A. Claim Register

Four kinds of claim. Only two are bets.

The essay separates four levels of claim.

B. Simulation and Instantiation

Whether a substrate represents a property or instantiates it depends on what the property is.

John Searle used simulations of storms, digestion, and fire to distinguish formal computation from the causal powers of the physical system represented.^[1] That distinction begins the analysis.

A simulated stomach does not digest. A simulated calculator does calculate. An artificial heart can pump blood. An artificial wing can fly.

David Chalmers’s organisational-invariance argument is the strongest clean counterposition. If two systems share sufficiently fine-grained causal organisation, he argues, their conscious experience should not differ. Otherwise gradual replacement of biological components would produce implausible “fading” or “dancing” qualia that the subject could never notice.^[20]

The wetness analogy locates the claim without deciding it. Which kind of property consciousness is remains the open question. The essay wagers that it is the stomach’s kind: substrate-sensitive, and not supplied by conventional digital implementation. The evidence for that reading is the biology of Appendix C and the abstraction argument of Appendix D. The evidence against it is Chalmers’s invariance argument.

Coarse-graining is the strongest physical counterargument. A microscopically irreducible system can remain predictable at a higher descriptive level.^[21] Cognition can survive that reduction. The essay’s position is that consciousness does not. Abstractions designed to preserve cognitive output discard the physical continuity that produces experience.

Physical detail can be dispensable to the output yet constitutive of the subject. That is the hinge. Output equivalence does not entail phenomenal equivalence.

Current AI-consciousness frameworks remain theory-relative. They update credence from theory-derived indicators and treat behavioural imitation as one signal among others.^[22]

A 2025 adversarial collaboration tested Integrated Information Theory against Global Neuronal Workspace Theory. The results supported parts of each account and challenged central predictions of both.^[23] Consciousness science has no settled substrate test.^[24]

C. The Biological Architecture

Twenty watts, and the body is in the loop.

The 17–20 W estimate is whole-brain metabolic power. Accelerator-equivalent computation is a different quantity. Levy and Calvert’s audit estimates about 17 W and argues that communication costs dominate narrowly defined cortical computation.^[3]

The brain is hybrid rather than purely analogue. Human neurons combine discrete-like spikes with continuous voltages, currents, chemical concentrations, and dendritic processes. Gidon and colleagues found graded calcium-mediated dendritic action potentials in human cortical neurons. Detailed modelling by Beniaguev, Segev, and London required a deep temporal convolutional network to approximate the millisecond input-output behaviour of a model layer-5 pyramidal neuron.^[4]

The embodiment claim is operational. Large distributed networks regulate the body’s internal state through allostasis and interoception.^[6] Infants also structure their own learning by directing attention and intervening on the environment.^[8]

The linguistic data comparison is deliberately narrow. Michael Frank estimates that major language models receive roughly four to five orders of magnitude more linguistic input than children. A child simultaneously receives continuous multimodal and social input, inherits evolved priors, and pursues different objectives.^[7]

Ephaptic coupling supplies local causal evidence. Endogenous-strength extracellular fields produce small membrane shifts and can entrain spike timing, particularly at low frequencies.^[14] Its demonstrated scale is local.

D. Continuum, Chaos, Quantum State, and Copy

A copy can shadow the orbit. It cannot be the orbit.

The argument has two layers: a mathematical limit on exact digital continuation, and a physical claim about what consciousness is. The first is proved. The second is wagered.

First, every finite bit string and every finite program belongs to a countable set. Turing’s computable numbers are therefore countable, while the real numbers are uncountable.^[25] A short algorithm can specify $\pi$, but finite digital descriptions cannot specify every point of the continuum.

Digital representation turns that limit into an engineering principle. It maps many lower-level physical states onto one logical state. Every voltage inside a noise margin becomes the same bit. This equivalence class gives software its portability, reliability, and perfect reproducibility.

