through the day. Because they made it possible to love, to create, to struggle, to hope. And without them, everyday life becomes direct exposure to the cold wind of cosmology: you are small, you are alone on a scale that has no adequate vocabulary, and the universe that produced your consciousness has no special obligation toward it.&lt;/p&gt;
&lt;p&gt;There is an implicit step in this observation that needs to be unfolded with care: if depression sees things that conventional health does not, and if what it sees is real, then the way forward cannot be simply to restore the illusions. It cannot be merely to rebalance the serotonin and send the person back into the same system that produced them. The way — if there is one — has to pass through what depression revealed, not through forgetting what was seen.&lt;/p&gt;
&lt;p&gt;This does not mean that medical treatment is wrong. It means that medical treatment, when it works, creates conditions — a window of functional capacity — within which a different kind of work becomes possible. The work of building ways of inhabiting the human condition that are more honest about it than the conventional ones. Less dependent on the great illusions that eventually collapse, more able to find a reason to go on from what is real and present rather than what is promised and transcendent.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;iii-pain-as-the-structure-of-the-real&quot;&gt;III. Pain as the Structure of the Real&lt;/h2&gt;
&lt;p&gt;There is a thought that systems of positive thinking, the culture of gratitude and the wellness industry conspire to suppress because they are economically dependent on suppressing it: suffering is not an anomaly. It is the texture of conscious existence in a universe of entropy.&lt;/p&gt;
&lt;p&gt;This is not pessimism in the sense of a choice of perspective that could be swapped for a cheerier one. It is an observation about the structure of experience. Every pleasure is transitory. Every attachment ends in loss. Every love contains, within it, the seed of pain — because everything that is loved is mortal, and love knows it before the rational mind admits it. Every project of meaning ultimately confronts the silence of a universe that does not answer. Every achievement ages. Every memory fades.&lt;/p&gt;
&lt;p&gt;Buddhism, which is in many respects the most honest philosophy of suffering the species has produced, begins with the declaration that life is dukkha — often translated as suffering, but more precisely as unsatisfactoriness, incompleteness, the quality of being fundamentally unable to remain in a state of complete satisfaction. Not because we are weak or mistaken, but because we are creatures of desire in a universe that does not permanently sustain any object of desire.&lt;/p&gt;
&lt;p&gt;What Buddhism does with this is its own discussion. But the diagnosis is of interest independently of the prescription: unsatisfactoriness is not the problem to be solved. It is the baseline condition of experience. And any system that promises definitive resolution is, at some level, lying.&lt;/p&gt;
&lt;p&gt;But there is another dimension of suffering that is not merely Buddhist unsatisfactoriness, and which Western thought captures in a way Buddhism sometimes softens: pain as a signal of how much we matter. Pain does not exist without attachment. A being that cares about nothing does not suffer. And a being that does not suffer — fully immune to loss, to disappointment, to grief — cannot love, cannot create, cannot find anything genuinely beautiful, because beauty requires the awareness that it will be lost.&lt;/p&gt;
&lt;p&gt;Here we arrive at one of the most disturbing and most true ideas philosophy has produced, rigorously embodied by Robert Nozick in a thought experiment that has haunted ethical thinking ever since it was formulated. The question is simple: would you plug yourself into a machine capable of simulating any experience you desire, with perfect realism, forever? An existence of complete happiness, constant pleasure, uninterrupted satisfaction — all of it fabricated, none of it real, but indistinguishable from the real from inside the experience.&lt;/p&gt;
&lt;p&gt;Most people, when honestly confronted with this question, answer no. And this refusal reveals something fundamental about being human. We do not merely want positive experiences. We want reality. We want our achievements to be real, our relationships to be real, our suffering, when it occurs, to be the price of something genuinely lived. We prefer authentic pain to fabricated pleasure. We prefer real mourning to simulated joy.&lt;/p&gt;
&lt;p&gt;What does this mean? It means that happiness is not the supreme good our culture proclaims. It means there is something we value more than feeling good — and that this something is, at its core, the authenticity of experience, contact with the real, the assurance that what we feel is a response to something that actually exists and matters. It means we prefer to be creatures of genuine suffering rather than simulacra of happy creatures.&lt;/p&gt;
&lt;p&gt;And this — this refusal of the experience machine — is the most honest confession the species has made about itself. We are creatures that need reality more than happiness. That need things to matter, even if mattering means hurting. That prefer real tragedy to fabricated comedy.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;iv-ernest-becker-and-the-price-of-self-knowledge&quot;&gt;IV. Ernest Becker and the Price of Self-Knowledge&lt;/h2&gt;
&lt;p&gt;Becker died in 1974, weeks before receiving the Pulitzer. He was being interviewed as the cancer advanced. He spoke about death with a calm his interlocutors described as unsettling — not the calm of someone who has made peace with the situation, but the calm of someone who has looked at it so directly for so long that the tremor moved somewhere deeper, less visible, perhaps more permanent.&lt;/p&gt;
&lt;p&gt;Becker’s intellectual project was, at bottom, an attempt to answer a simple and impossible question: why do we do what we do? Why does the human being, who has the rational capacity to recognize that violence is destructive and that hatred generates suffering, keep producing violence and hatred on a civilizational scale? Why did the species that wrote the Goldberg Variations and formulated general relativity also design gas chambers and build gulags?&lt;/p&gt;
&lt;p&gt;The answer he found — and which distinguishes him from more optimistic social psychologists and more systematic philosophers — is that the root of human evil does not lie in wickedness, ignorance or corruption by deficient institutions. It lies in terror. In the terror of death that is constitutive of human consciousness and that demands, in order to be made tolerable, a set of strategies that inevitably involve harm to others.&lt;/p&gt;
&lt;p&gt;We build for ourselves projects of meaning that promise to make us larger than our biological death. Ideologies, nations, religions, arts, sciences, families, legacies — all of them are ways of inscribing our existence in time so that it transcends it. And anything that threatens these projects activates a terror that is, at bottom, the terror of extinction. The ideological enemy is not merely someone we disagree with; he is someone whose mere existence, holding alternative beliefs, suggests that our construction of meaning may be wrong — and if it is wrong, we die in a way that is not only biological.&lt;/p&gt;
&lt;p&gt;The scapegoat — that mechanism by which we locate the filth, the threat, the proximity to death in the body and existence of another — is not a pathology of monsters. It is the ordinary defense of ordinary people whose construction of meaning is under enough pressure. It is not the serial killer who embodies the principle; it is the normal citizens who take part in collective processes of dehumanization because the alternative — recognizing that the other is as terrified and as vulnerable as they are — would undo the certainties on which they depend in order to function.&lt;/p&gt;
&lt;p&gt;“It is the evil ones who are the most afraid.”&lt;/p&gt;
&lt;p&gt;This sentence of Becker’s has the disturbing quality of truths we would rather not meet. It does not allow us the comfortable division of the world into predators and prey, monsters and innocent victims. It puts everyone on the same side of a different divide: all terrified, all building defenses, all capable of producing evil from the same emotional substrate of vulnerability and fear. The difference between what we call good and what we call evil is, to a great extent, a matter of which defenses were built and how much terror was needed to build them.&lt;/p&gt;
&lt;p&gt;This does not dissolve moral responsibility. It does not make the aggressor and the victim equivalent. But it transforms the way we see the origin of harm, and that transformation is necessary if we want anything resembling wisdom about how to treat the problem.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;v-the-universe-as-indifference-the-cold-beauty-of-what-does-not-care-for-you&quot;&gt;V. The Universe as Indifference: The Cold Beauty of What Does Not Care for You&lt;/h2&gt;
&lt;p&gt;There is a philosophical tradition, running from Lucretius to Spinoza to Camus, that finds a peculiar kind of freedom in the indifference of the cosmos. Not the freedom of the absence of obligations — that is a cheap freedom, which any lazy nihilist can claim. The harder and more interesting freedom that emerges from confronting cosmic indifference is a different one: the freedom that the meaning you build does not need to be approved by any transcendent authority in order to be real.&lt;/p&gt;
&lt;p&gt;The universe did not ask you to exist. It owes you nothing. It will not notice when you go. The stars you can see on a clear night are at distances that make the concept of contemporaneity biologically absurd — the light you see left those bodies years, decades, millennia ago, and many of them no longer exist. You are looking at ghosts of fire on a scale that dissolves any illusion of centrality.&lt;/p&gt;
&lt;p&gt;And this is, at once, devastating and strange and — if you are willing to sustain the observation long enough — incredibly beautiful.&lt;/p&gt;
&lt;p&gt;The beauty is not in spite of the indifference. It is within it. The fact that the universe does not need you in order to keep turning is the same structure that makes it possible for it to turn with a regularity that can be described mathematically, that can be anticipated, that can be understood by arrangements of matter complex enough to be you. Indifference and regularity are the same phenomenon. And this regularity — the law that governs the motion of the planets and the fall of bodies and the propagation of light — is in itself one of the most astonishing things human consciousness has ever encountered.&lt;/p&gt;
&lt;p&gt;Becker was right that we cannot live completely exposed to cosmology. We cannot go through every day fully conscious of the vastness and the entropy and the statistical insignificance of our existence. We need narratives, projects, belonging. We need the functional illusions that make the day bearable and love possible.&lt;/p&gt;
&lt;p&gt;But there is a difference — small in appearance, enormous in consequence — between illusions built on the denial of the real and ways of inhabiting existence that do not need to deny what is. The person who loves knowing that the beloved will die, who creates knowing that the work will be forgotten, who seeks knowledge knowing they will never reach the ultimate foundation — that person is not less capable of love, of creation, of seeking. They are more capable. Because what they do is real, weighs exactly what it weighs, is not inflated by false promises of permanence.&lt;/p&gt;
&lt;p&gt;The indifference of the universe is not a message about your worth. It is simply the way the universe is. And the way the universe is includes, among its countless properties, that of having produced creatures capable of contemplating it with wonder. This is not cosmic meaning. It is not a design argument. It is just one more remarkable property of a cosmos that never stops surprising whatever is left of our capacity for surprise.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vi-mark-fisher-and-the-present-without-exit-the-death-of-cultural-death&quot;&gt;VI. Mark Fisher and the Present Without Exit: The Death of Cultural Death&lt;/h2&gt;
&lt;p&gt;There is a specific kind of suffering that the contemporary world has produced and that does not yet have a completely adequate name. It is not clinical depression, although it overlaps with it. It is not grief, although it shares its quality of irrecoverable loss. It is not simply the existential ennui that earlier centuries already knew. It is something specific to our time, something Mark Fisher tried to articulate until the point at which he could no longer go on.&lt;/p&gt;
&lt;p&gt;Fisher called capitalist realism the inability to imagine alternatives — not as an intellectual position, but as an emotional and perceptual fact. The visceral sense that the way things are is the only way they could be. Not the worst possible choice, not the fairest, not the most humane, but simply the only real one. As if we had quietly crossed some imperceptible border beyond which utopian imagination ceased to be possible, not because its arguments were refuted, but because the very capacity to conceive radically different worlds was progressively erased.&lt;/p&gt;
&lt;p&gt;But what matters here is not the political dimension of this diagnosis — it is the existential dimension. What happens to the pain of living when the future disappears as a real horizon? When tomorrow is foreseen as more of the same, more accelerated, more exhausted, more precarious — not different in structure, only more intense in its repetition?&lt;/p&gt;
&lt;p&gt;Fisher wrote about the hauntology of contemporary culture — a culture that does not advance but circulates among its own ghosts, that recycles the forms of the past without managing to produce anything genuinely new, that experiences the present as absence rather than presence. The music that sounds as if it came from another decade. The films that are remakes of remakes. The retro aesthetic that is not nostalgia for a time that existed, but for possibilities that never came to be realized — futures that were promised and did not arrive, and that we keep haunting because the memory of their possibility is more vivid than any present we have managed to build in their place.&lt;/p&gt;
&lt;p&gt;For those who live with severe depression, this cultural structure has a particularly cruel resonance: the outside world seems to confirm what depression says internally. The system produces and distributes, the platforms capture and anesthetize, content flows at industrial volume — and none of it seems to move the needle on what fundamentally matters. Life goes on, time passes, and the sense that something essential is missing, that there is a quality of presence and meaning that should be here and is not, persists beneath everything like a background noise that no amount of entertainment can completely silence.&lt;/p&gt;
&lt;p&gt;Fisher died in 2017, in circumstances his friends described as the accumulated product of decades of depression and of a relationship with suffering that was as intellectual as it was utterly visceral. He had written about it with an honesty that remains disturbing precisely because it asks for neither pity nor admiration — it simply names what is.&lt;/p&gt;
&lt;p&gt;What Fisher understood, and what the wellness industry has an interest in obscuring, is that certain forms of suffering are not malfunctions of the individual. They are rational responses to structures that systematically produce the conditions for suffering while selling the instruments for not noticing it.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;vii-art-as-scar-what-remains-after-everything-passes&quot;&gt;VII. Art as Scar: What Remains After Everything Passes&lt;/h2&gt;
&lt;p&gt;If life is fundamentally the experience of loss — and it is; any honest analysis eventually arrives at that — then art is the species’ attempt to convert loss into something that remains after what was lost is gone. Not the thing itself, but the mark it left. Not the beloved, but the poem about the beloved. Not the joy that passed, but the music that captured its specific texture at a specific moment that never repeats.&lt;/p&gt;
&lt;p&gt;Art, in its deepest and most necessary sense, is not entertainment. It is not an ornament of existence. It is the only way we have found of making time carry weight. Of insisting that this specific moment — this particular light of this particular afternoon, the exact way this person laughed, this exact knot in the throat of this grief — not simply be swallowed by oblivion without leaving any trace.&lt;/p&gt;
&lt;p&gt;It does not matter whether the trace survives. The act of creating it is already what matters. Dostoevsky did not write his novels for posterity, at least not primarily. He wrote because there was inside him a pressure of meaning that needed form, that would explode if it found no outlet, that destroyed him from within when he tried not to write. Genuine art is not a rational choice among options; it is a necessity, as physical as the need to breathe, that emerges from an intensity of experience that cannot resolve itself internally.&lt;/p&gt;
&lt;p&gt;And that is why mass art — in the sense Fisher diagnosed with surgical precision — is not simply art of inferior quality. It is the negation of art as a function. Commodified art does not emerge from an intolerable internal pressure; it is produced in response to market demands. It does not convert pain into a form that endures; it converts attention into measurable engagement. It does not demand of its audience the work of presence that transforms; it demands the passive consumption that transforms nothing.&lt;/p&gt;
&lt;p&gt;There is no overstating how much this matters existentially. Because if art is the form the species found of converting the experience of time into something that weighs, that marks, that resists oblivion — then art that does not do this does not merely fail aesthetically. It deprives its consumers of something human experience would need in order to remain bearable.&lt;/p&gt;
&lt;p&gt;Contemporary culture is full of content that occupies time without weighing anything. That fills space without leaving a mark. That entertains without transforming. And the accumulation of this content does not add up to more meaning; it produces a specific noise, a kind of surface anesthesia that prevents both the deepest pain and the deepest joy. A constant affective middle ground where nothing quite hurts and nothing is quite genuinely beautiful.&lt;/p&gt;
