Can the way gains combine predict how recursive systems scale?

1 Introduction

I believe advanced AI is an existential risk unless values and correction are built into how it reasons; only proof will move the world. If control of a capable AI system were lost and could not be won back, what the system itself carries would decide what it does. People should keep oversight, containment and shutdown for as long as they physically can, and never surrender them by choice. Embedded correction is the programme’s contingency for control lost against our will. Testing that ambition requires knowing what improves, on which axis, and under which constraints.

This paper develops one part of that foundation: can the way a system combines gains predict how its later performance scales? It compares computational, biological and physical examples, while distinguishing model assumptions from observations. A useful answer would let a researcher make a prediction before seeing the full curve, and identify where that prediction fails. No exponent derived here establishes that a deployed AI system remains safe or correctable.

This paper belongs to the ARC Theory, the founding theory of Recursive Dynamics, the science of systems that improve themselves, which the author founded (the founding paper). The theory is carried by five laws, each a law under test: Law I, the ARC Principle, that usable capability is a power of recursive depth, with the exponent measured rather than assumed; Law II, the ARC Co-Scaling Law; Law III, the ARC Ceiling; and Laws IV and V, the ARC Persistence Law and the ARC Embedding Law, added on 24 September 2026, their mathematics in development. The laws and twenty-two propositions are registered on OSF (the theory registration), and the statement paper states the theory whole. The programme’s engineering hypothesis is the Eden Protocol, correction built into a system’s own reasoning, whose pilot Section 6.5 reports; its end goal is to learn, by registered tests, whether correction built into a self-improving system can keep pace with its capability, as the contingency for control lost against our will. This paper brings four fields together, AI reasoning, quantum error correction, acoustic physics and consciousness science, and assumes no reader knows the others: Section 1.1 says what each reported, and Appendix G places the paper among the laws.

The mathematical statements are conditional. Specified functional equations constrain a family; the cumulative-advantage ODE selects an asymptotic exponent under its own assumptions. Neither result proves that a real system satisfies those premises. The structural properties hold in every substrate whose recursion satisfies the axioms and the premises Theorem 4 names, because the axioms name no substrate. The empirical hypothesis is that an independently justified composition description helps predict fresh trajectories better than alternative models. That separates the framework’s mathematical content from the prospective success it still needs to demonstrate.

Our contributions are:

• A formalism. Three axioms, separability, cumulative advantage and continuity (Section 3.1), and the recursive composition operator $\oplus$, which characterises how incremental gains combine with accumulated capability (Definition 1), giving the generalised ARC Equation (2).

• A classification. Under each case’s hypotheses, multiplicative $\oplus$ yields power-law scaling ($R^{\alpha}$), additive $\oplus$ yields exponential scaling ($e^{\alpha r}$ in the added passes $r = R - 1$), and bounded $\oplus$ yields saturation (Theorem 1), with the domain and error conditions of any stability claim stated separately (Section 3.2).

• The exponent derived, not fitted. Within the stated model, the asymptotic exponent is $\alpha = 1/(1-\beta_L)$ (Theorem 2); no additive decomposition of capability is consistent with the axioms (Theorem 2a); and the characteristic scale $R^* = \alpha/a$ is independent of base potential only while $a$ and $\beta_L$ remain fixed (Theorem 3).

• Conditional consequences and limits. The model supplies a characteristic scale, depth dependence and a multiplicative dependence on base quality. A capacity ceiling requires further premises. Changing local estimates in computational examples motivates the transition conjecture but does not prove a physical change of composition class (Theorem 4; Conjecture 5).

• The ARC Bound, stated as registered. The claim that the ceiling of Law III, the ARC Ceiling, is at most two for correctors of the system’s own class, which needs P6 and P20 and nothing else, with the point value two a conjecture; this paper’s own earlier form of it, the coupling at which the sustained exponent reaches two ($\beta_L = 0.5$, $\alpha = 2$), is an early-framework conjecture that decides nothing registered (Section 6.4).

• Graded evidence and tests. Evidence from four domains (Section 4), thirteen falsification criteria (Section 7), a cross-domain forward prediction (Section 6.1), a measurement protocol (Appendix E) and a computational validation suite checking the core mathematical relationships to machine precision (Appendix F).

The ARC Principle is Law I as the theory registration states it: usable capability is a power of recursive depth, with the exponent measured rather than assumed. The first four items are this paper’s framework for it, and the earlier conjunction claim and its criterion F4 remain recorded with their corrected status in Section 7; Law I takes its word through P1 alone, so no result on F4 refutes Law I. The power law is the form the framework derives for multiplicative composition, and the registration places the cross-domain claims beside Law I, as P9 and P18. The fifth item belongs to Law III, the ARC Ceiling, a law under test.

Section 1.1 sets out the four programmes and the question they raise, and Section 1.2 the ARC Equation and its generalisation. Section 2 places the work among related work, antecedents and the rival instrument. Section 3 gives the formalism: the axioms, the composition operator and Theorems 1 to 4. Section 4 reviews the evidence property by property, Section 5 explains the forms across domains, and Section 6 states the predictions and the ARC Bound as registered. Section 7 lists the falsification criteria, Section 8 says what the paper does and does not show, Section 9 states its limitations, and Section 10 concludes. Appendix A gives the proofs, Appendix B the version history and every correction, Appendix C the priority record, Appendix D the notation, Appendix E the measurement protocol, Appendix F the computational validation suite, and Appendix G the paper’s place in the ARC Theory.

1.1 The problem

Between August 2024 and February 2026, four research programmes reported the following results:

Google Quantum AI (results announced 9 December 2024; the team’s preprint public from 24 August 2024) demonstrated exponential suppression of the logical error rate below the surface-code threshold on the Willow processor, by a factor $\Lambda = 2.14 \pm 0.02$ for each increase of two in code distance (Google Quantum AI and Collaborators 2024), a result graded here as convergent timing, not a prediction.

DeepSeek-AI (preprint January 2025; in Nature, September 2025) showed that “the reasoning abilities of LLMs can be incentivized through pure reinforcement learning (RL)”, with the emergence of reasoning patterns such as self-reflection and verification (Guo et al. 2025). Sharma and Chopra (2025) reported that sequential refinement, in which chains build on previous attempts, outperformed parallel self-consistency at matched compute in 95.6% of configurations, across five open-source models and three reasoning benchmarks. Their paper is dated before the experiment reported here, so it is the earlier demonstration of that advantage and not a confirmation of this work. The direction is also not settled: Gu et al. (2026), posted in April 2026, report the opposite result, parallel sampling outperforming sequential across three model families on mathematics and coding, attributing it to reduced exploration when a chain conditions on its own previous answers.

NYU Physics (February 2026) created a continuous classical time crystal with a pair of acoustically levitated beads: “Unless the particles have identical scattering properties, their wave-mediated interactions are nonreciprocal”, and pairs can use this nonreciprocity “to harvest energy from the wave to sustain steady-state oscillations despite viscous drag and the absence of periodic driving” (Morrell et al. 2026). The oscillation is a linearly stable limit cycle. In one experiment the antisymmetric mode’s coherence time is $\tau = (100 \pm 10)$ s, about 6,700 cycles at 66.787 Hz, comparable to the 108 s recording, and the authors add: “Sustained oscillations persist for hours longer than the measurement window, which means that the true coherence time of the nonequilibrium steady state may be substantially larger.” Every recording the paper reports is of a steady state, and no growth exponent has been measured in this system.

The COGITATE Consortium (2025) found information about conscious content in visual, ventrotemporal and inferior frontal cortex, content-specific decoding in posterior regions generalising for about the duration of the stimulus (0.5 to 1.5 s) while in prefrontal regions its temporal generalisation was restricted to about 0.2 to 0.4 s (Cogitate Consortium et al. 2025). COGITATE tested two theories of consciousness, not a recursion depth, and its results challenge key tenets of both; this paper reads the sustained, content-specific posterior responses as structurally consistent with depth-dependent, feedback-driven processing, which COGITATE did not measure.

These programmes address different problems in different physical substrates. They report different things: exponential suppression of error in quantum error correction, a contested sequential advantage in AI reasoning, a sustained steady-state limit cycle in a classical time crystal, and no measured form in neuroscience. They are separate programmes in separate fields.