Second, the brain leaves those physical variables active. Classical fields vary over continuous coordinates. Wavefunctions and even finite-dimensional quantum states are continuously parameterised. Continuous-variable quantum information works directly with observables whose spectra are continuous.^[26] The Bekenstein bound limits the distinguishable information in a bounded finite-energy system without turning its evolution into a finite automaton.^[27]

That bound is also the strongest objection to this layer. If continuum distinctions ever affect behaviour or report, they are measurable. What is measurable in a bounded finite-energy region is finitely describable. What is finitely describable can be digitally represented. The essay accepts each step. Its reply is the hinge of Appendix B: a representation of a distinction is not its occurrence. The continuum argument therefore does not prove the constitution wager on its own. It shows what digital abstraction discards, and leaves open whether the discarded motion is what experience is made of.

A digital computer is itself a continuous quantum object. Its logical architecture suppresses the causal relevance of most lower-level motion. A brain computes through membrane voltage, chemical concentration, field interaction, and bodily recurrence. The substrate thesis places consciousness in that active physical trajectory.

Digital and quantum computers can approximate broad classes of physical dynamics to any chosen accuracy.^[28][29] That is why the function survives. Approximation relates a model to a process. Identity belongs to the continued process itself.

Continuous dynamics span both computable and noncomputable cases. Polynomial ordinary differential equations of polynomial length characterise polynomial-time computation.^[30] Specially constructed wave equations can also evolve computable initial data into a noncomputable solution, with the result turning on how the initial field is represented.^[31] The essay places consciousness in the causally active physical distinctions that conventional digital implementation throws away.

Third, long-horizon identity has an exact limit. A closed deterministic machine with $N$ possible states repeats within $N+1$ transitions and cycles thereafter. An aperiodic continuum trajectory does not. Where the dynamics have sensitive dependence, small differences in initial state grow until the trajectories separate macroscopically.^[32]

The theorem is precise:

A fixed finite-state digital implementation cannot remain state-for-state equivalent for unbounded time to an aperiodic physical trajectory containing causally relevant continuum distinctions.

One constitutive continuum degree of freedom is enough to break exact instantiation. Sensitive dynamics make the difference compound through time. For computable dynamics, greater precision extends the chosen tracking horizon. The model still begins as a new physical process rather than the continued occurrence of the old one.

The theorem rules out exact continuation. It does not rule out organisational equivalence at a coarser grain, which is the functionalist’s actual claim. That dispute returns to Appendix B.

Shadowing explains why the copy can remain functionally excellent. A numerical pseudo-orbit can follow a true orbit near the intended one for long periods.^[33] Coarse-grained predictions can survive microscopic irreducibility.^[21] The output survives because it belongs to the equivalence class. The original subject does not, because it belongs to the trajectory.

Fourth, quantum mechanics supplies a physical precedent for limited copyability. Suppose one device clones two possible pure states, $|a\rangle$ and $|b\rangle$. Quantum evolution preserves their overlap. Perfect cloning would square it. The device therefore requires $|\langle a|b\rangle| = |\langle a|b\rangle|^2$, which holds only when the states are identical or orthogonal.^[34]

A device can copy a specified orthogonal set. A known state can be prepared again. Approximate copies are possible. One operation cannot perfectly clone every possible pure quantum input. Its role here is analogy: physical state is richer than universally clonable description.

Parfit’s branching cases complete the identity argument. Two successors can inherit one psychological history. Numerical identity cannot branch.^[35] The essay locates survival in the unbroken causal process. A conscious copy would be a new subject. It would not be the continued life of the person copied.

Capability belongs to the pattern. Experience belongs to the process. Identity belongs to the trajectory.

E. The Riemann Register

A new framework, and a new result within it.

Anthropic reported the zeta result on 10 August 2026. Its experimental system established a lower bound of 67.25 per cent for the relevant proportion of nontrivial zeros that are simple and lie on the critical line, up from 41.6 per cent.^[12]

The run used approximately 31 million output tokens, 650 unsuccessful initial ideas, roughly 60 subagents, 2,400 shell commands, hundreds of Python programs, and thousands of numerical checks. Two subagents produced the key ideas. Other agents explored failed routes, criticised, validated, and wrote. The result draws heavily on earlier human work but introduces a new rank–trace argument. It was examined by external mathematicians and formalised in Lean. As of this essay’s date, it remains a recent result rather than settled long-term consensus.