&lt;p&gt;This is what Becker would call alienation of a specific depth: not the worker’s alienation from the product of his labor, but the conscious being’s alienation from its own experience. The impossibility of being fully present in any moment, because the system is designed to capture and redistribute attention before it can deepen enough to produce something irreversible, like meaning.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;viii-transhumanism-the-last-bet-against-decay&quot;&gt;VIII. Transhumanism: The Last Bet Against Decay&lt;/h2&gt;
&lt;p&gt;There is a current of thought that emerges at the intersections of analytic philosophy, biology, neuroscience and technology, and that proposes something that would have sounded like messianic delusion in any other historical context, but that today is debated in laboratories and foundations with real funding: the idea that death is a technical problem, that suffering has identifiable biological causes, and that both can, in principle, be solved through the extension and intensification of the scientific project.&lt;/p&gt;
&lt;p&gt;Transhumanism, in its most rigorous forms — distinct from the naive technological optimism that is simply capitalism with a better narrative — starts from an observation Becker would have recognized: the human condition as currently configured imposes suffering that is not necessary, that is the product of contingent biological limitations, not of any higher moral law requiring that we live short, painful and forgotten lives. Finitude is not sacred. It is simply the current state of a still incomplete process.&lt;/p&gt;
&lt;p&gt;The question transhumanism poses — and which deserves to be taken seriously before being accepted or rejected — is: if we could remove the biological causes of suffering, should we? If we could extend life so as to make possible the full development of the capacities that premature death interrupts, would that be a betrayal of the human condition or the fulfillment of what is most genuinely human in us — the refusal to accept the limits that chance imposed?&lt;/p&gt;
&lt;p&gt;Here Nozick’s experience machine reappears, but now in a different and more disturbing configuration. Because the transhumanist proposal is not exactly the experience machine — it does not offer fabricated happiness in place of reality. It offers, in principle, more reality. More time to have real experiences. More cognitive capacity to process real experiences more deeply. Less suffering of biological origin that contributes nothing to meaning — the pain of arthritis, the degradation of dementia, the terror of premature death — in favor of forms of suffering that emerge from conscious activity in the world itself and that are therefore, in the sense Nozick suggests, sufferings we would choose over their absence.&lt;/p&gt;
&lt;p&gt;But there is a tension in this proposal that cannot easily be resolved. Becker showed that the terror of death is not only suffering to be eliminated; it is the engine that drives all human creation of meaning. Our symbolic immortality — art, science, philosophy, love, children, the projects that outlive us — exists precisely because we are mortal. The awareness of finitude is what gives existence its urgency. What makes us love with the intensity with which we love, create with the desperation with which we create, seek with the hunger with which we seek.&lt;/p&gt;
&lt;p&gt;If death were removed — and this is a genuinely disturbing thought, not an abstract exercise — what would happen to meaning? Would the urgency that finitude lends to love and creation depend on some other source of existential pressure? Or would we be closer to the experience machine than we would like to admit — long, comfortable, technically enriched existences, but deprived of the specific weight that limited time lends to everything it contains?&lt;/p&gt;
&lt;p&gt;I do not know the answer. I suspect no one does. But the question matters because it reveals that the transhumanist project, even in its most rigorous and least naive forms, has to confront the possibility that part of what is most valuable in human experience — not merely tolerable, but genuinely valuable — is inseparable from the suffering it aims to eliminate.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;ix-the-neuroscience-of-suffering-and-its-limits&quot;&gt;IX. The Neuroscience of Suffering and Its Limits&lt;/h2&gt;
&lt;p&gt;Contemporary neuroscience has mapped the neural correlates of suffering with growing precision. It has identified networks, neurotransmitters, activation patterns that correspond to states of pain, anxiety, depression. And from this mapping it has produced pharmacological and technological interventions that can alter these states with an efficacy that would have been indistinguishable from magic to any earlier generation.&lt;/p&gt;
&lt;p&gt;This is genuinely important. Genuinely valuable. Low serotonin is not a metaphor; it is a biological state that produces real suffering, that can be treated, that does not need to be romanticized as a clear view of the real. Suffering of purely neurochemical origin — what arises not from confronting real loss, not from confronting the existential condition, but from the simple dysfunction of circuits that can be rebalanced — is suffering that does not need philosophy. It needs medicine.&lt;/p&gt;
&lt;p&gt;But neuroscience has a limit that its most enthusiastic defenders often underestimate, and which is precisely the limit relevant to any discussion of existential suffering: it can describe the neural correlates of an experience without describing what it is like to have that experience. It can map brain activity during grief without capturing what it means to lose someone who was a constitutive part of your way of inhabiting the world. It can identify the activation patterns of the amygdala during existential terror without explaining what it means to know that you will die and that there is no guarantee it will matter to anything after it happens.&lt;/p&gt;
&lt;p&gt;The hard problem of consciousness — the problem of explaining why there is something it is like to be this specific person, with this specific subjective experience, now — remains the most intractable problem science has ever encountered. And until it is solved, or until we have solid reasons to believe it cannot be solved, neuroscience is a description of suffering from the outside, and what we need in order to inhabit it is an understanding from the inside.&lt;/p&gt;
&lt;p&gt;Free science — and here the word free is the key — is not afraid of this limit. Science that is not captured by the imperative of producing marketable interventions can look at the hard problem of consciousness and say: we do not know. And it can keep investigating without needing the answer to confirm any prior ideology about what the human being is or ought to be.&lt;/p&gt;
&lt;p&gt;This epistemological honesty is rare. It is rarer still in a culture that demands ready answers, immediate solutions, commercially exploitable certainties. But it is the only form of inquiry that can eventually arrive at something worthy of the name of knowledge about the condition we are trying to understand.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;x-what-is-human-art-science-and-technology-as-a-definition-in-progress&quot;&gt;X. What Is Human: Art, Science and Technology as a Definition in Progress&lt;/h2&gt;
&lt;p&gt;All technology is an extension of human capacity. The hammer extends the arm. Writing extends memory. The telescope extends the eye. The internet extends communication. And each extension redefines, subtly or dramatically, what counts as human — what lies within the scope of what we are, and what lies outside.&lt;/p&gt;
&lt;p&gt;This redefinition is permanent and has no natural endpoint. We are, more than any other animal, the species that defines itself through the capacity to redefine what it is. We have no fixed ecological niche. We have no instinctive behaviors specific enough to determine what we are without our being able to modify them. What we are is largely the product of what we choose to do with the capacities we have — and those capacities increasingly include the capacity to modify the capacities themselves.&lt;/p&gt;
&lt;p&gt;Artificial intelligence represents the most extreme point this process has reached so far: not the extension of a specific capacity, but the possibility that capacities we have always defined as specifically human — language, reasoning, creation, learning — can be produced by systems that are not made of flesh, that have no subjective experience (as far as we know), that have no fear of death because they have no death in the sense that matters.&lt;/p&gt;
&lt;p&gt;And here existentialism and technology meet in a question with no obvious answer: if a system produces language indistinguishable from human language, poetry indistinguishable from human poetry, reasoning indistinguishable from human reasoning — but without the suffering that has always been, in the Beckerian analysis, the engine of these products — what does that say about suffering? Was it an implementation flaw, a bug to be fixed in the next version? Or was it what made language, poetry and reasoning genuinely human — not in spite of being painful, but precisely because it was?&lt;/p&gt;
&lt;p&gt;The question is not rhetorical. It has enormous practical consequences for any project of understanding what we are and what we can be. If suffering is a defect, then transhumanism is right, and the project of eliminating it is the noblest we can conceive. If suffering is constitutive — if it is what gives weight to experience, what makes love love and not a simulation of love, what makes beauty hurt in the specific way we need it to hurt — then any project that eliminates it completely is, in Nozick’s refusal of the experience machine, a project of making us less than we are in the name of making us more comfortable.&lt;/p&gt;
&lt;p&gt;The most honest position, though also the most uncomfortable, is that we do not know. That we are, as a species, at an inflection point where the choices we make about how to use and develop these technologies will define what will be human from here on — without a guide, without a guarantee, without a sufficiently relevant precedent.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xi-the-suffering-body-and-the-observing-mind&quot;&gt;XI. The Suffering Body and the Observing Mind&lt;/h2&gt;
&lt;p&gt;There is a specific way of experiencing severe depression that deserves to be described precisely, not because the description helps anyone inside it — it does not, at the moment when it is deepest — but because it contains information about the structure of human experience that conventional health erases.&lt;/p&gt;
&lt;p&gt;In this state, the mind observes the suffering body with a kind of cold distance. Not the relief of dissociation, but something crueler: the clarity of perceiving that the suffering is real and that the observer who perceives it is powerless to stop it. Thought works — crookedly, slowly, deformed by the emotional demands of the state, but it works. It can perceive. It can reason in short lines. It can, at times, articulate precisely what is happening, where it comes from, how it relates to the broader structure of the human condition.&lt;/p&gt;
&lt;p&gt;And what it perceives, when it perceives with this bitter clarity, is this: suffering does not need an external justification in order to exist. It is not a response to any specific loss, any identifiable event, any cause that, once removed, would restore the state of well-being. It is there like a quality of the system, like a background noise that all surface activity tries to mask and that silence, or exhaustion, or the simple interruption of the usual noise, reveals with a clarity that is in itself yet another form of suffering.&lt;/p&gt;
&lt;p&gt;This confirms, in a way no philosophical argument can do as efficiently, the Buddhist diagnosis of dukkha: suffering is not in the circumstances. It is in the structure of the system that experiences the circumstances. It precedes any specific content. It is the texture of conscious experience in an organism that desires, that attaches, that fears loss — and that cannot help desiring, attaching and fearing, because these are the ways in which experience becomes vivid.&lt;/p&gt;
&lt;p&gt;And here the most sophisticated transhumanist proposal finds its most honest version: not the elimination of suffering, which would imply the elimination of everything that suffering makes possible, but the reduction of purely mechanical suffering — the kind that arises from no genuine contact with the real, that is simply the system spinning its wheels, producing pain without producing meaning — in favor of forms of existence in which the suffering that remains is proportionate, is a genuine response to something that genuinely matters.&lt;/p&gt;
&lt;p&gt;This distinction — between suffering that informs and suffering that merely consumes — is not easy to operationalize. There is no clinical test that measures it. But it exists, it is experienced, and it is the difference between the pain of grief that transforms and the pain of the deepest depression that only wears down. Grief honors the loss by making it present, heavy, irreversible. The deepest depression produces only absence, emptiness, the sense that nothing that should matter manages to impose its weight.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xii-technology-as-an-extension-of-terror-what-we-build-with-fear&quot;&gt;XII. Technology as an Extension of Terror: What We Build with Fear&lt;/h2&gt;
&lt;p&gt;We inevitably return to Becker because his diagnosis has a prophetic quality that deepens over time rather than resolving. Every large-scale technological system modernity has produced embodies, in its structure and its effects, the same existential terror Becker identified as the driving force of human evil. Not because its creators are monsters, but because they are human, and humans build with the material they have — and the material they have is composed, in significant proportion, of fear.&lt;/p&gt;
&lt;p&gt;Artificial intelligence as a field is no exception. The races to develop ever more capable systems, the speed that sacrifices considerations of impact in favor of competitive advantage, the concentration of this development in a tiny number of entities with the power to shape the conditions of existence of billions of people — all of this has the structure Becker describes as characteristic of the great projects of symbolic immortality when they operate at civilizational scale.&lt;/p&gt;
&lt;p&gt;The corporations leading this development are not merely building tools. They are building the infrastructure of the future, the environment within which the next generations will exist, love, suffer, create meaning and die. And this project is motivated — not exclusively, but significantly — by the same Beckerian terror: the terror of being left behind, of being surpassed, of not being what will define what comes next. The technological race is, among other things, the competition between projects of symbolic immortality of unprecedented scale.&lt;/p&gt;
&lt;p&gt;And the product of this race is not neutral. The tools that emerge from this process carry the emotional structures of the process that produced them. Recommendation systems that optimize for engagement and not for well-being, because engagement is measurable and well-being is not. Platforms that amplify conflict because conflict — the activation of existential terror — is the state that most efficiently captures attention. Interfaces designed to produce dependence because dependence is profitable.&lt;/p&gt;
&lt;p&gt;It is not a conspiracy. It is the structural logic of what happens when existential terror and the imperative of accumulation operate together with no external force strong enough to impose other criteria.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xiii-free-science-as-heresy-knowledge-that-serves-no-one&quot;&gt;XIII. Free Science as Heresy: Knowledge That Serves No One&lt;/h2&gt;
&lt;p&gt;There is a form of intellectual activity that is, in strictly economic terms, parasitic. That produces no product, that generates no measurable return in the short term, that solves no problem anyone has identified as urgent on any spreadsheet. That exists simply because there are questions that need answers and minds that cannot stop asking.&lt;/p&gt;
&lt;p&gt;Pure mathematics. Theoretical physics at its most speculative reaches. Philosophy that does not double as corporate consulting. The evolutionary biology of species that will never be domesticated. The cosmology of phenomena that will never be harnessed for energy. The archaeology of civilizations with no living descendants. The study of dead languages no one speaks anymore.&lt;/p&gt;
&lt;p&gt;This activity — let us call it free science in a broad sense, including all inquiry driven by disinterested curiosity — is the clearest proof that the human being is something more than an organism of survival and reproduction. That there is something in us that wants to know not in order to use the knowledge, but simply to have the knowledge. That understanding has intrinsic value, regardless of its usefulness.&lt;/p&gt;
&lt;p&gt;And in Beckerian terms, this is significant. Free science is one of the few human activities whose project of symbolic immortality does not depend, in its normative structure, on the creation of scapegoats. The mathematician who solves a problem no one else had solved does not need any other mathematician to be destroyed for the achievement to be real. The scientist whose discoveries outlive them survives not through the elimination of those who disagree, but through the validity of what was discovered — a validity that requires, in principle, that anyone with the right instruments can verify it.&lt;/p&gt;
&lt;p&gt;This is a model of symbolic immortality that does not contaminate the world with its terror. That can be shared without being reduced. That grows rather than shrinks when others take part. And that is, in this sense, the cleanest form the human project of transcending biological death has found.&lt;/p&gt;
&lt;p&gt;It is not enough. It does not solve the existential problem. But it is what we have that comes closest to an answer to the human condition that does not need to sacrifice anyone on the altar of its own permanence.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xiv-the-impossible-dignity-of-carrying-on&quot;&gt;XIV. The Impossible Dignity of Carrying On&lt;/h2&gt;
&lt;p&gt;There exists, somewhere between the pessimism that honesty demands and the surrender that nihilism offers, a stance that has no convenient name but that we recognize when we meet it. It is the stance of someone who has seen what there is to see — the entropy, the terror, the structural suffering, the indifference of the cosmos, the inevitability of loss — and who has chosen to go on not because they were persuaded that things will improve, but because the alternative is a capitulation before something that, although true in its coldness, is not the whole truth.&lt;/p&gt;