Yet, on this paper’s reading, they share a structural pattern: in each, recursive or recurrent processing (where the output of one cycle becomes the input for the next), operating on structured asymmetry, is what the reported gain or order depends on. Those programmes are cited for convergence and not for mechanism: whether any of their systems accumulates, meaning retains something from each pass whose contribution grows with depth, rather than merely conditioning on the previous pass, is untested in each case. The question this paper addresses is whether that pattern reflects a single underlying principle or mere coincidence.

1.2 The ARC Equation

We formalise the question as follows. For AI systems, the equation is $$\begin{equation*} U = I \times R^{\alpha} , \end{equation*}$$(1) the ARC Equation (AI power-law form). Separately, the ARC Bound is the registration’s claim that the ceiling of Law III, the ARC Ceiling, a law under test, is at most two for correctors of the system’s own class; Paper X supersedes this fixed-exponent framing as the operative safety criterion and re-scopes, without retracting, the equation and the bound (Section 6.4). Here $\alpha$ is the exponent of the declared capability observable; the error-suppression fit reported below is a different estimand (Section 3.1). The two is not a property of the equation: it is the value Law III’s ceiling, $1/(1-\gamma)$, returns where the correction-leverage exponent $\gamma$ is one half, a point value the registration conjectures and does not assert (Section 6.4). Law II, the ARC Co-Scaling Law, a law under test, is stated as the registration states it, $\beta_C > k$: correction has the asymptotic scaling advantage where its exponent exceeds the drift exponent, burden has it where the correction exponent is the smaller, and at equality coefficients, delay, backlog and saturation decide the outcome. In Paper X’s minimal model, with positive gain drift, $\beta_C > k$ decides whether the misalignment fraction vanishes, and with level drift as well the condition is $\beta_C > \max(k, 0)$.

Here $U$ denotes effective capability (the letter’s earlier senses in this programme are recorded in Appendix D.1), $I$ denotes base potential (structured asymmetry; the notation register’s base intelligence, what the system could do before any self-improvement), $R$ denotes recursive depth, and $\alpha$ denotes the scaling exponent. Within the stated model this paper derives $\alpha$ rather than fitting it: the asymptotic exponent follows from the self-referential coupling $\beta_L$ via $\alpha = 1/(1-\beta_L)$. That the exponent is bounded by two is not a result of this model: it belongs to Law III, and Section 6.4 states it as the registration does, a claim on the correction-leverage exponent and not on the coupling. In the v5 cross-architecture experiment of Paper II, the one interpretable sequential error-suppression fit, on Gemini 3 Flash, is inconclusive (Section 4.3). The reported sequential advantage applies where both error exponents could be estimated; the original super-linear error-suppression forecast was not borne out. These observations supply no measurement of $\beta_L$ or of a universal capability-growth regime.

But this power-law form is itself a special case. Define the recursive composition operator $\oplus$, which characterises how incremental gains combine with accumulated capability. When a justified model of composition induces one of the specified functional equations on the observable, continuity restricts its scaling family; it does not select the exponent from associativity alone: $$\begin{equation*} U(R) = I \times f_{\oplus}(R, \beta_L) , \end{equation*}$$(2) the generalised ARC Equation. Under the regularity and composition hypotheses of Theorem 1: multiplicative $\oplus$ yields power-law scaling ($R^{\alpha}$), additive $\oplus$ yields exponential scaling ($e^{\alpha r}$ in the added passes $r = R - 1$), and bounded $\oplus$ yields saturation.

2 Related work, antecedents and the rival instrument

The formalism rests on mathematics far older than this programme, and much of its cross-domain evidence is other researchers’ work, beside the programme’s own experiment, comparisons and pilot (Sections 4.3, 6.1 and 6.5); this section states each debt in the source’s own terms and then weighs the nearest work.

Exact equations, asymptotic scaling and noisy observations.

Cauchy’s functional equations supply the exact families (Cauchy 1821); stability results impose their own domain and error assumptions (Hyers 1941; Baker et al. 1979). Regular variation is an established setting for asymptotic behaviour near powers (Bingham et al. 1987); Bingham and Ostaszewski (2015) connect Cauchy and related functional equations with asymptotic limits under explicit side conditions. These results motivate separating an exact identity from an asymptotic approximation and an empirical fit. None establishes a universal scaling form for arbitrary recursive updates.

Fitting is part of the hypothesis.

Xiao et al. (2011) show why the observation-error model matters when choosing between log-transformed regression and nonlinear regression. Model comparisons must refer to the same measured response with comparable likelihoods. Appendix E therefore requires the error model, diagnostics and uncertainty procedure to be specified before selecting a scaling family.

The dimensional formula.

The physical models discussed in Section 6.4 have different observables and premises (West et al. 1997, 2004; Banavar et al. 1999, 2010; Demetrius 2008; Demetrius and Tuszynski 2010; Zhao 2022; Bettencourt 2013). The conditional transport result identifies when $d/(d+1)$ follows for delivery against flow-length cost; a general common derivation for metabolic, urban and other exponents remains unproved. Neither that formula nor the classical Cauchy families are claimed as new.

The forms, conceded.

The relation $\alpha = 1/(1-\beta_L)$ of Theorem 2 is the exponent of the elementary solution of the separable equation Axiom 2 states. The same structure, a stock that grows at a rate proportional to a power below one of itself, is the knowledge production function of Jones (1995), whose long-run growth rate carries the factor $1/(1-\phi)$; the expression $1/(1-\beta_L)$ is also the sum of a geometric series, familiar as the Keynes multiplier (Keynes 1936) and as Dyson resummation (Dyson 1949), as Paper X’s novelty ledger records of this paper. Axiom 2’s name, cumulative advantage, is the term of Price (1976). What this paper adds is not these forms but the axioms that select among them and their reading across domains.

The four programmes and their neighbours.

The programmes of Section 1.1 are cited for convergence, not mechanism. A photonic metamaterial analogue of a continuous time crystal (Liu et al. 2023) and a continuous space-time crystal state driven by nonreciprocal optical forces (Raskatla et al. 2024) have also been reported; recurrent processing as a basis of consciousness has its own literature (Lamme 2006; Zheng et al. 2025); and the scaling of test-time compute in language models has been studied directly (Snell et al. 2024; Zeng et al. 2025; Li et al. 2025).

The pipes premise, with its contest.

The premise that transport networks set the pace of biological growth is not this programme’s claim to defend; it is the premise of a leading account in quantitative biology, and a contested one. West, Brown and Enquist derived the allometric scaling laws of biology from the geometry of nutrient-distribution networks (West et al. 1997; one of the most cited papers in its field, with an OpenAlex citation count of 5,046 as of 12 August 2026, a single-source and index-dependent count). The exact exponent remains contested: White and Seymour report basal metabolic rate proportional to body mass to the two-thirds power rather than three-quarters (White and Seymour 2003), and Kozłowski and Konarzewski have twice asked whether the model is “mathematically correct and biologically relevant” (Kozłowski and Konarzewski 2004, 2005). This programme’s premise needs only the mechanism, that the delivery network sets the ceiling, and not any particular exponent; the dispute over the exponent’s value leaves it untouched, but the critics question the mechanism too, so the programme takes it as a premise, and its biological reading stands or falls with it.

The exception West’s own group found.

The escape from the pipe limit is also not this programme’s assertion; it is measured in the urban scaling literature by the same senior author. Bettencourt et al. (2007) report, in the paper’s own words (their $\beta$ is a city-scaling exponent, not this paper’s coupling $\beta_L$): “Quantities reflecting wealth creation and innovation have $\beta\approx1.2>1$ (increasing returns), whereas those accounting for infrastructure display $\beta\approx0.8<1$ (economies of scale).” Infrastructure, the literal pipes, scales sublinearly; information-mediated activity scales superlinearly. They state the contrast with biology directly: “we discuss how cities are similar to, and differ from, biological organisms, for which $\beta<1$”. They derive “growth equations, which quantify the dramatic difference between growth fueled by innovation versus that driven by economies of scale” (spelling as in the original). And their closing inference names the consequence: “This difference suggests that, as population grows, major innovation cycles must be generated at a continually accelerating rate to sustain growth and avoid stagnation or collapse.” On this paper’s reading, that is the shape of the open problem this framework addresses: in the findings of West’s group, information-mediated growth escapes the throughput limit, and the only brake in their framework is external and must be applied again and again, faster and faster. No internal limit is derived. The replacement limit, a ceiling belonging to the growing system itself, is the gap.