Anthropic explicitly says the technique is not expected to prove the Riemann hypothesis. The method is insensitive to an exceptional set of $o(N)$ zeros. Even an asymptotic density of one permits a sparse infinite exception, just as the perfect squares form a density-zero infinite subset of the positive integers.

The comparison with Riemann describes two architectures of discovery. The Stanford 2026 AI Index documents continued rapid capability gains, especially in mathematics, coding, and agents, alongside conspicuous failures on simple perceptual and embodied tasks.^[13] The profile is advancing and jagged.

F. Fixed Capability and Economic Use

A skill trained once is priced like a copy.

The 280-fold figure comes from Stanford’s comparison of the price of querying a model at approximately GPT-3.5-level MMLU performance: about 20 US dollars per million tokens in November 2022 and 0.07 US dollars in October 2024. Over roughly the same interval, the smallest model exceeding 60 per cent on MMLU fell from 540 billion to 3.8 billion parameters.^[17]

Epoch AI estimates benchmark-specific fixed-performance inference prices falling at annual rates ranging from about 9-fold to 900-fold, with a median near 50-fold across its selected thresholds and periods.^[36] Public API prices combine engineering cost with competition, margin, batching, and product strategy. They describe a historical rate rather than a perpetual law.

Current productivity evidence is bounded. The support study measured a 15 per cent average productivity increase, rising to 36 per cent for workers in the lowest skill quintile.^[37] The consultant experiment found higher speed and quality inside the tested model’s capability frontier, but users were 19 percentage points less likely to reach the correct answer on an outside-frontier task.^[38] These studies establish useful augmentation of specified work. Whole occupations require integrated coverage across the task frontier.

Task complementarity depends on prices, demand, workflow design, and the creation of new work.^[39][40] Human moral standing and political authorship are not comparative advantages at all. They belong to the welfare function and the theory of legitimacy.

G. The Falsifier Register

Recorded August 2026. The essay contains a substrate wager and a complementarity thesis. Neither can be dated the way a market can, so the register names the observable that comes first for each claim, where the claim lives, and what breaks.

G.1 — Digital prosthesis, at the first cortical replacements. The substrate wager predicts that a digital component can replace neural tissue, preserve the output, and remove the experience that tissue carried (Appendix B). Cochlear and retinal implants already replace transduction. The gate is cortex. If biological cortex is progressively replaced with digital prostheses while behaviour, introspective continuity, and every credible neural signature of conscious state remain intact, Chalmers’s fading-qualia argument wins as an experiment. The wager loses outright.

G.2 — Convergent indicators, from competing theories. Current frameworks derive indicators of consciousness from rival theories and update credence rather than verdicts (Appendix B). If a wholly digital system satisfies convergent, causally tested, difficult-to-game indicators drawn from otherwise competing theories, the wager falls from a position to a minority credence. The wetness analogy would then be a mood.

G.3 — Subtraction, in living tissue. Section II and Appendix C describe dendritic, chemical, interoceptive, and endogenous-field dynamics as the machinery of the subject. If those dynamics can be removed or radically altered while the relevant high-level organisation and conscious experience remain unchanged, the machinery was scaffolding. Section II would describe a brain. It would not describe a mind.

G.4 — Finite equivalence, from physics and neuroscience together. Appendix D already concedes half of this entry: the Bekenstein bound makes every causally relevant distinction in a bounded brain finitely describable. The live half is the second. If neuroscience shows that a finite digital equivalent preserves conscious state across substrate change, representation and occurrence coincide, and the second layer of Appendix D collapses into the first. The mathematics would stand. The wager would not.