&lt;p&gt;Because there is another truth that coexists with everything we have described. The truth that this specific morning existed and will never exist again with exactly these qualities of light and temperature and the specific way the coffee was hot. That this specific encounter between these two specific consciousnesses, even if brief, even if forgotten by both before the day was out, was real in a way the universe will never duplicate. That the equation the mind found, the sentence the pen produced, the gesture of care the body made toward another’s body — all these things occurred, and the fact that they occurred cannot be undone by subsequent forgetting.&lt;/p&gt;
&lt;p&gt;Buddhism calls impermanence what we are describing. But impermanence is not only loss. It is also the structure that makes it possible for each moment to be unrepeatable, for each encounter to be unique, for each experience to have the specific texture of its singular occurrence in time. Complete permanence would not be paradise; it would be the hell of immobility, the crystal world without variation that resembles death more than anything that contains change.&lt;/p&gt;
&lt;p&gt;Nozick was right to refuse the experience machine. But the deepest reason for this refusal is not that we want to struggle, to suffer, to strive instead of simply receiving. The deepest reason is that we want things to be real. We want love to be real, pain to be real, achievement to be real, loss to be real. And real means: it happened in the world, it left a mark, it was — and that was cannot be undone.&lt;/p&gt;
&lt;p&gt;This insistence on reality — on the irreversible quality of what happened, on the weight of what is real rather than what is comfortable — is, paradoxically, the most anti-depressive way of inhabiting existence that I know. Not because it produces happiness. But because it converts suffering into something that concerns the real world, instead of being simply the system spinning its wheels in a vacuum.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xv-epilogue-flesh-in-the-dark&quot;&gt;XV. Epilogue: Flesh in the Dark&lt;/h2&gt;
&lt;p&gt;The universe heads toward equilibrium. The stars will go out. The memory that there was life will last, at best, as long as it takes for the last structure organized enough to carry it to come apart. And after that, nothing that cares about it.&lt;/p&gt;
&lt;p&gt;You know this. The body knows it in a way that precedes any language — it knows it in the way the heart speeds up at the thought of its own extinction, in the way the mind recoils from the abyss when it leans too far over it, in the way the whole organism conspires not to look directly at what it is looking at when it looks directly at this.&lt;/p&gt;
&lt;p&gt;And in this refusal, in this conspiracy of the organism to carry on, there is something I cannot call anything other than beauty. Not the beauty that consoles. The beauty that haunts. The fact that this specific flesh, arranged in this specific way, insists on waking up tomorrow, on being hungry, on seeking human warmth, on asking questions about stars, on trying to understand why it suffers, on writing about all of this as if writing could change something fundamental in cosmology — that fact is, when seen without the usual defenses, one of the most extraordinary things in the universe that produced it.&lt;/p&gt;
&lt;p&gt;Not because it is special on a cosmic scale. But because it is the only point in the universe we know where the universe folded back on itself and began to ask why it exists. And this question — unanswered, with no guarantee of an answer, with no guarantee that it matters that it be asked — is in itself what we are, when we are most completely what we are.&lt;/p&gt;
&lt;p&gt;Suffering continues. Entropy continues. Death continues. Free science opens windows that do not close again once opened. Art converts the wound into a scar that can be touched, can be read, can say to another terrified being: you are not alone in this. Technology extends the reach of what the body can do and feel and understand, and each extension redefines what is possible without yet knowing what is desirable. And humans go on, provisionally, incompletely, with all the beauty and all the devastation that this provisionality and incompleteness imply.&lt;/p&gt;
&lt;p&gt;There is no resolution. There is only what is.&lt;/p&gt;
&lt;p&gt;And what is, sometimes, is enough to make it worth staying.&lt;/p&gt;
&lt;hr&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxii-the-experience-machine-revisited-on-what-we-refuse-and-why&quot;&gt;XXII. The Experience Machine Revisited: On What We Refuse and Why&lt;/h2&gt;
&lt;p&gt;Robert Nozick presented his experience machine in the context of a debate about philosophical hedonism — the theory that the supreme good is pleasure and that ethics should be oriented toward its maximization. His argument was simple in structure: if hedonism were correct, we should all want to plug into the machine. The fact that most people refuse, even when the experiment is presented with all the rigor that minimizes the usual response biases, suggests that we value something beyond pleasure — that the quality of reality of experience matters independently of its hedonic quality.&lt;/p&gt;
&lt;p&gt;But what Nozick perhaps did not fully explore is how much this refusal reveals about the structure of human suffering as a necessary condition of certain forms of good. We are not merely saying that we prefer reality to fabricated pleasure. We are saying that we prefer a world in which loss is real, because only in a world where loss is real can love be real. That we prefer a world in which failure is possible, because only in a world where failure is possible can achievement carry weight. That we prefer a world in which death is certain, because only in a world where death is certain can life have urgency.&lt;/p&gt;
&lt;p&gt;This is not masochism. It is the recognition that certain qualities of experience are inseparable from certain forms of vulnerability. Courage does not exist without fear. Compassion does not exist without the capacity to suffer. Beauty — especially the beauty that haunts, that stops the heart for an instant, that leaves a mark — does not exist without the awareness that what is beautiful is also passing.&lt;/p&gt;
&lt;p&gt;The experience machine offers us all the states without the conditions that make them possible. And this is not generosity; it is a sophisticated fraud. It offers the map without the territory. The symbol without the referent. The simulated experience of love without the real other who can die, who can disappoint you, whose love can be withdrawn, and whose presence is precisely for that reason irreplaceable.&lt;/p&gt;
&lt;p&gt;The transhumanism that takes Nozick seriously — and it should — is not the transhumanism that wants to plug us into the machine with more sophisticated implants. It is the transhumanism that wants to extend the reach of the real conditions of genuine experience: more time to love real things, more cognitive capacity to understand what is real more deeply, less suffering that is purely mechanical and that arises from no genuine contact with what matters. The distinction is subtle, but it is the difference between a project that deepens the human condition and a project that dissolves it into a more comfortable simulation.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxiii-on-depression-as-a-teacher-what-collapse-reveals&quot;&gt;XXIII. On Depression as a Teacher: What Collapse Reveals&lt;/h2&gt;
&lt;p&gt;There is a thought that needs to be said with care because it is dangerous if misunderstood and necessary if understood correctly: the deepest episodes of existential collapse — the moments when the system of meaning crumbles, when the usual mechanisms of symbolic immortality stop working, when the self finds itself exposed directly to the terror that normally dwells only at the edges of consciousness — these moments often produce a clarity about what is essential that periods of conventional functioning obscure.&lt;/p&gt;
&lt;p&gt;Not because suffering is pedagogical in the sense that the universe sends it to teach us lessons. The universe sends nothing; it is indifferent. But because the mechanisms that sustain ordinary life — the small and large functional illusions, the projects of importance that keep us moving, the usual ways of not thinking about what is most disturbing — when these mechanisms fail, what remains are the things that did not depend on them in order to be real.&lt;/p&gt;
&lt;p&gt;The residual affection for certain people. The curiosity that still works even when nothing else seems to work. The perception that certain forms of beauty — the light of this afternoon, this music, this sentence — can still cross the veil of anesthesia when everything else seems opaque. These things that persist in the collapse have a different weight from the weight they have in periods of conventional functioning, precisely because they have not been artificially inflated by the narrative of meaning that conventional functioning maintains.&lt;/p&gt;
&lt;p&gt;They are the things for which it would be worth going on, if anything is. And identifying them — not in theory, but in the raw experience of the moment when nothing else holds — is a kind of knowledge that no exercise in self-knowledge under comfortable conditions can produce with the same level of precision.&lt;/p&gt;
&lt;p&gt;This is not an argument for seeking collapse. It is an argument for not wasting what it reveals when it occurs. For paying attention to what survived, to what still weighed when everything that depended on narrative or illusion had evaporated. Because this — the irreducible residue — is where honest reconstruction, when it happens, has to be founded. Not on the old illusions recovered. On what was left after they went.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxiv-on-the-possibility-of-beauty-in-a-universe-of-entropy&quot;&gt;XXIV. On the Possibility of Beauty in a Universe of Entropy&lt;/h2&gt;
&lt;p&gt;We return to the beginning because cycles are the form honest thought often takes. There is no straight line from suffering to a way out. There is a return, with slightly different instruments, to the same territory — but seen now from angles that were not accessible before.&lt;/p&gt;
&lt;p&gt;The universe heads toward equilibrium. This is not pessimism; it is physics. And within this march toward equilibrium, every local structure of low entropy — every atom, every complex molecule, every cell, every organism, every consciousness, every culture, every star — is a pause in the dissolution. A temporary resistance to the inevitable. A parenthesis of organization within which things happen that would not happen at final equilibrium.&lt;/p&gt;
&lt;p&gt;You are one of those parentheses. More improbable than most. A parenthesis that can look at other parentheses and recognize them as such. That can appreciate the specific form of its own organization — the exact texture of how this specific system resists equilibrium at this specific moment. That can create records of this appreciation that other instances of temporary organization can access and that, in accessing them, produce something perhaps deserving of the name beauty.&lt;/p&gt;
&lt;p&gt;Beauty is not a property of the object independent of the observer. It does not exist in the universe itself. It exists in the relation between certain arrangements of matter organized enough to perceive other arrangements of matter. It is, in this sense, a completely local phenomenon, completely contingent, completely dependent on parentheses of organization that the universe did not need in any way to produce and that entropy will eventually dissolve.&lt;/p&gt;
&lt;p&gt;And this, paradoxically, is what makes it extraordinary. Not in spite of its contingency, but because of it. If beauty were necessary, if it were guaranteed in the structure of the cosmos, it would be just one more property of the universe — as remarkable as gravity or the speed of light, which is to say: completely unremarkable, simply the way things are. It is its improbability, its fragility, its dependence on arrangements that exist only while they exist and will not exist forever, that gives beauty its specific quality of making the heart stop for an instant.&lt;/p&gt;
&lt;p&gt;Suffering is the price of this structure. Not a price imposed by any moral authority. Simply the structural consequence of the fact that everything that can be beautiful can be lost, and that the system that perceives beauty is the same system that experiences loss. You cannot have one without the other. It is not possible to build the appreciation of beauty without building the capacity to suffer from its absence.&lt;/p&gt;
&lt;p&gt;And this is, perhaps, the most honest answer I can offer to the question of what we do with the human condition as it is: we do not deny suffering, we do not romanticize it, we do not manage it into invisibility. We recognize that it is the texture of what it means to exist with enough intensity for anything to matter. And, recognizing this, we try to exist in a way that is proportionate to what there is to be existed — with the attention the real deserves, with the presence that impermanence demands, with the honesty that is the only foundation on which anything genuine can be built.&lt;/p&gt;
&lt;p&gt;It is not enough. It is never enough. But it is what there is that is real.&lt;/p&gt;
&lt;hr&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxv-the-silence-before-and-after-on-the-symmetry-of-nonexistence&quot;&gt;XXV. The Silence Before and After: On the Symmetry of Nonexistence&lt;/h2&gt;
&lt;p&gt;Epicurus had an argument that should console and rarely does: before you were born, you did not exist, and that did not trouble you. After you die, you will not exist in the same way. The symmetry should produce equanimity. What did not exist before did not suffer. What will not exist afterward will not suffer.&lt;/p&gt;
&lt;p&gt;The argument is logically valid and existentially useless. Because the problem is not the state of nonexistence. The problem is the transition. The disappearance of something that is, at present, here. The erasure of this specific system which, however accidental its existence, does exist and has the particular quality of knowing that it exists.&lt;/p&gt;
&lt;p&gt;There is an asymmetry Epicurus ignores: before existing, I did not know that I would exist. There was no I anticipating existence with terror. But now that I exist, there is an I that knows existence will end. And this knowing — this anticipation of future absence from within present presence — is a form of experience that has no analogue in prior nonbeing. The nonbeing before birth was pure. The nonbeing after death is preceded by decades of a being that knew it was coming.&lt;/p&gt;
&lt;p&gt;This is the specific cruelty of human consciousness that Becker articulated more precisely than anyone else: not only that we will die, but that we know we will die, and that this knowing shapes every day of existence in a way no other animal experiences. The animal that does not know it will die can live completely in the present, entirely in what is. The animal that knows carries the future of its extinction into every present moment like a weight that cannot be set aside.&lt;/p&gt;
&lt;p&gt;It is this difference that makes the question of symbolic immortality so central. We do not need symbolic immortality because we are egocentric or cowardly. We need it because we are conscious. Because the weight of knowing that this specific arrangement of matter that experiences being this I will cease is a weight that needs some form of counterbalance in order for existence to be livable. And belief systems, projects of legacy, rituals of permanence — all of this is the response the species developed to the specific weight that the awareness of death imposes.&lt;/p&gt;
&lt;p&gt;The problem — and Becker never moved away from this problem — is that this necessary response to the weight produces, as a systematic side effect, the most devastating forms of collective violence the historical record contains. The need for meaning generates the need for an enemy. The need for permanence generates the need for a scapegoat. The need for a narrative that justifies existence generates the need for a narrative that justifies the elimination of whoever threatens that justification.&lt;/p&gt;
&lt;p&gt;And so the cycle perpetuates itself. With new instruments, with new ideological garb, with new amplification technologies — but the same emotional structure, the same substrate of terror, the same mechanism of projection that turns internal vulnerability into external violence.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxvi-love-in-an-entropic-structure&quot;&gt;XXVI. Love in an Entropic Structure&lt;/h2&gt;
&lt;p&gt;There is something about love specifically that no philosophy of death and no cosmology of entropy can completely dissolve, and which therefore deserves particular attention in an essay that tries to be honest about the condition.&lt;/p&gt;
&lt;p&gt;Love — not attraction, not affection, not tribal belonging, but love in the most demanding sense of the word: the fundamental orientation of the self toward another specific being, whose existence matters in a way that cannot be reduced to utility, to reciprocity or to any other calculation — is the most rebellious experience in the universe of entropy that we inhabit.&lt;/p&gt;
&lt;p&gt;Because love does not accept impermanence. Not in the sense of denying it rationally — the lover knows that the beloved will die, that he will die, that the time between them is finite. But at the visceral level, the level at which experience is lived before it is thought, love resists the temporal structure of dissolution. It wants the beloved permanent, it wants what exists now to exist forever, it wants the specific texture of this encounter between these two systems of consciousness never to cease to have existed.&lt;/p&gt;
&lt;p&gt;And this resistance is condemned to fail. It fails biologically, when death occurs. It fails relationally, when love ends. It fails even symbolically, when memories fade and records are lost and the civilization that might have kept them eventually disappears, as all civilizations disappear.&lt;/p&gt;
&lt;p&gt;But here is what matters: while it lasts, it is real. And what was real cannot be made unreal retroactively by later forgetting. That this specific love, at this specific moment, was — that is a fact of the universe as definitive as any physical law. The universe that contains that moment is different from the universe that would not have contained it. And that difference does not come undone.&lt;/p&gt;
&lt;p&gt;It is little. It is all there is. And sometimes — not always, not as a permanent resolution, but sometimes, in the moments when the system manages to sustain the observation — it is enough for the weight of existential terror to ease for long enough that life keeps being lived.&lt;/p&gt;