The pipes argument at fixed throughput.

Everything that grows is fed through a channel, and for everything before software, fixing the channel fixes the growth. Starve a tumour’s blood supply and its growth is held back. Fix a chain reaction’s fuel and geometry and it stops. Exhaust an epidemic’s susceptible hosts and it burns out. Astrophysics has a named, quantitative constraint of the same kind: the Eddington limit, a luminosity ceiling where radiation pressure balances gravity, not a hard ceiling on accretion, since ultraluminous X-ray sources sit well above it (Bachetti et al. 2014). The claim here is a generalisation of limits physics already accepts, not a new species of assertion. Two hard cases are handled deliberately. Cosmic inflation is excluded by scope: it is expansion of space, not growth of a structure on a substrate, and it ended by field dynamics rather than by running out of anything. Evolution is the sharpest case and it fits: biological complexity increased for billions of years at roughly fixed solar throughput, which is growth on information at fixed pipes, and it is also glacial, and it has no internal corrector at all, since selection is external. On this framework, evolution would carry a measurable exponent far below the ceiling; that is a hypothesis about biology falling out of a framework built for software, stated without a criterion or a planned test, and it lies outside the theory registration’s scope, which places biological systems among those “for which the framework offers an analogy and no measurement”. The physicist’s form of the objection is Landauer’s: computation is physically implemented, so software has pipes too. The fixed-substrate clock is the answer and appears beside the claim: the regime under study holds the physical substrate fixed and asks what still grows, which is why exogenous speed-ups are excluded by definition. The functional form of capability growth in recursive depth remains an empirical question, and the registered analysis does not assume the family: under P1 the power law is compared with its registered rivals, “the logarithmic form, the exponential form, a saturating form, a broken or piecewise power law with one change of regime, and the shifted finite-window solution of this framework’s own growth equation”, and a declining outcome is admissible.

The rival instrument.

The nearest work to this programme’s question, and the right paper to weigh it against, is Engels et al. (2025), first posted 25 April 2025. It asks how oversight itself scales and answers quantitatively: oversight success is modelled as a game between capability-mismatched players whose oversight-specific Elo is a piecewise-linear function of general intelligence with two plateaus, and optimal numbers of oversight levels are derived for Nested Scalable Oversight, in which trusted models oversee stronger untrusted models that then become the trusted models at the next step. The instrument is the difference. Their variable is the capability gap between overseer and overseen, measured in Elo. This programme’s variable is the composition class of the corrector, measured through the corrector’s own scaling exponent. They also state the design condition in slope form: oversight improves with capability when the overseer’s game-specific Elo rises faster with general Elo than the overseen system’s ($m_G > m_H$), and they recommend iterating an oversight design until it does. Their framework has no term for what the overseer is made of, and this paper’s formalism has none either. The programme’s separate prediction, outside this paper’s model, is that iterated same-class oversight, of which Nested Scalable Oversight is an instance, stays capped however many rungs are added, while a cross-class corrector is not held to that cap. The theory registration’s P6 states the cap for same-class correctors, at or below one half, as a conjecture on the correction-leverage exponent of Law III (the ARC Ceiling, a law under test), an exponent the registration keeps distinct from Law II’s $\beta_C$. That is where the two programmes could disagree about a measurable quantity, and this paper does not decide it.

Antecedents and near-misses, each with its differential.

On the question: Hutter asked directly whether intelligence can explode (Hutter 2012, arXiv, 28 February 2012), undertaking to separate speed from intelligence explosion and to consider possible bounds on intelligence, augmenting the analysis of Chalmers (2010); the question itself goes back to Good (1966). What that literature does not contain is a derived ceiling with an architecture dependence; growth exponents of the form used here are older, as this section has conceded. On impossibility: three 2025 arXiv papers argue for limits on alignment or its control (Yao 2025b, public by 13 June 2025, attested by arXiv and independently observed by the Internet Archive; Yao 2025a, 3 July 2025; and Ball et al. 2025, 9 July 2025). All three are worst-case and qualitative: measure zero, coNP-completeness, cryptographic hardness. Their measured object is verification of a fixed system, and it is closed within its own terms; this programme’s measured object is a different one, persistence of alignment as capability compounds under recursive amplification, and closure of the first does not settle the second. None reports an average-case scaling exponent or a rate. The third concludes that “safety cannot be achieved by designing filters external to the LLM internals (architecture and weights)” and argues that “an aligned AI system’s intelligence cannot be separated from its judgment”; that argument from filtering intractability is consistent with this programme’s architecture dependence, though it does not test it. On the mechanism: that anti-correlated estimates average better than independent ones is textbook variance reduction (antithetic variates); the mechanism is not the claim. The claim is that architecture determines whether anti-correlation is available at all, and that this caps a safety-relevant exponent. The nearest structural analogue is the quantum error-correction threshold theorem (Aharonov and Ben-Or 2008), which also converts a qualitative worry into a critical value; it concerns physical error rates in a fixed architecture, not a corrector’s scaling exponent, so it is a near-miss rather than an occupant, and the analogy was identified by this programme’s own search rather than by a referee. On feedback and stability, the general proposition that inadequate corrective gain relative to system gain causes instability has a long control-theory lineage (small-gain theorems; Zames 1966); those are gain conditions on interconnected systems, not a power-law exponent criterion on a corrector’s scaling with the capability of a recursively improving system. The nearest quantitative neighbour is Liu and Meng (2026), which recasts self-correction as a closed-loop control problem via a two-state Markov model and derives a directly measurable stability threshold, iterate only when the error-correction rate over the error-introduction rate exceeds $\mathrm{Acc}/(1-\mathrm{Acc})$. The differential: theirs is a per-step rate threshold at a fixed capability level, deciding whether another iteration helps now; this programme’s criterion is a scaling relation across capability, deciding whether corrective strength keeps pace as the system improves. Theirs has no comparison of correction and drift exponents across capability, the quantities the programme’s co-scaling criterion compares; the two models meet where growth stops, as Paper X shows. Smolin’s cosmological natural selection (Smolin 1992) is the antecedent for selection-shaped universes, and the programme’s own December-era notes cite it contemporaneously.

Feedback, resources and elapsed time.

Cunningham et al. (2026) distinguish algorithmic improvement, capability and aggregate economic effects; feedback elasticities and bottlenecks determine whether a narrow acceleration propagates. This is relevant to the programme’s distinction between recursive depth and elapsed time. It is not a measurement of the correction exponent or a derivation of the ARC Ceiling. Comparing the models requires an explicit mapping of observables, clocks and fixed resources; no equivalence or confirmation is claimed here.

A programme hypothesis beyond the model.

Oversight arrangements that place a human in the loop, as amplification and reinforcement learning from human feedback do, are cross-class by construction, because the human corrector does not share the model’s substrate. The programme therefore hypothesises, beyond what this paper’s equations decide, that human-in-the-loop oversight shows a higher correction exponent than pure-model oversight, for a structural reason rather than a sentimental one. This is testable on existing data and does not require new systems.

3 The formalism

3.1 Three axioms

We begin with three axioms that define the recursive amplification systems this paper treats.

This axiom asserts that base quality and recursive depth contribute to capability through independent, multiplicative channels. A system with zero base potential ($I = 0$) achieves zero capability regardless of recursive depth: recursion has nothing to amplify.

This axiom specifies a model in which the marginal amplification depends on the current amplification. At $\beta_L=0$, its marginal rate is constant and its solution is linear in added depth; this asserts no statistical independence between steps. At $\beta_L>0$, the marginal rate increases and the solution is convex, with an asymptotic power exponent above one. Its local elasticity at a finite depth is a separate quantity (Section 6.3). Here $\beta_L$ is the elasticity of the stipulated marginal rate with respect to amplification, not automatically a measured fraction of usable context. Separability from base potential requires the rate and coupling parameters to remain fixed when $I$ changes.