G.5 — Sample efficiency, on the child’s budget. Section II rests the intuition advantage on embodied, active, continuously plastic learning at twenty watts. Data-limited training benchmarks and energy per task are published quantities. If digital systems reach human-like open-ended learning, linguistic sample efficiency, persistent embodied adaptation, and comparable energy cost without reproducing biological dynamics, the advantage becomes history. Twenty watts would be a curiosity rather than an argument.

G.6 — Conjecture, under scarcity. Section III and Appendix E contrast Riemann’s compression with the zeta run’s bandwidth. If artificial systems repeatedly generate major new conceptual frameworks under novel, underspecified conditions and severe information constraints, compression has a second architecture. The complementarity thesis weakens. The constitution question is untouched.

G.7 — Authorship, in the market. Section VII claims that relationship, provenance, responsibility, contestability, and legitimate authorship keep human judgement in the loop (Appendix F). If AI-only systems durably outperform human–AI systems in domains where those goods are part of what is valued, with no residual demand for human judgement, complementarity was a preference rather than an economics. Standing would remain. It belongs to experience, not to comparative advantage.

Each of these observations would expand the class of architectures that instantiate experience, or redraw the division of work. Experience would remain the source of standing.

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[2] Sutton, R. S. (2019). “The Bitter Lesson.” (The historical claim that general methods exploiting computation, especially learning and search, repeatedly defeated hand-engineered domain knowledge.)

[3] Levy, W. B. & Calvert, V. G. (2021). “Communication consumes 35 times more energy than computation in the human cortex, but both costs are needed to predict synapse number.” Proceedings of the National Academy of Sciences 118(18), e2008173118. (An updated whole-brain power estimate of roughly 17 W; the familiar 20 W is an order-of-magnitude statement.)

[4] Gidon, A. et al. (2020). “Dendritic action potentials and computation in human layer 2/3 cortical neurons.” Science 367(6473), 83–87. See also Beniaguev, D., Segev, I. & London, M. (2021). “Single cortical neurons as deep artificial neural networks.” Neuron 109(17), 2727–2739.e3. (Active dendritic computation and the complexity hidden by the artificial point-neuron abstraction.)

[5] Horn, A. (2015). “The Human Connectome.” Wikimedia Commons. CC BY-SA 4.0. (A group connectome rendered from diffusion MRI across twenty subjects. White-matter fibres are coloured by traversing direction. The displayed version uses a black background.)

[6] Kleckner, I. R. et al. (2017). “Evidence for a large-scale brain system supporting allostasis and interoception in humans.” Nature Human Behaviour 1, 0069. (A distributed system linking internal bodily regulation with networks involved in cognition and emotion.)

[7] Frank, M. C. (2023). “Bridging the data gap between children and large language models.” Trends in Cognitive Sciences 27(11), 990–992. (The four-to-five-order gap in linguistic input, with the necessary caveats about multimodal grounding, priors, and social interaction.)

[8] Smith, L. B., Jayaraman, S., Clerkin, E. & Yu, C. (2018). “The developing infant creates a curriculum for statistical learning.” Trends in Cognitive Sciences 22(4), 325–336. See also Poli, F. et al. (2020). “Infants tailor their attention to maximize learning.” Science Advances 6(39), eabb5053. (Infant learning as active selection and construction of experience, not passive exposure to a fixed corpus.)

[9] Holtmaat, A. & Svoboda, K. (2009). “Experience-dependent structural synaptic plasticity in the mammalian brain.” Nature Reviews Neuroscience 10, 647–658. (How experience can produce persistent structural change in neural circuits.)

[10] Nader, K. & Hardt, O. (2009). “A single standard for memory: the case for reconsolidation.” Nature Reviews Neuroscience 10, 224–234. (Memory retrieval can destabilise and update a stored trace rather than simply reading an immutable record.)

[11] Riemann, B. (1859). “On the Number of Primes Less Than a Given Magnitude.” Monatsberichte der Berliner Akademie. See also Riemann, B. (1854, published 1868). “Über die Hypothesen, welche der Geometrie zu Grunde liegen.” Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13. (The 1859 paper and manuscript in which Riemann formulated the hypothesis, and the 1854 habilitation lecture on the foundations of geometry whose curved manifolds became the mathematics of general relativity.)