&lt;p&gt;Love is the most honest answer to the experience machine. We would not plug in not only because we want reality in general, but because we want that specific one — that specific person whose love is possible only because she is real, vulnerable, mortal, and therefore genuinely other than I am. The machine can simulate love without the real other. And that is exactly why it is unacceptable.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxvii-technology-art-and-science-as-a-tripod-not-salvation-but-extension&quot;&gt;XXVII. Technology, Art and Science as a Tripod: Not Salvation, but Extension&lt;/h2&gt;
&lt;p&gt;If there is no salvation — and this essay does not believe in salvation, at least not in the sense of a definitive resolution of the existential problem — there is still the question of what we do with the time we have. And here, with all the caution that honest pessimism demands, there is something that can be said positively.&lt;/p&gt;
&lt;p&gt;Free science, genuine art and technology applied with wisdom — not as ends in themselves, not as rituals of immortality disguised as projects of knowledge, but as ways of inhabiting time with the intensity it deserves — form a tripod that is, so long as it does not collapse under the weight of its own contradictions, the most sophisticated answer the species has found to its condition.&lt;/p&gt;
&lt;p&gt;Free science extends the reach of perception. It allows the human system of consciousness to perceive dimensions of reality that the biological senses alone would never reach. The infrared and the ultraviolet, gravitational waves and quantum fields, the structure of DNA and the composition of nebulae billions of light-years away — all of this lies outside the direct reach of experience, but within the reach of the extended instrument that science builds. And each extension of perceptual reach is an extension of the capacity to be astonished, to have the heart stopped by something that was not visible before.&lt;/p&gt;
&lt;p&gt;Genuine art — not entertainment, not content optimized for engagement, but art that emerges from the need to convert lived experience into a form that can be shared — does something different but complementary. It does not extend the reach of external perception. It extends the reach of internal perception, the capacity to recognize in another what one had only recognized in oneself. Each work that genuinely works expands the cartography of the human that the reader or listener or viewer can inhabit. It makes the inner world of another being — a being dead for centuries, a being who never existed except in the imagination of its creator — temporarily habitable. And in this temporary inhabiting, the structural solitude of consciousness is not dissolved, but it endures contact with its own universality in a way that changes something permanently.&lt;/p&gt;
&lt;p&gt;Technology, when applied with wisdom — which is to say, when applied with full awareness of the questions that Nozick and Becker and Fisher force us to ask about what we actually want to be and to have — can extend both the physical reach and the temporal duration within which science and art can happen. It can reduce the purely mechanical suffering that consumes capacity without producing meaning. It can create conditions of access to knowledge and expression that were historically the privilege of a few. It can, in principle, make humanity as a whole more able to make full use of the little time it has.&lt;/p&gt;
&lt;p&gt;But only in principle. In practice, technology follows the logic of what Becker would call the ritual of capitalist immortality: it produces instruments that extend the reach of existential terror and of the scapegoating machinery far more efficiently than they extend the reach of wisdom, beauty or compassion. Not by destiny, not by necessity — but because that is what happens when powerful instruments are developed without the fundamental question of what they should serve.&lt;/p&gt;
&lt;p&gt;The fundamental question is: what do we want from existence, given the limits that existence imposes? Not what the market wants us to want. Not what our unexamined existential terror wants. What we want when we examine the condition with the honesty it deserves — when we know that we are mortal, that loss is real, that beauty is transient, that suffering cannot be completely avoided, and that Nozick’s experience machine is not paradise but the extinction of everything that makes paradise conceivable?&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxviii-the-last-honesty-no-conclusion&quot;&gt;XXVIII. The Last Honesty: No Conclusion&lt;/h2&gt;
&lt;p&gt;Essays are supposed to end with conclusions. That is the convention. The course of the argument culminates in an arrival point that retrospectively justifies the path. The reader closes the text knowing more than when they opened it.&lt;/p&gt;
&lt;p&gt;This essay does not end that way. Not because it failed to arrive somewhere, but because arriving somewhere would be a falsehood. The existential condition we have examined has no conclusion. You will close this text and go on being an animal that knows it will die, in a universe that does not need you, carrying the specific weight of this specific consciousness in this specific body that time will undo.&lt;/p&gt;
&lt;p&gt;That does not change. No philosophy changes it. What can change — and it is the only modest and real thing an essay can aspire to do — is the relationship you keep with it. If you read it with enough attention. If you let it disturb you in a productive way instead of merely confirming a pessimism already installed. If it leads to questions you would not otherwise have asked.&lt;/p&gt;
&lt;p&gt;Existential pain is not the enemy. It is the sign that something matters enough to hurt. Existential emptiness — the anesthesia, the inability to feel deeply, life as a performance of life — is the real enemy. And the real enemy does not come from outside. It is built into every system that promises comfort in exchange for presence, every platform that offers connection in exchange for attention, every ideology that offers certainty in exchange for honesty about uncertainty.&lt;/p&gt;
&lt;p&gt;Refusing these exchanges is not heroic. It is merely the consequence of taking seriously what Nozick revealed: we prefer the real to the comfortable. We prefer the suffering that arises from contact with what matters to the absence of suffering that arises from contact with nothing.&lt;/p&gt;
&lt;p&gt;And if this is true — and most people, when sufficiently honest with themselves, recognize that it is — then the project is not to eliminate suffering. It is to ensure that the suffering we live is proportionate, is a response to something genuinely real, is the price of something that genuinely is worth it.&lt;/p&gt;
&lt;p&gt;That is all. It is little. It is, perhaps, exactly enough.&lt;/p&gt;
&lt;hr&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxix-on-what-the-dying-star-does-not-know&quot;&gt;XXIX. On What the Dying Star Does Not Know&lt;/h2&gt;
&lt;p&gt;A star of medium size like the Sun goes through a process of nuclear fusion for billions of years. It converts hydrogen into helium, helium into carbon, in a process that releases energy in quantities that define the scale by which all human energy is measured. Eventually, when the fuel runs out, the star collapses in on itself and expands into a nebula, dispersing into interstellar space the heavy elements it produced.&lt;/p&gt;
&lt;p&gt;The star does not know it is doing this. It does not experience the process. It does not anticipate the collapse, does not fear the dispersal, does not mourn the end of the fusion that kept it stable. It is simply a physical process of extraordinary duration following the laws that govern everything else.&lt;/p&gt;
&lt;p&gt;And the carbon you carry in your body — the same carbon that was in stars before it was in you, that will be in different forms after the arrangement you call yourself comes apart — that carbon does not know this either. Your molecules do not know they are you. Your neurons, individually, do not know they are part of a system that has experience.&lt;/p&gt;
&lt;p&gt;And yet — and this is the thing that haunts any honest philosophy of what we are — there is something it is like to be you. There is an interior. There is a point of view. There is a difference between what you see from outside a stone and what it is, from the inside, to be you.&lt;/p&gt;
&lt;p&gt;Where does this interior come from? No one knows. This is the most honest statement that neuroscience and the philosophy of mind have produced after decades of intensive work: we do not know how subjective experience emerges from physical processes. We know that it emerges. We know what kind of physical processes it seems to be correlated with. But the leap from the objective to the subjective — from process to experience — remains the most intractable problem in science.&lt;/p&gt;
&lt;p&gt;And this matters for the question of suffering in the following way: if we do not know where the interior comes from, we also do not know where it goes. If subjective experience is a physical phenomenon — if it is a process that happens in the universe just as nuclear fusion happens in a universe — then it is subject to the same laws. Including entropy. Including dispersal. Including the end.&lt;/p&gt;
&lt;p&gt;But while it lasts — while there is an interior, while there is a point of view, while there is something it is like to be this specific system — what happens in that interior is real in a way that nothing from outside can completely capture. The suffering is real. The love is real. The wonder before the cosmos that folds back on itself to observe itself is real. And this reality — fragile, contingent, cosmically peripheral and at the same time completely inaccessible to any external perspective — is the only place where anything worthy of the name of experience exists.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxx-the-art-that-can-still-be-made-a-note-on-the-margin&quot;&gt;XXX. The Art That Can Still Be Made: A Note on the Margin&lt;/h2&gt;
&lt;p&gt;Mark Fisher described a cultural world in a loop, with no exit, incapable of producing anything genuinely new. And this diagnosis is correct as an analysis of mass culture, of cultural production oriented toward the attention market. But Fisher also knew — and it is visible in the attention he gave to artists who worked at the margins of these structures — that the loop of mass culture is not the whole of culture.&lt;/p&gt;
&lt;p&gt;At the margins, in what is produced with no guarantee of an audience, with no optimization for the algorithm, with no funding that demands a measurable return — there, something different is still possible. Not always. Not consistently. Not in a way that can be scaled or systematized. But occasionally, with the irregularity that characterizes any genuinely emergent phenomenon, something happens that is not the loop. That is new in the relevant sense: not merely different in form, but different in function. That does to its interlocutor what art must do in order to deserve the name — that disturbs in a productive way, that expands what can be inhabited, that leaves the person who went through it different from what they were before.&lt;/p&gt;
&lt;p&gt;This kind of art exists at the margins of contemporary cultural production as it has always existed at the margins — because the center has always belonged to what is safe, to what confirms rather than disturbs, to what can be sold to the greatest number of people simultaneously. The contemporary difference is that the margins are more exposed, more accessible, more visible — and at the same time more prone to being captured and absorbed into the loop when they reach sufficient visibility.&lt;/p&gt;
&lt;p&gt;The challenge — for those who create and for those who seek — is to find and sustain contact with what has not yet been captured. That still preserves the disturbing function. That is still willing to disturb in a way that is not immediately resolvable by consumption, that demands the work of presence that the system is designed to make unnecessary.&lt;/p&gt;
&lt;p&gt;This is a work of attention. And attention — the capacity to be completely present in an experience for long enough for it to produce its slower and more lasting effects — is the scarcest and most systematically attacked resource in the contemporary world. Not by accident. By design.&lt;/p&gt;
&lt;p&gt;Recovering attention is not a political act. It is an existential act. It is the affirmation, small and everyday and never definitive, that what happens inside you matters more than what the platform wants you to feel matters. It is the most basic resistance possible: simply being present in your own experience in a way that noise cannot completely erase.&lt;/p&gt;
&lt;p&gt;This does not save the world. But it is what is available to be done, here, now, before the system reabsorbs the attention you briefly recovered.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxxi-a-final-word-on-what-was-not-said&quot;&gt;XXXI. A Final Word on What Was Not Said&lt;/h2&gt;
&lt;p&gt;This essay chose pessimism as an intellectual stance not out of temperament, not out of an appreciation of darkness as such, but out of a simple calculation: optimism that denies what is does not comfort. It produces the illusion of comfort while leaving intact the suffering it denies. The only genuine comfort — if the word fits here at all — is the one that emerges from recognizing the real, not from replacing it with more pleasant narratives.&lt;/p&gt;
&lt;p&gt;And the real includes things this essay could not completely capture. The specific quality of certain mornings. The weight of certain shared silences. The way certain songs arrive at the exact place where the pain was and produce something that is not a cure but is recognition, and recognition is enough for a moment to make the moment inhabitable. The texture of another person’s skin. The satisfaction of a problem that finally yields. The instant when an idea that had been only at the periphery of consciousness suddenly finds form and becomes sayable.&lt;/p&gt;
&lt;p&gt;These things prove nothing about the universe. They do not refute entropy, they do not answer the existential question, they do not solve the Beckerian problem. But they are the substance of any honest argument for going on. Not the philosophical substance — the existential substance. What actually weighs when everything else fails.&lt;/p&gt;
&lt;p&gt;The universe does not need you. But you are here. And while you are here, the universe is, in some tiny and irreplaceable sense, different from what it would be without you. This specific afternoon, these specific thoughts, these specific pains and beauties — all of this exists because you exist, and the existence of these contents of consciousness is a fact of the cosmos as real as any other.&lt;/p&gt;
&lt;p&gt;It is not enough to solve anything. But it is real. And the real, as Nozick reminded us, is what we choose over everything else when we are most completely what we are.&lt;/p&gt;
&lt;hr&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxxii-entropy-and-eros-why-we-keep-creating&quot;&gt;XXXII. Entropy and Eros: Why We Keep Creating&lt;/h2&gt;
&lt;p&gt;Freud’s psychoanalysis — however dated in its specific aspects — got something fundamental right in placing Eros and Thanatos as opposing forces that structure psychic life. Not as metaphysical principles, not as literal entities, but as vectors: the drive toward connection, toward increasing complexity, toward the creation of new arrangements — and the drive toward dissolution, toward the return to the inorganic, toward final simplification.&lt;/p&gt;
&lt;p&gt;What Freud perhaps did not fully explore is that these two drives are not simply opposed. They are complementary in a way that makes it impossible to have one without the other. Eros creates structures that are necessarily temporary. Thanatos dissolves structures that were necessarily created. The tension between them is the condition of possibility of anything that happens between formation and dissolution.&lt;/p&gt;
&lt;p&gt;Creation — of art, of knowledge, of relationships, of anything that bears the mark of a consciousness — is an essentially erotic act in the Freudian sense: it is the drive of organization resisting the return to equilibrium, the local and temporary resistance to universal entropy. And that is exactly why creating matters in a way that passively receiving does not matter in the same way. Creation is the active affirmation of the drive toward complexity against the universe’s march toward final simplicity.&lt;/p&gt;
&lt;p&gt;When a person in a state of deep suffering can still create — can still make something that did not exist before, that bears the mark of their specific consciousness, that inscribes something of their inner experience in the outer world in a way that can be found by another consciousness — that is Eros working against Thanatos at the most fundamental level. Not victory. Not resolution. But the affirmation that the drive of organization is still active, that the system still resists dispersal, that there is still something here that insists on being more than entropy.&lt;/p&gt;
&lt;p&gt;Art that emerges from genuine suffering is not art about suffering. It is art made despite suffering, or perhaps through it, using it as fuel instead of being consumed by it. And this distinction — between suffering without transformation and suffering with transformation — has nothing to do with the type of suffering or its intensity. It has to do with the relationship the system manages to keep with what is happening in it. Whether it can, even in the midst of what is hardest, keep asking: what form could this take? What structure could emerge from this? How could this become something another consciousness could find and recognize?&lt;/p&gt;
&lt;p&gt;This is the hardest existential work there is. Harder than any physical labor, more demanding than any intellectual discipline. It is the work of converting raw experience into something that transcends it without denying it. And there is no guarantee that it works. There is no method. There is only the attempt, repeated and often failed, to resist dispersal with the only thing it has against it: the insistence on giving form.&lt;/p&gt;
&lt;hr&gt;
&lt;hr&gt;
&lt;h2 id=&quot;xxxiii-cosmology-of-the-instant-what-resists-dissolution&quot;&gt;XXXIII. Cosmology of the Instant: What Resists Dissolution&lt;/h2&gt;
&lt;p&gt;There is a question that physics does not ask because it lies outside its domain, and that philosophy asks but rarely answers: what happens to the past? Not in the sense of its physical consequences, which keep propagating in the present in measurable ways. But the past as past — the fact that something occurred. Does that persist?&lt;/p&gt;