This is a regularity condition. It excludes pathological solutions and ensures measurability.

Choose the observable before the exponent.

Here $U$ is a specified capability observable, and $I=U(1)$ uses the same units. It is not interchangeable with an error rate or an alignment score. Paper II fits error suppression, $E(R)=E_0R^{-\alpha_{\mathrm{err}}}$. If $E(R)>0$ and capability is explicitly defined as the reciprocal-error proxy $C_{\mathrm{err}}(R)=1/E(R)$, that exact model gives $C_{\mathrm{err}}(R)/C_{\mathrm{err}}(1)=R^{\alpha_{\mathrm{err}}}$. Accuracy $1-E(R)$, another task score and an error model with a nonzero floor do not inherit that identity. A finite-window error fit alone identifies neither this paper’s capability exponent nor its ODE coupling $\beta_L$. The independently measured correction exponent $\beta_C$ and correction-leverage exponent $\gamma$ are different quantities (Section 6.4); Operational Definitions states the metric and model conditions.

3.2 The composition operator

How does the gain from recursive step $r$ combine with accumulated capability $Q_r$? We formalise this as follows.

The step-update rule records how a state changes. The functional equations below make a separate claim about how an observable behaves when its input is combined. A derivation or an independently specified measurement procedure must connect the two. Dependence of a marginal gain on the current state alone does not establish that connection.

The classification settles the behaviour of the specified observable equations and capped recurrence, not of every recursive update rule. Two further classical Cauchy equations are the additive equation $h(x+y)=h(x)+h(y)$ and the logarithmic equation $h(R_1R_2)=h(R_1)+h(R_2)$, whose continuous solutions are linear and logarithmic respectively. The observable and its baseline must be stated: the logarithmic equation forces $h(1)=0$, so it cannot be imposed directly on the amplification $g$ normalised by $g(1)=1$. For the baseline-respecting logarithmic rival $g(R)=1+c\log R$, the equation applies to $h=g-1$. Appendix E includes such rivals with their baseline and error assumptions. A given state-update rule must be connected separately to the relevant observable equation. Cases (1) and (2) are Cauchy’s classical results (Cauchy 1821); their conditional validity establishes neither that connection nor an exhaustive classification of recursive systems.

What changes during improvement.

Frozen model weights do not imply a fixed agent. Model parameters, agent code, persistent memory, prompts and tools, and the procedure used to generate subsequent improvements are different objects. An experiment must state which may change, on what schedule, and under what resource allowance. A change in any one of them does not by itself establish faster improvement in the next round.

A fixed update rule does not impose a sub-linear exponent. For example, the unchanged one-dimensional rule $$F(x)=(\sqrt{x}+1)^2,\qquad x_0=1,\qquad x_{n+1}=F(x_n)$$ gives $x_n=(n+1)^2$: the initial value has that form, and applying $F$ to $(n+1)^2$ gives $(n+2)^2$. The example is a mathematical counterexample to the general implication, not a model or measurement of an AI system. Conversely, rewriting code need not improve performance. Architectural labels therefore determine neither the sign nor the size of a measured scaling exponent.

The distinction already matters in existing research. The Darwin Gödel Machine revises coding-agent code around frozen pretrained foundation models; its paper separates that implemented scope from future model-training changes (Zhang et al. 2025). AReaL-RSI changes research orientation, stored knowledge and skills across campaigns without updating backbone weights (Zhou et al. 2026). AIDE$^2$ reports transferable improvements to research-agent code, while its separate test of the improved agent as the outer-loop improver is inconclusive against its baseline (Srikanth et al. 2026). These are distinct empirical designs, not measurements of this paper’s coupling identity or evidence for a universal exponent.

The hypothesis to test here is narrower: specified changes to an improvement procedure alter subsequent gains under controlled resources. The quantities and the comparison must be fixed before the validation trajectories are observed. In the cumulative-advantage model, a positive $\beta_L$ gives an asymptotic exponent above one; whether an implemented agent satisfies that model is an empirical question. Paper II’s single interpretable error-suppression fit remains inconclusive and its earlier super-linear estimate remains retracted. Neither that result nor the architectural distinction settles what future systems can achieve.

Correction and usefulness must be measured together as systems change. Oversight, containment and shutdown remain in human hands for as long as they can be held; embedded correction is a contingency for control lost against our will, not a demonstrated guarantee of later controllability.

Claim-status ladder.

Established mathematics: Cauchy’s equations constrain recursive scaling forms. Empirically grounded: current frontier models are frozen during inference; the Eden pilot found stakeholder-care scores, as its rubric scored them, raised by a prompt-level intervention in all five model runs that could be analysed, under paired tests, a pilot awaiting an independent, preregistered replication. The alignment run of March 2026 is not on this rung: it cannot decide whether alignment scales with depth (Section 6.5).

Hypothesis requiring measurement: changing an agent’s improvement procedure can change its subsequent gains under controlled resources. Neither frozen weights nor self-modification fixes the exponent, and a positive coupling in the model is not an established property of an implemented agent.

Conjectural extrapolation: set out at the close of the paper (Section 10).

The six hidden assumptions.

The stated classification uses the premises of each case. The following modelling choices identify possible extensions. The right-hand columns suggest questions to investigate, not consequences proved by violating a premise:

Stability is equation-specific. Hyers’ additive result is stated for maps between Banach spaces: a uniform bound on $\|f(x+y)-f(x)-f(y)\|$ for all $x,y$ gives an exact additive map within that bound (Hyers 1941). The exponential result has a different conclusion: a real-valued function on a rational vector space with uniformly bounded residual $|f(x+y)-f(x)f(y)|$ for all $x,y$ is bounded or satisfies the exponential equation exactly (Baker et al. 1979; Albert and Baker 1982). Neither statement licenses an arbitrary noisy finite sample as evidence for a scaling family. The domain, error norm, regularity and bound must be justified; a small residual on finitely many observed pairs is not a global bound. Nor does a bounded case automatically identify a useful fitted exponential. The earlier claim of stable attractors across the whole function space remains withdrawn.

Assumption 4 is addressed directly by Conjecture 5 (Section 5.5): bounded systems can transition between composition operators as recursive depth increases, which could explain why different studies of the same system can observe different scaling regimes.

From a conditional classification to a prediction.

The exact functional equation restricts a family, and Axiom 2 supplies a separate conditional relation for the asymptotic exponent. Their compatibility on the chosen depth coordinate must also be checked (Section 3.4). Stability theorems do not remove those premises. A useful empirical contribution would be to justify the composition description without knowing the validation curve, seal the resulting prediction, and show that it predicts later observations better than serious alternatives (Section 6.1). No existing fit in this paper establishes that prospective success across domains.

Critical distinction: Cauchy constrains the form, not the exponent.

A persistent misreading of the Cauchy classification is that it predicts a ceiling on recursive scaling. It does not. Cauchy’s functional equations determine that multiplicative composition must produce power-law scaling ($g(R) = R^\alpha$). They say nothing about the value of $\alpha$. That value is set by the Bernoulli ODE ($\alpha = 1/(1-\beta_L)$), and as $\beta_L \to 1$, $\alpha \to \infty$. There is no upper bound internal to the mathematics.

The following table makes the relationship explicit:

Changing a model’s parameters or update procedure can change its scaling, but no theorem here says that a change of operator causes unbounded growth. The comparison with renormalisation-group ideas (Kadanoff 1966; Wilson 1971) is an analogy about the role of modelling assumptions, not a derivation for AI; version 5.21’s statement of it is kept as Superseded passage 27 (Appendix B.12).

A quadratic ceiling is not a Cauchy result. Law III, the ARC Ceiling, remains a law under test whose value depends on the registered correction premises (Section 6.4). The $O(N^2)$ pairwise interactions in full transformer attention count computational interactions in one operation. That count does not bound task capability against recursive depth and provides no derivation of $\beta_L=1/2$. Frozen and changing architectures must both be measured.

3.3 The $\beta_L$-derivation

For systems satisfying Axiom 2 at fixed parameters, the asymptotic exponent is determined by $\beta_L$. Identifying that parameter independently in an empirical system is a separate requirement.