[12] Anthropic (2026). “Learning more about Claude’s mathematical capabilities.” 10 August 2026. See the research paper and Lean formalisation. (The 67.25 per cent zeta-zero result, method, run structure, validation, and explicit boundary relative to the Riemann hypothesis.)

[13] Stanford Institute for Human-Centered Artificial Intelligence (2026). “2026 AI Index Report: Technical Performance.” (Rapid gains in mathematics, coding, and agentic benchmarks alongside uneven performance on perception, computer use, and embodied tasks.)

[14] Anastassiou, C. A. et al. (2011). “Ephaptic coupling of cortical neurons.” Nature Neuroscience 14, 217–223. (Local endogenous extracellular fields can causally shift membrane potential and entrain spike timing independently of synapses.)

[15] Milinkovic, B. & Aru, J. (2026). “On biological and artificial consciousness: A case for biological computationalism.” Neuroscience & Biobehavioral Reviews 181, 106524. (A substrate-sensitive proposal centred on scale-inseparable multiscale processing and hybrid discrete–continuous biological dynamics.)

[16] Block, N. (2026). “Can only meat machines be conscious?” Trends in Cognitive Sciences 30(4), 298–308. (The case for investigating the subcomputational electrochemical mechanisms that realise neural computation rather than assuming abstract role is sufficient.)

[17] Stanford Institute for Human-Centered Artificial Intelligence (2025). “AI Index 2025: State of AI in 10 Charts.” (The decline in inference price at fixed MMLU capability and the reduction in the smallest model crossing a fixed benchmark threshold.)

[18] Peter, F. (2023). “Political Legitimacy.” Stanford Encyclopedia of Philosophy. (The distinction between effective or epistemically competent rule and the conditions under which authority is legitimate.)

[19] Tao, T. (2026). “Mathematics in the age of AI.” arXiv:2608.16753. (An ICM 2026 essay that conditions on research-capable AI and asks what goals and values mathematical research should serve.)

[20] Chalmers, D. J. (1995). “Absent Qualia, Fading Qualia, Dancing Qualia.” In Conscious Experience, ed. T. Metzinger, 309–328. Imprint Academic. (The argument that conscious experience is an organisational invariant under sufficiently fine-grained functional equivalence.)

[21] Israeli, N. & Goldenfeld, N. (2004). “Computational irreducibility and the predictability of complex physical systems.” Physical Review Letters 92, 074105. (A demonstration that microscopic computational irreducibility can coexist with predictable, reducible coarse-grained dynamics.)

[22] Butlin, P. et al. (2026). “Identifying indicators of consciousness in AI systems.” Trends in Cognitive Sciences 30(6), 488–501. (A theory-derived, credence-based framework for possible AI consciousness that makes uncertainty and the limits of behavioural imitation explicit.)

[23] Cogitate Consortium et al. (2025). “Adversarial testing of global neuronal workspace and integrated information theories of consciousness.” Nature 642, 133–142. (A preregistered, multimodal test that supported some predictions and challenged central claims of both theories.)

[24] Seth, A. K. & Bayne, T. (2022). “Theories of consciousness.” Nature Reviews Neuroscience 23, 439–452. (An authoritative review of major consciousness-theory families and their unresolved empirical relationships.)

[25] Turing, A. M. (1936–37). “On Computable Numbers, with an Application to the Entscheidungsproblem.” Proceedings of the London Mathematical Society s2-42(1), 230–265. (The formal basis of computable numbers; finite programs describe only a countable subset of the reals.)

[26] Weedbrook, C. et al. (2012). “Gaussian Quantum Information.” Reviews of Modern Physics 84, 621–669. (Continuous-variable quantum systems, including observables with continuous spectra and states in infinite-dimensional Hilbert spaces.)