&lt;p&gt;The physical answer is yes, in a strictly causal way: every event leaves traces in the present state of the universe, so that in principle, with enough information and enough computational capacity, it would be possible to reconstruct the past from the present. But that is not what the question really asks. The question is about the ontological status of the past. About whether what was continues to be, in some way, even after it has ceased to be present.&lt;/p&gt;
&lt;p&gt;The answer that seems most honest — and which has a strangely comforting quality that is not cheap consolation — is that what happened cannot be undone. Not in the sense that it can be reversed or erased. In the most fundamental sense: the fact that it occurred is a permanent fact about the universe. The universe in which this moment existed is different from the universe in which it would not have existed, and that difference is not undone by forgetting, is not undone by the death of the system that generated it, is not undone by the end of the civilization that recorded it.&lt;/p&gt;
&lt;p&gt;This is a consolation that is not a lie. It does not say that death does not hurt. It does not say that loss is not real. It does not say that forgetting does not erase. It says only that what was, was. And this having-been is irrevocable in a way no entropy can reach, because entropy operates in the present and the future, not in the past.&lt;/p&gt;
&lt;p&gt;Every moment of genuine love that existed exists as an irrevocable past fact. Every instant of beauty that was perceived was perceived irrevocably. Every act of creation that occurred occurred in a way the universe cannot undo by dissipating the energy that made it possible. The fact that the poem was written is as permanent as any physical law, even after the poem is forgotten, even after the paper rots, even after the last person who read it has died.&lt;/p&gt;
&lt;p&gt;This is not enough. It does not satisfy the desire for continuity, it does not replace presence with the record of presence, it does not convert longing into equanimity. But it is what is true. And truth — even when it is less than we wanted, even when it is only the irrevocable record of something that passed — has a weight that illusion does not have.&lt;/p&gt;
&lt;p&gt;The weight of the real. Which is the only weight that, deep down, when we are more honest than habitual comfort allows, we choose over any substitute.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;references-and-dialogues&quot;&gt;References and Dialogues&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;Ernest Becker&lt;/strong&gt; — &lt;em&gt;The Denial of Death&lt;/em&gt; (1973). The foundation of the whole existential analysis developed here on the terror of death, symbolic immortality and the origin of evil in fear. Becker died weeks after receiving the Pulitzer, which lends the project an irony he would have appreciated.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Robert Nozick&lt;/strong&gt; — &lt;em&gt;Anarchy, State, and Utopia&lt;/em&gt; (1974). The Experience Machine thought experiment, central to the argument that suffering is not a simple defect to be corrected, but a structural condition of everything we value as genuine.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Mark Fisher&lt;/strong&gt; — &lt;em&gt;Capitalist Realism&lt;/em&gt; (2009), &lt;em&gt;Ghosts of My Life&lt;/em&gt; (2014), &lt;em&gt;The Weird and the Eerie&lt;/em&gt; (posthumous, 2016). The hauntology of contemporary culture, the impossibility of imagining alternatives, and depression as a rational response to irrational structures. Fisher died in 2017.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Sigmund Freud&lt;/strong&gt; — &lt;em&gt;Beyond the Pleasure Principle&lt;/em&gt; (1920), &lt;em&gt;Civilization and Its Discontents&lt;/em&gt; (1930). Eros and Thanatos as structural vectors of the psyche, and civilization as the simultaneous production of well-being and endemic suffering.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Otto Rank&lt;/strong&gt; — &lt;em&gt;Art and Artist&lt;/em&gt; (1932), &lt;em&gt;Will Therapy&lt;/em&gt; (1929). Creativity as a specific form of symbolic immortality, and the artist as a being whose relationship with death shapes the whole of the work.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Albert Camus&lt;/strong&gt; — &lt;em&gt;The Myth of Sisyphus&lt;/em&gt; (1942), &lt;em&gt;The Stranger&lt;/em&gt; (1942). The absurd as the baseline condition and rebellion — not resolution — as the only honest stance. Held here in tension with the question of what it costs, in the daily life of real suffering, to make this choice again and again.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Søren Kierkegaard&lt;/strong&gt; — &lt;em&gt;The Concept of Anxiety&lt;/em&gt; (1844), &lt;em&gt;The Sickness Unto Death&lt;/em&gt; (1849). Anxiety as constitutive of conscious freedom. Despair as the specific relation the human being keeps with their own condition when unable to be what they are.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Thomas Ligotti&lt;/strong&gt; — &lt;em&gt;The Conspiracy Against the Human Race&lt;/em&gt; (2010). The most rigorous philosophical pessimism in contemporary literature, articulating without euphemism what academic philosophy hesitates to say: human consciousness may be the most unfortunate aberration the cosmos has produced.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;David Benatar&lt;/strong&gt; — &lt;em&gt;Better Never to Have Been&lt;/em&gt; (2006). Philosophical antinatalism as an extreme position that, even when refused, forces an honest confrontation with the amount of suffering existence entails and that we systematically tend to underestimate.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Derek Parfit&lt;/strong&gt; — &lt;em&gt;Reasons and Persons&lt;/em&gt; (1984). The philosophical dissolution of personal identity as a persisting entity and its disturbing implications for the terror of death: if there is no continuous self, what exactly dies?&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Maurice Merleau-Ponty&lt;/strong&gt; — &lt;em&gt;Phenomenology of Perception&lt;/em&gt; (1945). The irreducible corporeality of experience as the foundation of any existential analysis that wants to stay in the real instead of floating in abstractions about consciousness.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Emil Cioran&lt;/strong&gt; — &lt;em&gt;On the Heights of Despair&lt;/em&gt; (1934), &lt;em&gt;The Book of Delusions&lt;/em&gt; (1936), &lt;em&gt;A Short History of Decay&lt;/em&gt; (1949). Pessimism as a way of being honest, written with the ferocity of someone who realized that most thought is well-formulated anesthesia.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Friedrich Nietzsche&lt;/strong&gt; — &lt;em&gt;The Gay Science&lt;/em&gt; (1882), &lt;em&gt;Thus Spoke Zarathustra&lt;/em&gt; (1883). Amor fati as an alternative to existential denial — not optimism, but the affirmation of what is, including what hurts. And the death of God as a diagnosis of the collapse of the old symbolic immortalities without an adequate replacement.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;William Gibson&lt;/strong&gt; — &lt;em&gt;Neuromancer&lt;/em&gt; (1984), &lt;em&gt;Pattern Recognition&lt;/em&gt; (2003). Cyberpunk not as a genre, but as an emotional diagnosis of a world in which technology has advanced far beyond wisdom about how to inhabit it.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Nick Bostrom&lt;/strong&gt; — &lt;em&gt;Are You Living in a Computer Simulation?&lt;/em&gt; (2003), &lt;em&gt;Superintelligence&lt;/em&gt; (2014). Transhumanism in its most rigorous forms and the existential questions it raises about consciousness, suffering and what it means to extend or modify the human condition.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Benjamin Libet&lt;/strong&gt; — experiments on the timing of conscious decision (1983). The finding that neural activity precedes the conscious experience of decision, with disturbing implications for the nature of the lived present and of subjective agency.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Epicurus&lt;/strong&gt; — &lt;em&gt;Letter to Menoeceus&lt;/em&gt;. The argument of symmetry between nonbeing before birth and nonbeing after death, discussed here not as functional consolation but as an example of how logic can be correct and existentially useless at the same time.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Blaise Pascal&lt;/strong&gt; — &lt;em&gt;Pensées&lt;/em&gt;. The human being as a “thinking reed” — greatness in fragility, and the horror of the eternal silence of infinite spaces that Pascal himself confessed to feeling before finding faith where philosophy did not reach.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Ludwig Wittgenstein&lt;/strong&gt; — &lt;em&gt;Tractatus Logico-Philosophicus&lt;/em&gt; (1921), &lt;em&gt;Philosophical Investigations&lt;/em&gt; (1953). The limits of language as the limits of the communicable world, and the structural solitude of consciousness that any language circumscribes but does not dissolve.&lt;/p&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>philosophy</category><category>pessimism</category><category>science</category><category>civilization</category><category>featured essays</category></item><item><title>AI Should Learn Like Children</title><link>https://matheussorancophilomathes.com.br/en/essays/ai-should-learn-like-children-a-hypothesis-on-artificial-intelligence-cognitive/</link><guid isPermaLink="true">https://matheussorancophilomathes.com.br/en/essays/ai-should-learn-like-children-a-hypothesis-on-artificial-intelligence-cognitive/</guid><description>Why scaling passive, static language models will not yield general intelligence, and what the cognitive development of children — priors, curiosity, world models, continual learning, embodiment — suggests about the architectures that might.</description><pubDate>Sat, 21 Feb 2026 20:29:25 GMT</pubDate><content:encoded>&lt;p&gt;&lt;em&gt;A Hypothesis on Artificial Intelligence, Child Cognitive Development, and the Limits of the Statistical Mind&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;─────────────────────────────────────&lt;/p&gt;
&lt;p&gt;&lt;em&gt;“Intelligence is the efficiency with which a system acquires new skills outside its training domain.”&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;— François Chollet&lt;/p&gt;
&lt;p&gt;&lt;em&gt;“The problem is not to build a machine that thinks, but to build a machine that learns to think.”&lt;/em&gt;&lt;/p&gt;
&lt;p&gt;— Yann LeCun&lt;/p&gt;
&lt;h2 id=&quot;introduction-the-invisible-bottleneck&quot;&gt;Introduction: The Invisible Bottleneck&lt;/h2&gt;
&lt;p&gt;There is a scene that repeats itself, with variations, in artificial intelligence laboratories around the world. A state-of-the-art language model — trained on quantities of text that no human being could read in a thousand lifetimes — is presented with a simple visual reasoning problem: given a three-by-three grid with color patterns, infer the rule and complete the fourth example. The model fails. Not occasionally, not under specific adversarial conditions. It fails systematically, while a four-year-old child, armed only with her eyes and her tireless curiosity about how things work, would solve the same problem in a matter of seconds.&lt;/p&gt;
&lt;p&gt;This essay starts from that disconcerting asymmetry — not to denigrate the magnitude of what large language models have achieved, but to investigate what it reveals about the nature of intelligence, the limits of current architectures and what a serious hypothesis about Artificial General Intelligence should take into account. The central thesis I propose here is neither a formalized academic dissertation proposal nor an engineering roadmap with verifiable milestones. It is a philosophical-technical hypothesis: the idea that AGI cannot emerge from architectures of passive learning and static structure, and that the most promising path lies in emulating the cognitive mechanisms of child development — intrinsic curiosity, evolutionary priors, the dynamic world model, and continual learning anchored in embodiment.&lt;/p&gt;
&lt;p&gt;To develop this argument, I will traverse four large territories. First, I will offer an honest diagnosis of the current state of LLMs and deep-learning architectures, identifying their real achievements and their structural bottlenecks. Second, I will revisit the psychology of child cognitive development, not as an inspirational metaphor, but as a technical reference model. Third, I will map the alternative and hybrid architectures that try, from different angles, to bring machine learning closer to biological learning. Finally, I will present the thought experiment that gives this essay its title — the Room of the Curious Child — as a philosophical tool for illuminating what separates statistical competence from genuine intelligence.&lt;/p&gt;
&lt;p&gt;A caveat before beginning: this is not a text of technological pessimism. Transformers and LLMs represent a genuine civilizational leap. Language is, as I will argue, one of the most sophisticated forms of knowledge compression that humanity has produced, and models capable of operating in that space with the fluency we observe today are extraordinary instruments. The problem is not what they do. The problem is what they will continue to be unable to do as long as the underlying architecture remains fundamentally passive, static and disembodied.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-i-the-triumph-and-the-ceiling--what-llms-have-conquered-and-where-they-stopped&quot;&gt;Part I: The Triumph and the Ceiling — What LLMs Have Conquered and Where They Stopped&lt;/h2&gt;
&lt;h3 id=&quot;11-the-revolution-of-language-as-latent-space&quot;&gt;1.1 The Revolution of Language as Latent Space&lt;/h3&gt;
&lt;p&gt;To understand where large language models have failed, one must first understand — with the seriousness the feat deserves — what they have achieved. Language is not merely a means of communication. It is, possibly, the most powerful cognitive technology the human species has developed: a symbolic system of compression that allows concepts to be transmitted, combined and recombined across time and space. When a child learns the word ‘gravity,’ she does not learn merely a label. She gains access to a dense network of conceptual relations that other humans took centuries to formalize.&lt;/p&gt;
&lt;p&gt;LLMs, by learning to model the statistical distribution of language sequences at industrial scale, did something that should not be underestimated: they captured, in their weights, a functional shadow of this conceptual network. When we ask GPT-4 or Claude Sonnet about the relationship between entropy and information, or ask them to explain the intuition behind Bayes’ Theorem, these models often produce answers that are not only correct but pedagogically sophisticated — answers that reflect a structural understanding of the relations between concepts, not just a retrieval of superficial patterns.&lt;/p&gt;
&lt;p&gt;This matters because language, as the linguist and philosopher Dan Sperber observes, is the medium through which cultural knowledge accumulates and is transmitted from generation to generation. A model trained on the entire digitally available textual output is, in a sense, distilling centuries of collective cognition. It is an unprecedented feat in the history of computing — and perhaps of civilization.&lt;/p&gt;
&lt;p&gt;The transformer architecture that makes this possible deserves a few paragraphs. The multi-head attention mechanism, introduced in 2017 in the seminal paper ‘Attention Is All You Need,’ solves a problem that had frustrated decades of research in natural language processing: how to capture long-range dependencies in sequences. Instead of processing tokens sequentially, as RNNs did, the transformer computes the relevance of each token with respect to all the others simultaneously. The result is a contextualized representation in which the meaning of each word is a function of its complete context — which is, curiously, what linguists have always said about how humans process language.&lt;/p&gt;
&lt;p&gt;Scaling, then, did something extraordinary: as we increased the number of parameters, the quality of reasoning seemed to scale non-linearly. Smaller models seemed merely to retrieve patterns; larger models exhibited something resembling the emergence of new capabilities — analogical reasoning, rule generalization, basic arithmetic — without these capabilities having been explicitly trained. The scaling laws of Kaplan et al. described a predictable progression: more data, more parameters, more computation resulted in better performance, measured by perplexity over language sequences.&lt;/p&gt;
&lt;h3 id=&quot;12-the-scaling-barrier-and-the-arc-agi-problem&quot;&gt;1.2 The Scaling Barrier and the ARC-AGI Problem&lt;/h3&gt;
&lt;p&gt;But the scaling laws are coming to an end. Not abruptly — there is no cliff. There is an increasingly perceptible plateau. The marginal returns of adding parameters and data are diminishing. Models that cost tens or hundreds of millions of dollars to train are improving less and less on relevant benchmarks. And there is one benchmark in particular that captures the problem with surgical precision: ARC-AGI, the Abstraction and Reasoning Corpus for Artificial General Intelligence, developed by François Chollet.&lt;/p&gt;
&lt;p&gt;ARC-AGI is elegantly simple in its structure and devastating in its implications. The system receives pairs of input-output examples — typically colored grids demonstrating a visual transformation — and must induce the rule that governs the transformation and apply it to a new input case. The examples are few: usually three or four. The rules involve concepts such as symmetry, counting, topological correspondence, spatial continuity — concepts that a four-year-old child masters intuitively.&lt;/p&gt;
&lt;p&gt;The best language models, even the most advanced, perform modestly on ARC. This is not a problem of a poorly calibrated benchmark; it is a precise diagnosis. ARC measures exactly what Chollet defines as intelligence: the efficiency of acquiring new skills from very little information. An adult human, without specific training, solves most ARC problems with ease. A five-year-old solves a significant proportion. Models trained on trillions of tokens — with implicit access to all the physics, geometry and logic that human text contains — fail consistently.&lt;/p&gt;
&lt;p&gt;The technical diagnosis is clear: LLMs have learned to interpolate within the distribution of their training data with extraordinary sophistication. But when confronted with situations that require synthesizing new rules from a few examples — what cognitive psychologists call ‘few-shot inductive learning’ — the statistical mechanisms that underpin the transformer are simply not the right instruments for the job. It is like trying to cut wood with a scalpel: the tool is extraordinarily precise for what it was designed for, but it is the wrong tool for this task.&lt;/p&gt;