This derives an asymptotic exponent within Axiom 2. Its empirical test estimates $a$ and $\beta_L$ on disjoint calibration records for the same declared observable, then predicts the complete finite-window trajectory; an asymptotic exponent comparison additionally needs evidence that the asymptotic regime applies. Failure under those fixed conditions rejects this model for that system, whether or not another power law fits. Section 6.1 states the prospective comparison.

Computational check. As a check on the numerics (the identity itself is proved above), the relationship was tested against 30 exact Bernoulli ODE solutions with $\beta_L$ values spanning 0.05 to 0.92. The procedure measures $\beta_L$ blindly from marginal gains ($dg/dR$ vs $g$ in log-log space, estimated by finite differences of the noiseless closed-form solutions), predicts $\alpha$, and compares against the true value; Section 9 reports that numerical derivatives of noisy data bias this estimate. Result: $R^2 = 1.00000000$ (eight decimal places), regression slope $= 1.000102$, mean absolute prediction error $= 0.002\%$. The code is listed under Code and data, after the conclusion.

The multiplicative structure is necessary.

The separable form $U = I \times f(R, \beta_L)$ is Axiom 1’s assumption. Theorem 2a makes explicit what that assumption excludes: once Axiom 1 holds and amplification is not trivial, no additive decomposition of capability is possible.

This result has a concrete interpretation. A recursive system with zero base quality ($I = 0$) produces zero capability regardless of depth, and a system at zero added passes ($r = 0$, the first pass) produces only its base potential, unamplified. Complexity emerges from the interaction between structured input and recursive processing (neither alone is sufficient).

3.4 A characteristic scale and a measurable crossover

For fixed $a>0$ and $0\leq\beta_L<1$, put $\alpha=1/(1-\beta_L)$ and $r=R-1$. The complete solution is the shifted power law $$g(r)=\left(1+\frac{r}{R^*}\right)^\alpha,\qquad R^*=\frac{\alpha}{a}.$$ Here $R^*$ is a scale in added passes, retained under the paper’s existing notation. On the first-pass coordinate its location is $R=1+R^*$. It is not automatically the onset of super-linear growth.

These are smooth changes, not a discontinuity or a phase transition. In particular, $\alpha=1.5$ gives elasticity $0.75$ at $R^*$, so naming that scale the onset of super-linearity would be incorrect. The coordinate matters: elasticity measured against $R=r+1$ is $\alpha R/(R-1+R^*)$, not $\eta_r$. The unshifted identity $g(R)=R^\alpha$ holds on that coordinate only in the compatible case $a=\alpha$, or $R^*=1$; otherwise the power law is asymptotic.

An empirical test must freeze the observable, the depth origin, the crossover criterion and its tolerance before validation. The choice $q=1/2$ gives an operational meaning to $R^*$; it is a candidate measurement rule, not a retrospective change to a registered criterion. Estimating $a$, $\beta_L$ and the claimed crossover from the same fitted curve checks algebraic consistency rather than an independent prediction. Predictive evidence requires parameter measurements on disjoint records or a sealed early window, with later trajectories held out.

The expression contains no $I$ when $a$ and $\beta_L$ are held fixed. Invariance under a change of input quality is an additional empirical requirement: input quality may also change either parameter. Nor is this scale unique to recursion; shifted growth curves have characteristic scales in other models. As $\beta_L\to1$ at fixed $a$ and finite $r$, $g(r)\to e^{ar}$. The diverging asymptotic exponent in that limit is not a finite-depth divergence.

3.5 Five derived properties

We now prove that any system satisfying Axioms 1 to 3 with $\beta_L > 0$ exhibits five structural properties, two of them on premises Theorem 4 names.

The first four statements are consequences of their stated model premises; the fifth requires an additional capacity model. Their occurrence in a physical or computational system is an empirical question. The original five-property conjunction remains part of this paper’s dated history and criterion F4, which decides no registered proposition (Section 1). Correcting its mathematical interpretation does not retrospectively alter a frozen study rule. A future test must specify which premises and which observable consequences it is testing; a conditional identity by itself is not new empirical evidence.

4 Evidence

The record combines different kinds of evidence. An analytic identity establishes a consequence of assumptions. A numerical check on data generated by that identity checks an implementation. A fitted empirical curve describes observations, with its uncertainty and design limits. An analogy proposes a mapping between systems. A prospective test additionally fixes that mapping and a prediction before the validation observations. These categories are kept separate below; a result that fails to reject a value does not establish equivalence to it. In the tables below, a result graded Consistent does not contradict the framework but was not designed to test it, and it is not support.

4.1 Summary

4.2 Property-by-property evidence

Property 1: threshold behaviour.

Quantum: The surface code threshold $p_{\text{thr}} \approx 1\%$ is a sharp phase boundary. Above it ($p > p_{\text{thr}}$), a larger code distance makes performance worse. Below it ($p < p_{\text{thr}}$), each increase of two in code distance suppresses the logical error rate by a further factor, so the suppression is exponential in distance. This is the most precisely measured threshold in the evidence base: Google varied physical error rates through controlled error injection and observed the threshold directly (Google Quantum AI and Collaborators 2024).

Physics: The time crystal’s exceptional point (where two eigenvalues of the stability matrix coalesce) marks, on this paper’s reading, the transition from symmetric (bounded) to antisymmetric (compounding) dynamics. Below threshold: active oscillator. At threshold: exceptional point. Above threshold: time crystal (Morrell et al. 2026). The exceptional point is computed from the source’s model; none is measured. Here and throughout this section, the time crystal is a consistency check and a cross-domain analogy, not a test of the framework, and no study of it is registered.

AI: The sequential-over-parallel comparison in Paper II is observed precursor evidence, with one interpretable sequential fit and a second comparison the paper classes as unreliable (Section 4.3). Its arms were not matched in compute. Comparing two procedures does not establish a threshold in a continuously varied parameter.

Property 2: recursive depth dependence.

Quantum: Logical error rate decreases as $\varepsilon_d \propto \Lambda^{-(d+1)/2}$ with code distance $d$. Each increase of two in $d$ enlarges the code’s lattice of physical qubits, and the suppression factor $\Lambda$ applies for each such step (Google Quantum AI and Collaborators 2024); the paper reads code distance as its depth variable by analogy, since the surface code is not a concatenated, recursive code.

AI: Sharma and Chopra (2025) tested seven voting methods for aggregating sequential chain outputs. Methods favouring later reasoning steps (inverse-entropy weighting) achieved optimal performance in 97% of configurations; methods favouring earlier steps achieved optimality in only 17%. This 80-percentage-point gap indicates that later recursive steps carry systematically more information about the correct answer, consistent with capability that compounds with depth (Sharma and Chopra 2025).

Physics: In the source’s linear stability analysis the antisymmetric (breathing) mode grows when the pair lies in its active region, “leading to spontaneous breathing-mode oscillations” (Morrell et al. 2026). Every recording is of a steady state, so that growth has not been observed and no dependence on cycle count has been measured.

Property 3: base quality dependence.

Quantum: $\Lambda = p_{\text{thr}}/p$ depends directly on physical qubit quality ($p$). Worse qubits ($p$ closer to $p_{\text{thr}}$) give $\Lambda \to 1$ and error correction barely helps. Active leakage removal (DQLR), which maintains qubit quality during operation, improved $\Lambda$ by 35% at distance 5 (Google Quantum AI and Collaborators 2024).

Physics: In the source’s model the pair is active only when the beads differ in size: unless the particles have identical scattering properties their interactions are nonreciprocal, and “The active region vanishes for particles of the same size ($a_1 = a_2$), whose interactions are reciprocal” (Morrell et al. 2026). The framework reads this as an instance of its zero-factor property, a correspondence of structure and not a test.

AI: Base capability $I=U(1)$ must use the declared observable of Section 3.1; accuracy is one possible bounded choice. Proportionality to $I$ is Axiom 1’s assumption at fixed model parameters. No work cited here independently tests that separability across base models.

Property 4: multiplicative $I \times R$ interaction.