[27] Bekenstein, J. D. (1981). “Universal upper bound on the entropy-to-energy ratio for bounded systems.” Physical Review D 23(2), 287–298. (A bound on the distinguishable information available to a bounded finite-energy physical system; an important limit on naive infinite-information arguments.)

[28] Deutsch, D. (1985). “Quantum theory, the Church–Turing principle and the universal quantum computer.” Proceedings of the Royal Society A 400(1818), 97–117. (Why perfect simulation is a physical principle rather than merely a property of classical Turing machines, and the proposal of a universal quantum simulator.)

[29] Lloyd, S. (1996). “Universal Quantum Simulators.” Science 273(5278), 1073–1078. (Local quantum systems can be simulated by a quantum computer to any specified accuracy; accuracy is not identity.)

[30] Bournez, O., Graça, D. S. & Pouly, A. (2017). “Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length.” Journal of the ACM 64(6), 38:1–38:76. (A continuous-time ordinary-differential-equation model and ordinary polynomial-time computation are equivalent at the relevant computability and complexity level; continuity alone does not imply hypercomputation.)

[31] Pour-El, M. B. & Richards, J. I. (1981). “The wave equation with computable initial data such that its unique solution is not computable.” Advances in Mathematics 39(3), 215–239. See also Weihrauch, K. & Zhong, N. (2002). “Is wave propagation computable or can wave computers beat the Turing machine?” Proceedings of the London Mathematical Society 85(2), 312–332. (A constructed continuous system in which computable initial data can yield a noncomputable later solution, and the representation-sensitive conditions under which wave propagation is computable; neither establishes physical hypercomputation in the brain.)

[32] Lorenz, E. N. (1963). “Deterministic Nonperiodic Flow.” Journal of the Atmospheric Sciences 20(2), 130–141. (The canonical demonstration that small differences in initial conditions can grow into large trajectory differences in a bounded nonlinear system.)

[33] Hammel, S. M., Yorke, J. A. & Grebogi, C. (1987). “Do numerical orbits of chaotic dynamical processes represent true orbits?” Journal of Complexity 3(2), 136–145. (Numerical error destroys agreement with one chosen chaotic orbit, while shadowing can preserve proximity to a true orbit for long periods in relevant classes of systems.)

[34] Wootters, W. K. & Zurek, W. H. (1982). “A single quantum cannot be cloned.” Nature 299, 802–803. (The original proof that linear quantum evolution forbids one fixed apparatus from perfectly copying every possible pure quantum input; copying a specified orthogonal set remains possible.)

[35] Parfit, D. (1984). Reasons and Persons. Oxford University Press. (Branching cases in which psychological continuity survives while one-to-one numerical identity does not.)

[36] Cottier, B. et al. (2025). “LLM inference prices have fallen rapidly but unequally across tasks.” Epoch AI. (Benchmark-specific fixed-performance price declines, with large variation by threshold and period.)

[37] Brynjolfsson, E., Li, D. & Raymond, L. R. (2025). “Generative AI at Work.” Quarterly Journal of Economics 140(2), 889–942. (A field study of 5,172 customer-support agents; average and skill-heterogeneous productivity effects.)

[38] Dell’Acqua, F. et al. (2026). “Navigating the Jagged Technological Frontier: Field Experimental Evidence of the Effects of AI on Knowledge Worker Productivity and Quality.” Organization Science 37(2), 403–423. (A preregistered experiment with 758 consultants showing gains inside an AI capability frontier and a correctness penalty outside it.)

[39] Autor, D. H. (2015). “Why Are There Still So Many Jobs? The History and Future of Workplace Automation.” Journal of Economic Perspectives 29(3), 3–30. (Why automating tasks can raise the value of remaining bottleneck tasks, conditional on demand and substitution.)

[40] Acemoglu, D. & Restrepo, P. (2019). “Automation and New Tasks: How Technology Displaces and Reinstates Labor.” Journal of Economic Perspectives 33(2), 3–30. (Automation’s displacement effect and the contingent creation of new human tasks.)