&lt;h3 id=&quot;13-the-structural-problem-passivity-staticity-disembodiment&quot;&gt;1.3 The Structural Problem: Passivity, Staticity, Disembodiment&lt;/h3&gt;
&lt;p&gt;Beyond ARC, there are three structural limitations of LLMs that any hypothesis about AGI must confront directly.&lt;/p&gt;
&lt;p&gt;The first is passivity. An LLM learns by observing text — an extraordinary quantity of text, but text produced by others, capturing the thought of others, describing the experiences of others. The model never acts in the world. It never tests a hypothesis. It never discovers that its prediction was wrong through a real physical consequence. The difference between learning that ‘eggs break when they fall’ by reading that sentence in text and learning the same by letting an egg fall from one’s hand is, I will argue, a difference of kind, not merely of degree. Active learning — discovery through action and the observation of consequences — produces representations structurally different from those produced by passive observational learning.&lt;/p&gt;
&lt;p&gt;The second is staticity. The weights of an LLM, once trained, are fixed. The model you query today is identical, in terms of its internal parameters, to yesterday’s model — regardless of how many interactions it has had. There is research on continual fine-tuning and on reinforcement learning with human feedback, but these processes are slow, costly and subject to catastrophic forgetting: updating the weights to learn something new tends to destroy what was learned before. The human brain solves this problem elegantly, through mechanisms still incompletely understood that allow new memories to be consolidated without erasing old ones.&lt;/p&gt;
&lt;p&gt;The third is disembodiment. An LLM has no body. It has no proprioception. It does not experience resistance, gravity, temperature, texture. Its concepts of ‘heavy,’ ‘hot,’ ‘rough,’ ‘distant’ are statistical relations between symbols — patterns of co-occurrence in text — not representations anchored in sensorimotor experience. This matters because, as I will argue in the next section, there are both empirical and theoretical reasons to believe that a substantial part of human cognition — including high-level abstract reasoning — is built upon sensorimotor foundations. Removing those foundations may be not merely a pragmatic limitation; it may be a constitutive one.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-ii-what-a-child-knows--cognitive-development-as-model&quot;&gt;Part II: What a Child Knows — Cognitive Development as Model&lt;/h2&gt;
&lt;h3 id=&quot;21-being-born-with-priors-the-end-of-the-tabula-rasa&quot;&gt;2.1 Being Born with Priors: The End of the Tabula Rasa&lt;/h3&gt;
&lt;p&gt;The classical empiricist view of the infant mind, popularized by John Locke and perpetuated in various forms by behaviorism, is that of the tabula rasa: the child is born as a blank slate, and experience writes on it everything she will come to know. This view, although intuitively appealing in its egalitarian openness, is empirically false — and its falsity has profound implications for any attempt to emulate child learning in machines.&lt;/p&gt;
&lt;p&gt;The research of Elizabeth Spelke, Karen Wynn, Renée Baillargeon and others in the field of infant cognition has demonstrated, through decades of meticulous experiments with habituation and eye-tracking techniques, that newborn babies arrive in the world equipped with a rich set of expectations about how reality works. Spelke called these expectations ‘Core Knowledge’ and identified several domains in which it operates: physical objects (cohesion, continuity, contact), agents (intentionality, means-end rationality), number (comparison of quantities, principles of counting) and geometry (spatial navigation, topological relations).&lt;/p&gt;
&lt;p&gt;What makes this especially fascinating from the standpoint of AI is that this core knowledge is not learned from experience — it is present from birth, the product of millions of years of evolutionary selection. A four-month-old baby shows surprise when a solid object appears to pass through another solid object — a violation of the law of impenetrability that the baby never needed to learn explicitly. A six-month-old shows correct expectations about the addition and subtraction of small sets of objects, even before any formal mathematical instruction.&lt;/p&gt;
&lt;p&gt;This means that child learning does not begin from zero. It begins with a set of structural inductive biases — predispositions to organize experience in certain ways, to formulate certain hypotheses rather than others, to notice certain regularities while ignoring others. These biases are not limitations; they are what makes efficient learning possible. Without them, the space of hypotheses compatible with any finite set of observations would be infinite, and learning would be computationally intractable.&lt;/p&gt;
&lt;p&gt;The implication for AI is direct: architectures that initialize with random weights — with no structured prior about how the world works — are at a fundamental disadvantage relative to human learning. It is not a question of the quantity of data; it is a question of initial structure. More data trained on an architecture without priors will still, at some point, have to infer the regularities that the human baby brings with her from birth.&lt;/p&gt;
&lt;h3 id=&quot;22-active-learning-and-curiosity-as-algorithm&quot;&gt;2.2 Active Learning and Curiosity as Algorithm&lt;/h3&gt;
&lt;p&gt;Watch an eighteen-month-old child for an hour. What you will see, with individual variations, is something remarkable: an extraordinarily efficient learning system operating in a mode of active, self-directed search. The child does not wait to be taught. She explores. She manipulates objects, tests their properties, forms expectations, checks those expectations, and revises her mental model when the expectations are violated. She seems intrinsically motivated to understand — not merely to respond to stimuli, but to build predictive models of the world around her.&lt;/p&gt;
&lt;p&gt;Pierre-Yves Oudeyer and his collaborators formalized this process with the concept of ‘Learning Progress’ as a mechanism of intrinsic attention. The thesis is that children — and, hypothetically, AI systems inspired by them — do not distribute their attention randomly across the space of possible explorations. They focus on situations in which they are actively learning: where the prediction error is decreasing, where the internal model is being productively revised. Situations that are too simple (no novelty, no prediction error) bore. Situations that are too complex (pure noise, no detectable pattern) frustrate. The child gravitates toward the intermediate zone — what Vygotsky called the ‘zone of proximal development’ — where the challenge is calibrated to her current capacity.&lt;/p&gt;
&lt;p&gt;This mechanism has considerable mathematical elegance. If we formalize a system’s prediction error about its own world model as a loss function, then the intrinsic motivation to explore can be encoded as a reward proportional to the temporal derivative of the error — not to the absolute error, but to the rate of change of the error. The system is not rewarded for being confused; it is rewarded for becoming less confused. This generates exactly the observed behavior: gravitating toward situations of productive learning, avoiding both the trivial and the chaotic.&lt;/p&gt;
&lt;p&gt;There is something philosophically profound here. Curiosity, often treated as a mysterious emotion or a personality trait, reveals itself, in this formalization, as an optimization mechanism — a gradient signal that guides an agent’s exploration through the space of possible experiences so as to maximize learning efficiency. The curious child is not merely ‘being a child.’ She is executing, with an elegance that generations of AI engineers have not yet replicated, the optimal algorithm of active data collection.&lt;/p&gt;
&lt;h3 id=&quot;23-embodiment-as-foundation--not-as-detail&quot;&gt;2.3 Embodiment as Foundation — Not as Detail&lt;/h3&gt;
&lt;p&gt;Jean Piaget, whose work on the stages of child cognitive development remains a central reference in developmental psychology, made an observation that is often mentioned but rarely taken seriously in its radical implications: cognitive development begins at the sensorimotor stage. The child does not learn to think in abstractions and then apply that thinking to the physical world. She learns to think through interaction with the physical world. Abstract thought emerges from concrete action, not the other way around.&lt;/p&gt;
&lt;p&gt;Piaget described how an eight-month-old learns the concept of object permanence — the understanding that objects continue to exist even when they are not directly observed. She learns this not by reading a definition, not by hearing an explanation, but by repeating, hundreds of times, the experience of seeing an object hidden, searching for it, finding it. The mental representation of ‘permanent object’ is built, literally, from cycles of action-perception-prediction-surprise-revision.&lt;/p&gt;
&lt;p&gt;Contemporary neuroscience adds a layer to this observation: the neural representations of abstract concepts retain functional traces of their sensorimotor origin. When we think of ‘grasping,’ motor areas of the cortex activate. When we process metaphors of social temperature (‘she was cold to me’), the insular cortex — which processes physical temperature — shows activation. Embodied cognition, as a line of research, argues that cognition is not purely amodal computation over abstract symbols, but is fundamentally anchored in systems of action and perception.&lt;/p&gt;
&lt;p&gt;If this is correct — and there is substantial evidence suggesting that it is — then AI systems that process language without any sensorimotor experience are building conceptual edifices on missing foundations. They can learn the relations between symbols that refer to physical experiences, but without the physical experiences that anchor those symbols to the world. It is a distinction that analytic philosophy has explored under the name ‘symbol grounding problem’: how do symbols acquire meaning if there is no anchoring in non-symbolic experience?&lt;/p&gt;
&lt;h3 id=&quot;24-neurological-development-as-architectural-model&quot;&gt;2.4 Neurological Development as Architectural Model&lt;/h3&gt;
&lt;p&gt;The infant brain is not just a smaller version of the adult brain. It is a system in active transformation — and the processes of that transformation are as revealing as the final state. Two processes deserve special attention: neurogenesis and synaptic pruning.&lt;/p&gt;
&lt;p&gt;At the beginning of development, the brain produces synapses in enormous excess — an overabundance of connections between neurons that creates a highly plastic system, capable of capturing an extraordinary variety of patterns. As the child grows and learning occurs, synapses that are repeatedly activated are strengthened (long-term potentiation), while synapses that rarely fire are eliminated (synaptic pruning). The result is a network that becomes progressively more specialized and efficient — less plastic, but more effective at the tasks for which it was, through experience, calibrated.&lt;/p&gt;
&lt;p&gt;This process has an intuitive analogy with regularization in machine learning — techniques such as dropout or weight decay that penalize unnecessary model complexity. But there are crucial differences. Synaptic pruning is structural: it eliminates physical connections, altering the topology of the network. Regularization techniques in artificial neural networks modify weights but keep the topology fixed. And biological pruning is use-guided — it preserves what is used and eliminates what is not, creating a system whose structure literally reflects the individual’s history of experiences.&lt;/p&gt;
&lt;p&gt;There is also the critical period — a temporal window during which the brain is especially receptive to certain types of input. The famous research of David Hubel and Torsten Wiesel demonstrated that depriving a kitten of visual input during the critical period of development resulted in permanent deficits in visual processing, even if the input was later restored. During this window, the brain is not merely learning; it is configuring itself structurally to process certain types of information. Missing that period has irreversible consequences.&lt;/p&gt;
&lt;p&gt;These mechanisms suggest that efficient learning may require not only the right exploration strategies and the right data, but the right temporal sequence of exposures — an idea that curriculum learning in AI tries to capture, but in a much more rudimentary way.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-iii-the-thought-experiment--the-room-of-the-curious-child&quot;&gt;Part III: The Thought Experiment — The Room of the Curious Child&lt;/h2&gt;
&lt;h3 id=&quot;31-the-context-of-the-philosophical-rooms&quot;&gt;3.1 The Context of the Philosophical Rooms&lt;/h3&gt;
&lt;p&gt;The philosophy of mind has produced, in recent decades, a series of thought experiments that have become landmarks of the debate on consciousness, intentionality and cognition: John Searle’s Chinese Room, Frank Jackson’s Mary’s Room, Hilary Putnam’s Brain in a Vat, David Chalmers’ Philosophical Zombies. These experiments are not empirical arguments; they are conceptual devices — tools for testing intuitions, revealing hidden presuppositions and illuminating distinctions that everyday discourse tends to obscure.&lt;/p&gt;
&lt;p&gt;Searle’s Chinese Room, in particular, is relevant as a starting point. In it, a monolingual English speaker is confined to a room with a set of rules (in English) for manipulating Chinese symbols. Messages in Chinese enter through the slot; the occupant applies the rules, manipulates the symbols, and produces responses that, to external observers, seem to demonstrate fluent understanding of Chinese. Searle argues that the system — the man plus the rules plus the symbols — does not understand Chinese in any meaningful sense. It manipulates syntax without access to semantics. Form without content. Symbol without meaning.&lt;/p&gt;
&lt;p&gt;There is a vast philosophical debate about the validity of this argument. But what matters here is not whether Searle is right about consciousness, but what the experiment illuminates about a specific type of cognitive limitation: the possibility of producing correct outputs without possessing the internal representation that normally underpins those outputs. It is exactly this limitation that I want to explore in relation to LLMs — and I want to do so with a different thought experiment, which I call The Room of the Curious Child.&lt;/p&gt;
&lt;h3 id=&quot;32-the-room-of-the-curious-child-description-of-the-experiment&quot;&gt;3.2 The Room of the Curious Child: Description of the Experiment&lt;/h3&gt;
&lt;p&gt;Imagine two rooms. In each room there is a child — or what appears to be a child. But the rooms are structurally very different.&lt;/p&gt;
&lt;p&gt;The first room is the Room of the Statistical Child. In it, we find a system that was given, before being placed in the room, all the scientific and popular literature ever produced on physics, chemistry, biology, psychology and engineering. It received transcripts of lectures, instruction manuals, scientists’ diaries, accounts by explorers. If you ask this system what happens when a ball rolls down a ramp, it will produce a precise and sophisticated description of the relevant kinematics, mention the angle of inclination, the coefficient of friction, the gravitational acceleration, perhaps cite Galileo and his experiments in Pisa. The answer is impeccable.&lt;/p&gt;
&lt;p&gt;But the room is empty. There are no objects. There is no ramp, no ball, nothing to manipulate. The system has never pushed anything. It has never seen an object roll, never calibrated an expectation based on an observation, never felt the surprise of seeing something act differently from what was predicted. When you place a ball in the room for the first time and push it toward the system, what happens? The system does not catch it. It does not react adaptively. It does not explore its properties. It simply produces more text describing what should be happening.&lt;/p&gt;
&lt;p&gt;The second room is the Room of the Curious Child. In it, there is no library. There are no manuals. There is only an agent — endowed, like a human baby, with certain structural priors about reality (objects are solid, agents have intentions, causes precede effects) — and a room full of objects: wooden blocks of different sizes and weights, ramps of variable inclination, containers of different shapes, strings, springs, weights, pulleys. There are no instructions. There is no curriculum. Only the agent, the objects, and time.&lt;/p&gt;
&lt;p&gt;What does the Curious Child do? She explores. She picks up a block, drops it — observes. Picks up another, larger block — drops it — notes that it fell with a different sound. Stacks blocks, watches the tower collapse — rebuilds differently, more carefully. Places an object on the ramp — sees it roll — changes the angle of the ramp — notes that it rolls faster. In each cycle, she is doing something precise: formulating a hypothesis (implicit, non-verbal), testing it through an action, observing the result, and updating her internal model of how the world works.&lt;/p&gt;
&lt;p&gt;After a day in this room, the Curious Child will not be able to articulate Newton’s laws in formal language. But she will know, with a depth that no library can confer, that heavier objects are harder to move, that inclined surfaces change the trajectory of objects, that well-balanced structures are more stable, that certain pulley arrangements make it possible to lift weights with less effort. She will know this in the most robust sense: she will have internal representations anchored in experience, which will allow her to make correct predictions in new situations that were never explicitly demonstrated.&lt;/p&gt;
&lt;h3 id=&quot;33-the-decisive-test&quot;&gt;3.3 The Decisive Test&lt;/h3&gt;
&lt;p&gt;Now comes the decisive moment of the experiment. In each room you place a device that was never described in any text the Statistical Child read — and that the Curious Child has never seen before. A simple but novel mechanism: a lever with a fulcrum in an asymmetric position, connected by a cable to a closed box. The task: open the box.&lt;/p&gt;
&lt;p&gt;The Statistical Child will produce a textually sophisticated analysis. It will identify that there is a lever. It will mention the principle of the fulcrum. Perhaps it will cite Archimedes (‘give me a place to stand and I will move the world’). But when confronted with the need to interact physically with the mechanism — to calibrate how much force to apply, in which direction, what the expected behavior of the cable is when the lever moves — it floats in a space of uncertainty that its library cannot resolve. Not because it lacks information, but because the information it has is fundamentally of a different kind from what the situation requires.&lt;/p&gt;