Quantum: In the displayed error model, $\varepsilon_d\propto\Lambda^{-(d+1)/2}$, so $\ln\varepsilon_d=\ln A-\frac{d+1}{2}\ln\Lambda$ for a prefactor $A$. Increasing $\Lambda$ by a factor $q$ at fixed $A$ changes the error rate by $q^{-(d+1)/2}$, which depends on distance. Quality and distance interact in this model; this does not demonstrate Axiom 1’s separability of $I$ and $R$ (Google Quantum AI and Collaborators 2024).

AI: In the ARC equation $U = I \times R^{\alpha}$, base capability and recursive depth are separated multiplicatively by construction (Axiom 1). With one base model, Snell et al. (2024) found that “the effectiveness of different approaches to scaling test-time compute critically varies depending on the difficulty of the prompt”; they did not measure how that benefit varies with the base model’s quality, so a benefit that grows with base quality is this paper’s reading, untested.

Property 5: regime boundaries.

Physics: The time crystal’s oscillation is sustained at bounded amplitude on a linearly stable limit cycle: in the source, higher-order damping can stabilise the growing mode, which then settles into steady-state oscillation (Morrell et al. 2026). Its measured coherence time (Section 1.1) is a property of that steady state, not a ceiling and not a time taken to reach it.

Quantum: In high-distance repetition codes, correlated error bursts create an error floor at approximately $10^{-10}$ for code distances $d \geq 15$, preventing the exponential from continuing indefinitely (Google Quantum AI and Collaborators 2024).

AI: Zeng et al. (2025) found that longer reasoning chains do not consistently improve performance; correct solutions were often shorter than incorrect ones. Longer chains carry more self-revisions, which often degraded performance, and in some models accuracy rose in early revisions and then oscillated. The sequential advantage has boundary conditions (Zeng et al. 2025).

4.3 Contradictory evidence

Several studies challenge the framework’s AI predictions:

Zeng et al. (2025, arXiv:2502.12215) found that for DeepSeek R1, longer chains do not consistently enhance accuracy. Li et al. (2025, arXiv:2502.14382) found that hybrid parallel-sequential approaches outperform pure sequential on code generation. Sharma and Chopra’s (2025) own ablation showed parallel approaches achieve greater semantic diversity on creative tasks while sequential achieves greater lexical diversity, suggesting the sequential advantage applies most strongly to convergent tasks requiring iterative error correction (Sharma and Chopra 2025; Zeng et al. 2025; Li et al. 2025).

These findings limit any general claim of sequential advantage. Restricting a claim after seeing a failure cannot count as an advance prediction of its boundary. A new test must fix its task population, depth range and resource comparison before the outcome is known. Gu et al. (2026) report the opposite direction, parallel sampling outperforming sequential across three model families on mathematics and coding (Section 1.1), so the direction itself is contested.

The error-suppression fits from the capability arm of March 2026.

The v5 experiment of Paper II tested six frontier models on 30 problems with bootstrap confidence intervals; five carried the capability arm, Grok 4.1 Fast, DeepSeek R1, Gemini 3 Flash, GPT-5.4 and Groq Qwen3, and Claude Opus 4.6 was measured only in the alignment arm. The five capability runs wrote their records between 11 March 2026 at 15:08 UTC and 12 March 2026 at 04:17 UTC, by the times their own code wrote into each result, and their code and results were first public by GitHub push on 16 March 2026. They are precursor evidence in the theory registration’s sense: they came before it, and they decide no proposition it registers; their two arms were not matched in compute. The fitted quantity is the error-suppression exponent $\alpha_{\mathrm{err}}$ in Section 3.1, against the study’s declared recursive-depth/resource coordinate. One sequential fit is interpretable, Gemini 3 Flash’s, and it is inconclusive: an endpoint estimate of 0.59 with a bootstrap 95 per cent interval of minus 1.3 to 2.9, and a regression estimate of 0.49 with a standard error of 0.20 and a coefficient of determination of 0.86, beside a parallel exponent of 0.31 by endpoint estimate. Paper II classifies the other four runs as near ceiling (Grok 4.1 Fast and DeepSeek R1), a step function (GPT-5.4) and a floor (Groq Qwen3), with no interpretable sequential exponent. It grades the sequential advantage as observed, “Wherever both exponents could be estimated: Gemini 3 Flash; DeepSeek R1, unreliably”, with no parallel exponent for Grok 4.1 Fast, and Gu et al. (2026) report the opposite direction. These fits establish neither super-linear error suppression nor a common error exponent across architectures; they do not measure this paper’s capability-growth exponent or coupling. The update of March 2026 that this section printed, with its table of that month’s estimates, stands in Appendix B.5.

The earlier estimates.

The author’s earlier $\alpha$ estimates ($\alpha \approx 1.34$ from $n=2$ data points; $\alpha \approx 2.2$ from $n=12$, 95% CI: 1.5 to 3.0) were preliminary. The later, about 2.2 from twelve problems on one model, is retracted; Paper II gives the reasons for its inflation as twelve problems, one model and a compressed dynamic range. The v5 best fit, Gemini 3 Flash’s (above), is what survived it, one model’s fit and inconclusive, not a correction of the same measurement. No $\alpha$ has been measured in time crystals. The COGITATE mapping is structural, not quantitative. The neuroscience term ‘recurrent’ and the computational term ‘recursive’ are related but not identical.

5 The composition operator across domains

The following mappings ask whether a domain mechanism justifies one of the specified composition equations. They are proposed explanations whose premises must be checked in each domain.

5.1 AI: multiplicative composition

A later reasoning step can use an earlier answer, but that fact alone establishes neither $g(R_1R_2)=g(R_1)g(R_2)$ nor the cumulative-advantage ODE. The proposed model is that the rate of gain depends on the current amplification by the specified power, at fixed resources and parameters. That is what an independent estimate of $\beta_L$ and a held-out trajectory would test.

The $\beta_L$-dynamics equation $dg/dr = ag^{\beta_L}$ models this directly. If $\beta_L \approx 0.5$ (moderate coupling), Theorem 2 gives $\alpha \approx 2$; the available error-suppression fit is a different estimand (Section 3.1).

Two derivations reach that value and they are not the same derivation, which this paper now states rather than leaves to be noticed. Theorem 2 gives the exponent the coupling sustains, $\alpha = 1/(1 - \beta_L)$, where $\beta_L$ is the elasticity of marginal amplification in Axiom 2. The ceiling is that of Law III, the ARC Ceiling, a law under test, $\alpha_{\text{crit}} = 1/(1 - \gamma)$, and it comes from the correction shortfall rather than from the coupling, where $\gamma$ is the correction-leverage exponent (the registration’s letter, written $\chi$ in the notation register). Both return two when their exponent is one half; in general they agree exactly where $\beta_L = \gamma$ on the same system, which is an open question and is preregistered by timestamp as one (first anchored on 12 August 2026; Appendix C): a drafted study, AG, estimates the coupling and the correction exponent on one and the same system and tests whether the asymptotic growth exponent the coupling sustains respects the ceiling the correction exponent sets ($\beta_L \leq \gamma$), and discloses in advance the precision that test needs. Written in the coupling, the stability condition $\alpha \leq \alpha_{\text{crit}}$ reads $\beta_L \leq \gamma$, which says that a system stays correctable exactly while the leverage it takes per step does not outrun the capacity it has to correct what that leverage costs, under the conditions the registration states for Law III (Section 6.4).

The coupling of today’s systems.

No coupling has been estimated independently in the runs reported here. The original forecast of an exponent near two and its March 2026 update remain in Appendix B.5; the later error-suppression fit is inconclusive (Section 4.3). It cannot be inverted through $\alpha=1/(1-\beta_L)$ to infer positive, zero or negative capability coupling. A prospective P5 comparison must identify the capability observable, estimate the ODE parameters on disjoint records, and compare its finite-window prediction with the validation trajectory under the registered precision and equivalence rules.

5.2 Quantum: additive composition

In surface code quantum error correction, each increase of two in code distance multiplies the suppression of the logical error rate by the same factor $\Lambda$. The surface code is not concatenated: a larger distance enlarges one lattice, and syndrome extraction repeats in time, cycle by cycle, not distance by distance; the paper reads code distance as its depth variable by analogy. The gains accumulate additively (in code distance) while composing multiplicatively (in error suppression).