&lt;p&gt;The Curious Child looks at the mechanism. She has no vocabulary for ‘lever’ or ‘fulcrum.’ But she has something more powerful: a world model calibrated by experience. She sees a rigid object with a point of support. She already knows, from her explorations, that rigid objects supported at one point behave in a certain way when you push one of the ends. She sees the cable. She already knows that cables transmit force. She formulates a hypothesis, acts, observes the result, adjusts. In a few minutes, the box is open.&lt;/p&gt;
&lt;p&gt;This is the core of the thought experiment. It is not saying that the Statistical Child is stupid or useless. It is saying that it possesses a different kind of competence — extraordinary in its domain, but structurally inadequate for the task of novel generalization from a few examples. And it is suggesting that the difference is not simply quantitative (more data, more parameters) but qualitative: it is a difference in the kind of internal representation each system has built and in the process by which it built those representations.&lt;/p&gt;
&lt;h3 id=&quot;34-what-the-experiment-reveals--and-what-it-does-not&quot;&gt;3.4 What the Experiment Reveals — and What It Does Not&lt;/h3&gt;
&lt;p&gt;Before drawing conclusions from the experiment, it is necessary to be honest about its limitations. The Room of the Curious Child is not an argument about consciousness — I am not making the Searlean claim that the Statistical Child ‘does not understand’ in the phenomenological sense. The question of consciousness is separate, deeper, and deliberately set aside here. The experiment is about functional capacity: about which kinds of situations each system can navigate successfully.&lt;/p&gt;
&lt;p&gt;Nor am I arguing that language and text-based knowledge have no value. Clearly they do — the Statistical Child has access to centuries of distilled knowledge that the Curious Child would need multiple lifetimes to rediscover empirically. The argument is about complementarity: systems that can do only one of the two things are fundamentally limited for the task of AGI.&lt;/p&gt;
&lt;p&gt;What the experiment reveals is a distinction that cognitive psychology calls ‘declarative knowledge’ versus ‘procedural knowledge,’ or, in more philosophical terms, ‘knowing that’ versus ‘knowing how.’ Knowing that a lever with an asymmetric fulcrum amplifies force is different from knowing how to operate such a lever effectively in a concrete physical context. LLM architectures are optimized for the first kind of knowledge. The second requires something different: a world model anchored in experience, the capacity for planning based on that model, and the ability to learn efficiently from a few demonstrations.&lt;/p&gt;
&lt;p&gt;The experiment also points to an important temporal asymmetry. The Statistical Child was built through a single, massive learning process, followed by total freezing. The Curious Child learns continuously, updating her model with each interaction. The first is a finite product; the second is an ongoing process. The AGI we seek resembles the second: not a system that has been ‘trained to be intelligent,’ but a system that continues to become more intelligent through experience.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-iv-the-architectural-hypothesis--toward-a-model-of-child-learning&quot;&gt;Part IV: The Architectural Hypothesis — Toward a Model of Child Learning&lt;/h2&gt;
&lt;h3 id=&quot;41-the-five-pillars-of-the-hypothesis&quot;&gt;4.1 The Five Pillars of the Hypothesis&lt;/h3&gt;
&lt;p&gt;Based on the preceding analysis, I can now articulate the central hypothesis of this essay more precisely. The hypothesis is not that we should simply ‘emulate children’ in some vague and inspirational sense. It is that there are five specific mechanisms of child cognitive development that are computationally relevant and that any serious AGI architecture must take into account.&lt;/p&gt;
&lt;p&gt;The first pillar is initialization with structural priors. An AGI architecture should not start from absolute zero. It should be initialized with a set of inductive biases inspired by human Core Knowledge — structural predispositions for processing object cohesion, permanence, causality by contact, intentional agency, and basic numerical relations. These priors are not the final knowledge; they are the bias that makes efficient learning possible.&lt;/p&gt;
&lt;p&gt;The second pillar is self-directed exploration driven by intrinsic curiosity. Instead of being fed human-curated datasets, the agent should guide its own data collection through an intrinsic reward mechanism based on learning progress — the derivative of the world model’s prediction error. This generates an emergent curriculum, dynamically adapted to the system’s current capabilities.&lt;/p&gt;
&lt;p&gt;The third pillar is the predictive world model in latent space. Instead of learning to predict sequences of raw observations (pixels, tokens), the system should learn to build and maintain a world model in latent space — a mental simulator that allows counterfactual planning before action. This is the pillar most directly aligned with Yann LeCun’s proposals on JEPAs.&lt;/p&gt;
&lt;p&gt;The fourth pillar is continual learning without catastrophic forgetting. The system must be able to learn from new experiences without destroying what it learned from earlier ones. This requires architectural mechanisms that go beyond simple backpropagation — possibly inspired by the hippocampus-neocortex interaction, which neuroscience identifies as central to memory consolidation.&lt;/p&gt;
&lt;p&gt;The fifth pillar is embodiment — or, in its artificial analogue, anchoring in an environment of active interaction. The system cannot learn merely by observing; it must act and observe the consequences of its actions. This can be accomplished through physical robotics, high-fidelity simulations, or virtual environments with realistic physics — but the learning structure must be action-consequence, not observation-only.&lt;/p&gt;
&lt;h3 id=&quot;42-architectures-attempting-the-path&quot;&gt;4.2 Architectures Attempting the Path&lt;/h3&gt;
&lt;p&gt;The hypothesis I articulate does not exist in a vacuum. There are several research programs that, from different angles, try to advance toward some of these pillars. It is worth examining the most promising.&lt;/p&gt;
&lt;p&gt;Liquid Neural Networks, developed by Ramin Hasani and collaborators at MIT, represent an elegant approach to the problem of staticity. Instead of computational graphs with fixed weights, LNNs model neuron dynamics through nonlinear ordinary differential equations. The time constant of each neuron — analogous to the rate at which it integrates input over time — varies as a function of the input received. The result is a system that adapts its internal representations in real time, during inference, without the need for additional backpropagation. This does not solve the problem of continual learning in all its depth, but it represents a step toward plasticity during execution.&lt;/p&gt;
&lt;p&gt;State Space Models, and in particular the Mamba architecture developed by Albert Gu and Tri Dao, address the computational bottleneck of transformers for long sequences. By formulating sequence processing as a problem of continuous state spaces — mathematically related to linear differential equations — Mamba can process arbitrarily long sequences at linear cost in time and memory, instead of the quadratic cost of transformers. The innovation of dynamic parameter selection allows the system to ‘choose’ what to retain in working memory, functionally approximating the mechanism of selective attention. For time-series modeling and physical dynamics, SSMs show state-of-the-art performance with a fraction of the parameters transformers require.&lt;/p&gt;
&lt;p&gt;Neuro-Symbolic systems, particularly the research line of Josh Tenenbaum’s group at MIT, represent perhaps the most ambitious attempt to integrate representation learning with formal causal reasoning. The basic architecture combines a neural network that learns to map raw observations to structured representations with a probabilistic inference engine that performs causal and counterfactual reasoning on those representations. The goal is to create systems that can, like a three-year-old child, make robust causal inferences from a few examples — not through statistical correlation, but through structured generative models of reality.&lt;/p&gt;
&lt;p&gt;Active Inference, based on Karl Friston’s Free Energy Principle, offers a radically different perspective on what an intelligent agent is doing. Instead of maximizing external rewards (as in reinforcement learning) or minimizing prediction error on data (as in deep learning), an Active Inference agent minimizes ‘free energy’ — a quantity that works simultaneously as a measure of surprise (how unexpected the observations are given the agent’s model) and as an upper bound on the entropy the agent experiences. The consequence is that the agent acts both to confirm its predictions and to actively seek situations in which its predictions can be tested and refined. It is a sophisticated mathematical formalization of the intrinsic curiosity we observe in children.&lt;/p&gt;
&lt;h3 id=&quot;43-the-problem-of-robotics-and-hardware&quot;&gt;4.3 The Problem of Robotics and Hardware&lt;/h3&gt;
&lt;p&gt;Any honest hypothesis about embodied AGI must face the question of hardware. The human brain operates on approximately 20 watts — an incredibly modest energy budget for what it accomplishes. Modern language models require GPU clusters consuming megawatts for training and tens of kilowatts for continuous inference. This discrepancy is not an implementation detail; it is a fundamental obstacle for robotics.&lt;/p&gt;
&lt;p&gt;A humanoid robot operating on a battery of reasonable capacity cannot host, in its physical structure, the hardware needed to run a state-of-the-art foundation model in real time. The alternative of processing everything in the cloud and transmitting commands to the robot over the network introduces latencies that break real-time control requirements — physics demands responses in milliseconds, and network latency is rarely predictable and often exceeds that threshold.&lt;/p&gt;
&lt;p&gt;The most promising architectural solution, and the one the hypothesis articulated here favors, is cognitive-motor decoupling. The system is structured in two asynchronous control loops: a high-latency, high-abstraction loop (the ‘cortex’), running on potentially external hardware or on high-efficiency chips on the robot itself, which maintains the world model and performs high-level planning; and a low-latency, high-frequency loop (the ‘spinal cord’), running on ultra-low-power neuromorphic hardware directly on the robot, which manages fine motor control, balance and reflex reactions.&lt;/p&gt;
&lt;p&gt;Neuromorphic hardware — chips such as Intel’s Loihi 2 or Heidelberg University’s BrainScaleS — operates on a logic fundamentally different from conventional GPUs. Instead of processing dense matrices synchronously, these chips emulate networks of spiking neurons that compute asynchronously and in an event-driven way: energy is consumed only when there is a ‘firing,’ a spiking event, that propagates information through the network. For motor-control tasks with sparse sensory input, energy consumption can be orders of magnitude lower than the GPU equivalent, with response latency comparable to biological.&lt;/p&gt;
&lt;p&gt;The combination of an efficient world model (possibly based on SSMs or JEPA-inspired architectures) with motor policies implemented in neuromorphic hardware represents the most plausible direction for a system that is, at the same time, cognitively capable and physically viable in a robotic chassis.&lt;/p&gt;
&lt;h3 id=&quot;44-the-emergent-curriculum-and-the-bootstrap-problem&quot;&gt;4.4 The Emergent Curriculum and the Bootstrap Problem&lt;/h3&gt;
&lt;p&gt;There is a subtle philosophical problem in the hypothesis that deserves attention: how to begin? If the system is to learn through self-directed exploration driven by intrinsic curiosity, what is the initial state from which this exploration starts? The Core Knowledge priors provide a partial answer, but there is a bootstrap question that must be faced.&lt;/p&gt;
&lt;p&gt;A human child does not begin her exploration in a completely unstructured environment. She begins in a dense social environment — surrounded by caregivers who respond to her actions, who provide feedback, who build scaffolding for learning. The mother who imitates the sound the baby just made, who names the objects the baby is looking at, who grades the complexity of the challenges she presents — is functioning as an external adaptive curriculum that complements and amplifies the child’s intrinsic exploration.&lt;/p&gt;
&lt;p&gt;For AI systems, this role can be played by an initial stage of light supervised learning — not the massive, passive training of LLMs, but a minimal set of demonstrations that anchors the structural priors to specific instances of the world. Instead of training a model on terabytes of text so that it learns, by statistical correlation, that ‘apple’ refers to a round red object that falls when released, the agent could receive a small number of demonstrations of physical apples — falling, bouncing, rolling — that anchor this concept in real sensorimotor representations, and then be released to explore autonomously from there.&lt;/p&gt;
&lt;p&gt;This approach is radically more data-efficient because the structural priors and the intrinsic curiosity mechanism do the heavy lifting of generalization. The problem of few-shot learning — learning new concepts from a few examples — ceases to be a problem when the system has the right inductive biases and a world model that allows it to simulate unobserved consequences.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-v-chollet-lecun-and-the-philosophy-of-efficient-intelligence&quot;&gt;Part V: Chollet, LeCun and the Philosophy of Efficient Intelligence&lt;/h2&gt;
&lt;h3 id=&quot;51-chollets-definition-as-touchstone&quot;&gt;5.1 Chollet’s Definition as Touchstone&lt;/h3&gt;
&lt;p&gt;François Chollet, in the seminal paper ‘On the Measure of Intelligence’ (2019), proposed a definition of intelligence that is at once simple and revolutionary in its implications: intelligence is the efficiency with which a system acquires new skills in situations not anticipated in its training. It is not the absolute level of performance. It is the relation between what the system learns and what it needed in order to learn.&lt;/p&gt;
&lt;p&gt;This definition has profound consequences. It implies that a system that achieves exceptional performance in a specific domain after extensive training in that domain may be, by Chollet’s criterion, less intelligent than a simple system that quickly learns to navigate new domains with minimal exposure. Intelligence, on this view, is not a product; it is a capacity for acquisition. It is not what you know; it is how much you learn per unit of experience.&lt;/p&gt;
&lt;p&gt;This definition is deliberately calibrated to capture what is most distinctive about human cognition. Humans are not the best at any specific task when trained specifically for it — AlphaGo plays Go better than any human, AlphaFold predicts protein structures with superhuman accuracy, LLMs produce text more fluent than most writers. But none of these systems can, on waking up tomorrow, learn to play a game it has never seen, or figure out how to use a novel tool, or adapt to a radically different environment with the fluency and speed with which an average human would.&lt;/p&gt;
&lt;p&gt;The hypothesis articulated in this essay is fundamentally Cholletian. It is not seeking systems that achieve high performance on specific benchmarks. It is seeking systems with the capacity to acquire new skills efficiently — which requires the right learning mechanisms, not merely the right data and parameters.&lt;/p&gt;
&lt;h3 id=&quot;52-lecuns-world-model-and-the-question-of-self-vision&quot;&gt;5.2 LeCun’s World Model and the Question of Self-Vision&lt;/h3&gt;
&lt;p&gt;Yann LeCun, in a series of papers and presentations on his JEPA (Joint Embedding Predictive Architecture), articulated a critique complementary to autoregressive generative models. The central problem, for LeCun, is not that LLMs are dumb — it is that they were trained to predict the next token, and this training objective forces the model to learn representations that include all the noise, all the ambiguity, all the irrelevant specificity of the linguistic surface.&lt;/p&gt;
&lt;p&gt;The alternative LeCun proposes is learning representations in latent space — learning to predict not the future observation in its raw details, but the future abstract state in a low-dimensional representation space. A JEPA architecture does not predict which exact pixels will appear in the next frame of a video; it predicts what latent representation the next frame will have. This eliminates the need to learn irrelevant details (such as the exact texture of light on a surface) and forces learning to focus on the causal regularities that really matter.&lt;/p&gt;
&lt;p&gt;The connection with child development is direct. When a two-year-old sees a glass being pushed to the edge of a table, she does not predict the exact position of each fragment of glass when it breaks. She predicts, in the latent space of her world model: ‘object falls, object breaks, loud noise.’ It is an abstract prediction, incomplete in the details, but correct in the aspects that matter for action — and sufficient to guide the decision to step back, alert an adult, or simply watch with fascination.&lt;/p&gt;
&lt;p&gt;LeCun further argues that LLMs are incapable of true planning because they have no world model that can be interrogated counterfactually. When we ask an LLM ‘what would happen if you removed the support of a table?’, it produces an answer based on statistical correlations of how that question is typically answered in text. It does not run a mental simulation of the physical scenario. A four-year-old child, confronted with the real physical situation, would do exactly that simulation — she would test hypotheses in her world model before acting.&lt;/p&gt;
&lt;h3 id=&quot;53-language-as-tool-not-as-foundation&quot;&gt;5.3 Language as Tool, Not as Foundation&lt;/h3&gt;
&lt;p&gt;An honest synthesis of Chollet’s and LeCun’s positions, in relation to what LLMs have achieved, suggests the following perspective: language is an extraordinarily powerful tool for communicating, refining and transmitting knowledge — but it is a tool, not the foundation of cognition.&lt;/p&gt;