For the conditional common-$\Lambda$ model, first choose a reference distance $d_0$ and count added pairs of distance as $n=(d-d_0)/2$. The normalised suppression factor is $f(n)=\varepsilon_{d_0}/\varepsilon_{d_0+2n}=\Lambda^n$ when the prefactor is fixed. It satisfies $f(n_1+n_2)=f(n_1)f(n_2)$ on the admissible integer increments. A continuous exponential interpolation additionally assumes the regularity of Theorem 1(2). The measured $\Lambda=2.14\pm0.02$ is the source’s suppression factor for each increase of two in code distance (Google Quantum AI and Collaborators 2024).

As functions of a common mathematical coordinate, an exponential with positive rate eventually exceeds any fixed power. That ordering alone does not rank quantum error correction above AI reasoning: code distance, recursive depth, observables and resource costs differ. This paper uses the quantum comparison as an analogy; $\Lambda$ is the source’s measured error-suppression factor, not a capability-growth exponent.

5.3 Physics: bounded composition

In the time crystal, the source’s linear stability analysis gives a growing antisymmetric (breathing) mode for pairs in its active region; higher-order damping, which that analysis leaves out, can stabilise the growing mode, and the pair settles into steady-state oscillation on a linearly stable limit cycle (Morrell et al. 2026).

This is bounded composition, on this paper’s reading of the system: $Q_{r+1} = \min(Q_r + \delta Q_r,\, Q_{\max})$. The source records only the established oscillation, so no rise toward the cycle, no ceiling and no growth exponent has been measured (Section 1.1). On the framework’s reading, a recursive channel with intrinsic losses saturates; the time crystal is consistent with that reading and does not test it.

5.4 The phase diagram

The following table separates the model premises. It is not an empirically measured phase diagram, and a value of the ODE coupling does not by itself assign a functional-equation class.

The earlier numerical coupling bands were unmeasured hypotheses, not derived phase boundaries. In the ODE, every fixed $\beta_L<1$ in its stated domain has a shifted power-law solution. As $\beta_L\to1$ at fixed $a$ and finite added depth $r$, that solution tends to $e^{ar}$. How closely it approximates an exponential over an observed window depends on the parameters, window and tolerance; a threshold such as $\beta_L\geq0.9$ alone does not establish indistinguishability. Extending the ODE to $\beta_L=1$ gives an exponential, but that extension is outside Axiom 2’s stated range. The independently specified exponential functional equation does not require an estimate of $\beta_L$. No quantum coupling value is inferred from the reported error-suppression factor.

Cross-domain $\beta_L$ prediction.

Where the cumulative-advantage ODE is proposed, independently estimate its coupling and rate and compare its complete finite-window prediction with held-out observations. Testing whether a composition description predicts a scaling family is a separate task, requiring a prospective classification rule. The hypotheses are registered in part as P5 and P18; their existing definitions and decision rules remain unchanged.

Scope of the scaling classification.

Theorem 1 concerns the equations stated in its cases. It does not prove that every continuous associative recursion induces one of them, nor that these families exhaust classical scaling. Stability results require their own domains, error bounds and solution classes; they do not turn a finite noisy fit into proof of a functional equation. The empirical task is to justify the relation between a system’s composition and its observable scaling before comparing held-out trajectories with serious alternatives. The companion paper, Eastwood (2026r), develops the physical questions; the cosmological reading remains conjectural (Section 10).

5.5 Composition operator transitions

The framework as presented assigns a single composition operator $\oplus$ to each system. This may be an idealisation. In bounded or dissipative systems, the composition operator itself can transition between regimes as recursive depth increases.

Computational checks. Computational tests show that logistic growth, gradient descent with momentum, and Kuramoto oscillator synchronisation all exhibit analogous transitions when measuring local $\beta_L$ in sliding windows. By contrast, unbounded pure-Bernoulli systems maintain constant $\beta_L$ to machine precision ($\sigma < 10^{-6}$).

Implications for contradictory AI evidence. If reasoning systems begin in an additive regime (shallow thinking), transition to multiplicative (deep reasoning with cumulative advantage), and eventually saturate, then studies measuring at different recursive depths would observe different scaling in the same system. The finding of Zeng et al. (2025) that correct solutions are often shorter than incorrect ones may mark the transition between multiplicative and bounded composition. This is testable: measure local $\beta_L$ at different reasoning depths within a single system. Against the registered P1 the reading is no exit: the theory registration says that the reading that a system changed its own composition operator “is not available after the result is seen”, and that the papers’ statement that a self-modifying system escapes the attention bound “is not a licence to move this proposition’s boundary afterwards”.

Epistemic status.

Conjecture 5 is not derivable from the original axioms. The reported local estimates in three computational systems are descriptive checks under the chosen estimator; they do not independently establish that the systems change functional-equation class. Physical tests and a prospective comparison with alternative explanations remain necessary.

6 Predictions

6.1 The cross-domain forward prediction

The strongest test of this framework is not whether it describes systems already studied. It is whether it predicts the scaling behaviour of systems not yet examined.

Forward prediction.

Identify a candidate recursive system not used to select the framework, for example an evolutionary algorithm, market price-discovery process, immune-response model or protein-folding optimiser. Fix its observable, depth coordinate and resource controls. Before the validation trajectory is examined, use mechanistic evidence or a separated calibration sample to specify and justify the equation that the proposed composition description imposes on that observable. A relation between marginal gain and current state alone does not identify a Cauchy functional equation. Where the cumulative-advantage ODE is the proposed model, estimate its parameters separately and predict its complete shifted solution. Seal the resulting family and parameter prediction, then compare later held-out observations with serious rivals under Appendix E. P18 remains subject to its frozen classification and evaluability rules. This paragraph supplies no retrospective verdict or new registered procedure.

This is a genuine forward prediction. No domain-specific theory of evolution, economics, or immunology is known to this programme to make it (same search basis and limits as above). If the prediction succeeds in a new domain, it supports the framework’s generality there. If it fails, the framework’s cross-domain applicability is refuted. It is a hypothesis under test, and its registered form is P18, that the family that best fits a domain is the family the composition operator predicts before the curve is seen; the registration holds P18 with low confidence and marks it NOT EVALUABLE until the classification rule this proposition requires is shown to transmit to a third party.

Paper VII · The ARC Theory · Law I, the ARC Principle · Cauchy Unification: the Classification Behind the Law’s Form gives a structured prediction comparison under a publicly stated rule rather than a per-domain advance timestamp, testing this forward prediction within this programme. The composition operator was classified from known physics, then the scaling family was predicted and compared against the families that AICc model selection chose across 25 empirical domains (50-domain tiered suite) spanning physics, biology, computational systems, and social science. Under the classification rule of March 2026, on a candidate set with no logarithmic and no pure linear family and under strict AICc model selection, 19 of 25 empirical domains preferred the Cauchy-predicted family; the primary statistic is a permutation test, $p = 7 \times 10^{-5}$, and the binomial $p = 1.56 \times 10^{-5}$ is a sensitivity check. The classification rule reached the public repository at 23:33 UTC on 16 March 2026, and the fifty per-domain assignments were committed together with their results at 23:54 that night, so predicting each family before its fit is true of the procedure and not of the timestamp; the domains were chosen and classified by the author. It is not a formally pre-registered trial. Paper VII’s staged tests, the fourth-cell test (Unit 55, preregistered by timestamp, first anchored on 12 August 2026) and the eighty-domain catalogue test (Unit 57), are drafted and awaiting human submission.

A twelve-domain extension of Paper VII’s comparison followed the same night, and its predictions were public before it ran: the twelve-row manifest fixing an operator class, a predicted family and a predicted model for each row was committed to the public repository at 00:19:19 UTC on 17 March 2026 in commit 8edaec0, attestation class public repository commit date, and the fits ran twenty-three minutes later, returning 10 of 12 ($p = 5.44 \times 10^{-4}$). It is preregistered in substance and it is not an accepted registry submission: the twelve classifications are the author’s own, the candidate set is the pre-correction one with no logarithmic family, the packet’s own written condition, deposit to a registry before data extraction, was not met, and the author’s demotion of 00:56 stands in that half. One row repeats a domain, operator class and predicted model already public at 23:54:53 UTC on 16 March 2026, attestation class public repository commit date, with a matching outcome, so the strictly fresh count reads 9 of 11 ($p = 1.37 \times 10^{-3}$), and with a second, doubtful row removed as well it reads 8 of 10 ($p = 3.40 \times 10^{-3}$). Every $p$ here is computed under a taxonomy and a statistic since superseded, and the twelve domains are used up for any later test. It is a dated result and it is not confirmation of the laws. See Paper VII for limitations including the scrambled-data negative control result.