&lt;p&gt;Human cognition developed, evolutionarily, long before language. Our pre-linguistic ancestors already possessed sophisticated world models, capacity for instrumental planning, recognition of the intentions of other agents, and learning by observation and imitation. Language is a technology of compression and transmission that was superimposed on this existing cognitive substrate — it did not create it.&lt;/p&gt;
&lt;p&gt;LLMs inverted this sequence. They started with language and tried to derive cognition from it. The result is a system that is extraordinarily competent in the linguistic layer — in communication, information retrieval, the articulation of reasoning — but that lacks the base on which, biologically, language rests. It is like building the upper floor of a building without the lower floors. The upper floor may be impressive; but it is suspended in the air.&lt;/p&gt;
&lt;p&gt;The hypothesis proposed in this essay argues for the need to build the lower floors. Not to abandon LLMs — they are the most impressive upper floor we have ever built, and their integration into a more complete cognitive system would be extraordinarily powerful — but to recognize that they are a piece of a larger puzzle, not the whole puzzle.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vi-implications-and-objections&quot;&gt;Part VI: Implications and Objections&lt;/h2&gt;
&lt;h3 id=&quot;61-the-objection-of-continued-scaling&quot;&gt;6.1 The Objection of Continued Scaling&lt;/h3&gt;
&lt;p&gt;The most immediate objection to the hypothesis articulated here comes from the field of scaling optimism: the idea that the current limitations of LLMs are temporary, and that sufficiently large models, trained on sufficiently rich data, will eventually exhibit the capabilities that the hypothesis says require different architectures.&lt;/p&gt;
&lt;p&gt;This objection should be taken seriously. There are historical cases in which apparently fundamental limitations were overcome by scale and data engineering — the emergence of multi-step reasoning capabilities in GPT-4 models and their successors surprised many researchers who had argued that these capabilities required fundamentally different architectures.&lt;/p&gt;
&lt;p&gt;But there are reasons to resist unrestricted scaling optimism. First, the empirical evidence of a plateau: the marginal returns of scaling are documentably diminishing, and benchmarks of novel reasoning such as ARC do not show improvement proportional to the increase in parameters. Second, there is a theoretical argument from Chollet: if intelligence is the efficiency of novel learning, then a system that requires more data and parameters to solve the same problem is, by definition, becoming less intelligent as it scales — even if its absolute outputs improve. Third, there is the symbol grounding argument: it is not clear how scaling a system that processes only linguistic symbols can, in principle, produce the anchored sensorimotor representations that the Room of the Curious Child experiment identifies as necessary.&lt;/p&gt;
&lt;h3 id=&quot;62-the-virtual-world-objection&quot;&gt;6.2 The Virtual World Objection&lt;/h3&gt;
&lt;p&gt;A second objection questions the need for physical embodiment. If the argument is that agents need to learn through active interaction with a physical environment, why not a virtual one? High-fidelity physics simulators — such as NVIDIA’s Isaac Sim or MuJoCo — provide environments in which agents can learn through millions of action-observation cycles. There is no need for physical bodies, neuromorphic hardware, or any of the complex engineering challenges that real robotics imposes.&lt;/p&gt;
&lt;p&gt;This objection has considerable merit, and the hypothesis proposed here does not exclude learning in simulation. Indeed, training in simulation followed by transfer to the real world (sim-to-real) is probably a necessary component of any practical system. But the objection underestimates two problems.&lt;/p&gt;
&lt;p&gt;The first is the reality gap: simulations, however faithful, differ from the real world in respects that are often difficult to identify and correct a priori. A robot that learned to grasp objects in simulation often fails when attempting the same with real physical objects, because the simulation captured the most obvious properties (shape, approximate weight) but failed to capture the subtler ones (texture, deformability, variation in friction). Systems with the continual adaptation mechanisms proposed in the hypothesis — connected here to Liquid Neural Networks and Active Inference — can learn to calibrate the difference; but this requires real interaction, even if brief.&lt;/p&gt;
&lt;p&gt;The second is the deeper problem of semantic anchoring. If the goal is a system with representations genuinely anchored in physical reality — not symbols that correlate with other symbols, but representations that correspond to real causal structures of the world — then experience with the real world, with all its unpredictability and causally relevant richness, is probably necessary to some extent. Simulation can compress learning time, but it cannot completely replace reality as a source of calibration.&lt;/p&gt;
&lt;h3 id=&quot;63-the-objection-of-irreducible-complexity&quot;&gt;6.3 The Objection of Irreducible Complexity&lt;/h3&gt;
&lt;p&gt;A third objection is more philosophical: even if all the mechanisms proposed in the hypothesis are successfully implemented — Core Knowledge priors, intrinsic curiosity, a world model in latent space, continual learning, embodiment — there are reasons to doubt that the result will be what we call AGI, or that it will be sufficient for agentic capability in the most complex domains.&lt;/p&gt;
&lt;p&gt;The human brain is extraordinarily complex. It contains approximately 86 billion neurons with between 100 and 500 trillion synapses. It operates on multiple time scales simultaneously — from milliseconds (action potentials) to years (memory consolidation). It is massively parallel, asynchronous, and plastic in ways we do not yet fully understand. It may be that this complexity is necessary, not contingent — that there is no architectural shortcut that reproduces human cognitive capabilities without reproducing something of its structural complexity.&lt;/p&gt;
&lt;p&gt;The honest answer to this objection is: we do not know. It may be that an architecture with the five identified pillars, implemented on adequate hardware, is sufficient for general agentic capabilities far beyond what LLMs exhibit, even if it does not achieve everything human cognition accomplishes. It may be that additional components, not yet identified, are necessary. The hypothesis is not a claim of sufficiency; it is a claim that the identified mechanisms are necessary — that any path to genuine AGI must pass through them.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;part-vii-the-horizon--what-would-be-different&quot;&gt;Part VII: The Horizon — What Would Be Different&lt;/h2&gt;
&lt;h3 id=&quot;71-a-hypothetical-system&quot;&gt;7.1 A Hypothetical System&lt;/h3&gt;
&lt;p&gt;To make the hypothesis concrete, it is worth sketching how a system implementing the five pillars might work in practice — not as a detailed engineering proposal, but as an exercise in conceptual clarification.&lt;/p&gt;
&lt;p&gt;Imagine an agent — whether a physical robot or an agent in a high-fidelity simulation — that begins its existence with a minimal set of architecturally encoded structural priors: biases for segmenting sensory input into cohesive objects, for detecting cause-effect relations by contact, for distinguishing agents from inanimate objects based on movement patterns. These priors are not specific knowledge about the world; they are the ‘lens’ through which the agent observes the world.&lt;/p&gt;
&lt;p&gt;The agent is placed in an environment and begins to explore, guided by an intrinsic motivation function that optimizes learning progress. It moves objects, observes what happens, forms latent representations of what it observed, and continually updates a predictive world model that allows it to anticipate the consequences of future actions. When its model errs — when the world behaves differently from what was predicted — the error signal is both a penalty (the model needs to be updated) and an intrinsic reward (this is interesting; worth exploring more here).&lt;/p&gt;
&lt;p&gt;As the agent explores, it develops a growing vocabulary of concepts — not linguistic concepts, but prototypes of situations, objects and actions, represented in the latent space of the world model. A concept such as ‘ramp’ emerges as a prototype of a spatial configuration associated with a pattern of behavior of sliding objects. It does not need a word to have the concept; the concept is the latent representation.&lt;/p&gt;
&lt;p&gt;When, eventually, this agent is connected to a language system — whether a fine-tuned LLM or a smaller language module — language can be anchored to these pre-existing concepts. ‘Ramp’ becomes a symbol that maps to the latent representation the agent has already built through experience. This is symbol grounding in its most direct form: the symbol acquires meaning because it is connected to a representation that already has causal content.&lt;/p&gt;
&lt;h3 id=&quot;72-what-this-system-could-do&quot;&gt;7.2 What This System Could Do&lt;/h3&gt;
&lt;p&gt;Such a hypothetical system would have capabilities qualitatively different from today’s LLMs in the domains where the hypothesis predicts they fail. Faced with a novel mechanism — a simple machine that was never described in any training dataset — the system would not need a textual description to understand it. It could observe it briefly, mentally simulate its possible operations using the world model, test one or two hypotheses through physical interaction, and then operate the mechanism correctly. It is the behavior of the Curious Child in the thought experiment.&lt;/p&gt;
&lt;p&gt;This system would also be capable of genuine continual learning. Each new experience would refine the world model without erasing earlier representations — analogously to how humans learn new motor skills or new concepts without forgetting the skills and concepts they already possessed. The fundamental distinction between ‘training’ and ‘inference’ would become more fluid: the system would always be, in a sense, learning — updating its world model with each new observation.&lt;/p&gt;
&lt;p&gt;In domains of formal reasoning — mathematics, logic, science — the architecture would allow a deeper integration between intuition (world model in latent space) and formal verification (symbolic module). The system could formulate hypotheses based on its world model, express those hypotheses in formal language, verify them with a symbolic inference engine, and update the world model based on the results. It is the computational formalization of the scientific cycle of observation, hypothesis, prediction and test.&lt;/p&gt;
&lt;h3 id=&quot;73-robots-and-the-question-of-physical-agency&quot;&gt;7.3 Robots and the Question of Physical Agency&lt;/h3&gt;
&lt;p&gt;The hypothesis has specific implications for robotics, which is where the distinction between the two kinds of systems becomes most visible and most consequential. Today’s robots — including the impressive humanoids of companies such as Boston Dynamics, Figure and Tesla — have extraordinary bodies and relatively limited brains. They can execute sequences of complex movements with remarkable precision, but they are brittle in generalization. A robot that learned to pick up a specific box under specific conditions often fails when the box changes shape, or the environment’s lighting changes, or an obstacle not anticipated in training appears.&lt;/p&gt;
&lt;p&gt;A system based on the hypothesis proposed here would approach this problem in a fundamentally different way. Instead of learning motor policies that map directly from percepts to actions, it would maintain a world model that allows it to simulate the effect of possible actions before executing them, and to adapt its behavior in real time when the world behaves differently from the model. The world model is not merely a memory bank — it is an instrument of adaptive planning.&lt;/p&gt;
&lt;p&gt;The asynchronous control architecture — slow, abstract ‘cortex’ plus fast, reactive ‘spinal cord’ — allows this system to operate within the energy and latency constraints of real robotic hardware. The cognitive cortex does high-level planning with tolerable latency; the reactive spinal cord ensures that the body remains stable and responsive to physical perturbations in real time, without requiring the cortex to redo all the reasoning every millisecond.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;conclusion-the-intelligence-we-have-not-yet-built&quot;&gt;Conclusion: The Intelligence We Have Not Yet Built&lt;/h2&gt;
&lt;p&gt;We began with an asymmetry: a four-year-old child solves in seconds problems that challenge the most powerful AI systems in the world. This fact, apparently paradoxical given the computational power and the quantity of data of modern systems, points to something profound about the nature of intelligence — and about what we still lack in order to build it.&lt;/p&gt;
&lt;p&gt;The hypothesis articulated in this essay is not a complete research program. It is an orientation: the idea that AGI will not be achieved through the continued scaling of passive and static architectures, but requires a qualitative change in the way AI systems learn. That change must draw inspiration — not naively, but with technical rigor — from the mechanisms that make child learning extraordinarily efficient: evolutionary priors, intrinsic curiosity, predictive world models in latent space, continual learning, and anchoring in active experience.&lt;/p&gt;
&lt;p&gt;The Room of the Curious Child that I proposed as a thought experiment illuminates the central distinction: between statistical competence and genuine intelligence; between knowing that and knowing how; between correlation and causality; between interpolation and generalization. These are not problems that more data or more parameters will solve. They are problems that require different architectures, different training objectives, and a different conception of what it means for a system to learn.&lt;/p&gt;
&lt;p&gt;The tools are beginning to appear. Liquid Neural Networks offer plasticity at inference time. State Space Models offer computational efficiency for long sequences and physical dynamics. Neuro-symbolic systems integrate representation learning with causal reasoning. Active Inference provides a mathematical formalization of intrinsic curiosity. Neuromorphic hardware begins to make low-power computation in robotic chassis viable. None of these pieces, alone, is the solution; but together, integrated with architectural coherence and guided by the principles of child cognitive development, they point to a path.&lt;/p&gt;
&lt;p&gt;The AGI project, like any genuinely ambitious scientific project, may fail — for reasons we have not yet identified, because of limitations we do not yet understand, because of complexities that biology resolves but that engineering may be unable to reproduce. But failing while attempting the right thing is more promising than succeeding at scaling the wrong thing. And the right thing, the hypothesis suggests, is learning to learn — building systems that, like the children who inspire the hypothesis, are fundamentally ongoing processes and not finished products; systems that become more intelligent with each new experience, that explore with curiosity what they do not yet understand, and that build, one observation at a time, a model of the world that allows them to act upon it with efficiency and wisdom.&lt;/p&gt;
&lt;p&gt;The child playing with blocks in the room is not merely playing. She is building the architecture of thought — piece by piece, fall by fall, surprise by surprise. This is the learning we do not yet know how to replicate. And replicating it may be the most important task of the next century’s engineering.&lt;/p&gt;
&lt;hr&gt;
&lt;h2 id=&quot;notes-on-the-conceptual-references&quot;&gt;Notes on the Conceptual References&lt;/h2&gt;
&lt;p&gt;This essay is deliberately philosophical and exploratory. The references below do not constitute a formal academic bibliography, but a map of the intellectual traditions and specific works that inform the arguments developed.&lt;/p&gt;
&lt;p&gt;In the philosophy of mind, John Searle’s Chinese Room (Minds, Brains, and Programs, 1980) and Frank Jackson’s Mary’s Room (Epiphenomenal Qualia, 1982) provided the template for the proposed thought experiment. The debate on symbol grounding, originally articulated by Stevan Harnad (The Symbol Grounding Problem, 1990), is central to the section on the limitations of LLMs.&lt;/p&gt;
&lt;p&gt;In developmental psychology, the work of Elizabeth Spelke on Core Knowledge (Language and Conceptual Development, 1994; Core Knowledge, 2000) and of Karen Wynn on infant numerical cognition (Addition and Subtraction by Human Infants, 1992) underpins the section on evolutionary priors. Jean Piaget’s theory of sensorimotor development (The Construction of Reality in the Child, 1937) and Lev Vygotsky’s concept of the zone of proximal development (Mind in Society, 1978) inform the discussion of active learning and scaffolding.&lt;/p&gt;
&lt;p&gt;In artificial intelligence and cognitive science, François Chollet’s paper On the Measure of Intelligence (2019) is the central philosophical starting point. Yann LeCun’s work on JEPAs and world models — especially A Path Towards Autonomous Machine Intelligence (2022) — informs the section on alternative architectures. Pierre-Yves Oudeyer’s research on intrinsic motivation and developmental robotics, particularly the work on Learning Progress as an intrinsic reward signal (Intrinsic Motivation Systems for Autonomous Mental Development, 2007), is central to the discussion of curiosity as algorithm.&lt;/p&gt;
&lt;p&gt;Karl Friston’s Free Energy Principle (The Free-Energy Principle: A Unified Brain Theory, 2010) underpins the discussion of Active Inference. The work of Josh Tenenbaum and collaborators on probabilistic cognition and computational Core Knowledge (How to Grow a Mind, 2011) informs the section on neuro-symbolic systems. Ramin Hasani’s research on Liquid Neural Networks (Liquid Time-Constant Networks, 2020) and the work of Albert Gu and Tri Dao on Mamba and State Space Models (Mamba: Linear-Time Sequence Modeling with Selective State Spaces, 2023) underpin the discussion of alternative architectures to the transformer.&lt;/p&gt;
&lt;p&gt;Finally, research on embodied cognition — especially the works of Francisco Varela, Evan Thompson and Eleanor Rosch (The Embodied Mind, 1991) — underpins the argument for embodiment as a necessary, not contingent, condition of genuine cognition.&lt;/p&gt;</content:encoded><dc:creator>Matheus Soranço</dc:creator><dc:language>en</dc:language><category>artificial intelligence</category><category>cognition</category><category>science</category><category>philosophy</category><category>technology</category><category>featured essays</category></item></channel></rss>