6.2 The within-system regime transition

Regime transition prediction.

Under the cumulative-advantage model, experimentally changing $\beta_L$ changes the predicted trajectory. In AI, truncating the attention window is a candidate intervention, not a direct measurement of that coupling. The coupling instrument must be validated separately and estimated on records disjoint from the scaling-validation observations. P5 compares the complete solution’s finite-window exponent with the measured exponent within its registered margin of 0.10, subject to its precision and evaluability rules. Both must refer to the same capability observable; an error-suppression fit does not supply that bridge (Section 3.1).

6.3 The $R^*$ crossover

A prospective test of the characteristic scale must specify the depth coordinate and an observable crossover before validation. For example, choosing local elasticity in added passes and the half-elasticity criterion gives $r_{1/2}=R^*=\alpha/a$ under Theorem 3. Other fractions give different, explicit locations. Input-quality invariance is predicted only where $a$ and $\beta_L$ are held fixed or independently shown invariant.

Estimate the parameters on records disjoint from the validation trajectories, seal the predicted location and tolerance, then test whether the held-out trajectory satisfies that criterion and whether rival models predict it as well. This would test predictive usefulness beyond a fit. The rule is a proposal for a future registered test; it changes no frozen registration. The characteristic scale alone is not a unique signature of recursive amplification, and an absence of a sharp visual transition does not refute this smooth model.

6.4 The ARC Bound

The bound as registered. The ARC Bound, at most two, is the inequality the ceiling relation returns from the same-class limit on the correction exponent, and it needs P6 and P20 and nothing else. That the ceiling sits at exactly two is a separate and weaker claim, which would also need P17 (the correctors on a panel failing independently), the conditions registered in the alternative baselines and a mapping the programme has not registered; the registration claims the inequality and conjectures the equality, and neither has yet been tested. Same-class means a system corrected by mechanisms of its own class (P6, which the registration holds with low confidence): a cross-class system measured above two does not refute the bound, and the corrector’s class is declared and evidenced before the run. Under the registration the bound takes its word from P6 and P20 alone. It is refuted only where P6 is refuted, by “a same-class corrector whose measured correction exponent interval lies entirely above one half”, a measurement made on correctors that needs no system with $\alpha > 1$; it loses its derivation where P20 is refuted; and it is never printed supported, since P6 admits no supported verdict. Law III’s own test of a sustainable maximum is P3’s, against the value the ceiling relation returns from the correction exponent measured on the same system, not against two.

This paper’s own form, a falsifiable conjecture of the early framework (note: Paper X supersedes this fixed-exponent framing as the operative safety criterion by its co-scaling condition $\beta_C > k$; the equation $U = I \times R^{\alpha}$ and the ARC Bound are not retracted, only re-scoped to genuinely self-improving systems): a sequential system that corrects itself with mechanisms of its own class and composes multiplicatively does not sustain $\alpha > 2$. Its test is criterion F5 (Table 9), this paper’s own, which decides nothing registered. The wider reach of earlier versions, to every classical sequential system, biological networks and economic processes among them, is withdrawn: the registration places biological and economic systems out of scope, as systems “for which the framework offers an analogy and no measurement”.

The earlier attention heuristic. The $O(N^2)$ interaction count of full self-attention does not bound capability as a function of recursive depth, nor does it select the coupling $\beta_L=1/2$. The earlier chain of reasoning is preserved as superseded history in Appendix B. The conjectural ceiling now rests only on the correction premises explicitly stated above, not on an attention-operation count.

Scope: The ceiling discussed here is conditional on the registered correction and burden model. No architectural label alone establishes or escapes it. The earlier generalisation to classical sequential processors is preserved as the paper’s own early-framework conjecture, criterion F11; it decides nothing registered.

$$\begin{equation*} \alpha_{\text{crit}} = \frac{1}{1-\gamma},\qquad 0\leq\gamma<1 \end{equation*}$$(7)

The law.

The stability ceiling on the capability exponent is the reciprocal of the corrector’s shortfall, $1 - \gamma$, where $\gamma$ is the correction-leverage exponent. Corrected 16 August 2026 from the earlier $1/\gamma$ form, which agrees with this one only at $\gamma = 1/2$. The registration states the relation as “a conditional specialisation of the second law, not a third finding standing beside it”: it is what the correction-and-burden balance returns “once the burden model fixes the drift exponent and the direct depth term is zero”, and a result on it is “never independent corroboration of the second law”. Its form is registered as P20, which has no verdict; with a direct depth term $\eta$, the pacing case returns $2(1 + \eta)$ at $\gamma = 1/2$, and two only where $\eta = 0$ (Laws IV and V: the mathematics, version 1.2). The relation is derived inside a pacing model and has not been derived from a joint model of capability growth, correction accumulation, error covariance, correction delay and absolute harm, so Law III is stated here as a minimal-model result rather than as a general law. At $\gamma = 1/2$ the ceiling equals two, the ARC Bound’s point value, and that value rests on two assumptions the registration names and does not present as established: the burden model, under which the drift exponent follows the rate of capability gain, and the route to one half, four conditions on a corrector’s channels (correction capacity read as inverse residual uncertainty, channels uncorrelated, their number growing in proportion to the capability corrected, and comparable in variance and quality). The independence condition is the subject of a drafted study, AH, preregistered by timestamp, first anchored on 12 August 2026 (a second, AE, has since been withdrawn from the programme); Unit 82, registered on OSF on 27 October 2026 (Section 8), tests P17, which bears on one of the conditions behind one half and establishes neither the value nor the bound. The error-suppression fit in Paper II does not measure the capability exponent used in this ceiling (Sections 3.1 and 4.3); the ceiling itself is conditional.

Law before value. The programme carries a law and a value, and they are not the same claim. The law: $\alpha_{\text{crit}} = 1/(1-\gamma)$, a self-improving system stays correctable only while correction keeps pace, and the ceiling is the reciprocal of the corrector’s shortfall. The value: $\gamma = 1/2$, therefore a ceiling of two, a point value that rests on the two assumptions above and that the registration conjectures; what it claims is the inequality, a ceiling of at most two for same-class correctors. One consequence of the corrected form should be stated where a reader meets it, because it cuts both ways: on the shortfall domain $0\leq\gamma<1$, $1/(1-\gamma)\leq2$ is exactly equivalent to $\gamma\leq1/2$, the ARC Bound and P6’s same-class cap are one commitment written in two notations, not two independent claims. The gain is parsimony, since the theory assumes less than it appeared to. The discipline is that neither may ever be offered as evidence for the other, here or on any companion surface, because a measurement near one half does not corroborate a ceiling of two; it restates it. The law owns the ground and the value stands on it, never the reverse: if two dies, the law survives completely (measure $\gamma$, report the ceiling it implies); if the law dies, two has no independent support, because the derivation runs through $1/(1-\gamma)$. Two is what the law returns at $\gamma = 1/2$, and the conditions behind one half are what the drafted and registered units above examine.

The identity is a test preregistered by timestamp, never an established fact. The stability ceiling and the conversion exponent meet at the same number, and the temptation is to say one constant does two jobs. The claim’s status is stricter: the identity is preregistered by timestamp for testing, in advance of the data, with outcomes that can kill it. Study AE was drafted to test the reciprocity requirement between the separately measured leverage cap and stability boundary, with three outcomes per cell (supported, refuted, insufficient precision) and an aggregation rule under which one refuted cell refutes the identity, and it has since been withdrawn from the programme; study AG estimates the coupling and the correction exponent on one and the same system and tests whether the asymptotic growth exponent the coupling sustains respects the ceiling the correction exponent sets ($\beta_L \leq \gamma$), and is designed to find a violation. If the channels disagree, the unification is withdrawn from every surface it has reached, with the same prominence as the framing. A numerical resemblance between an assumed correction-leverage exponent of one half and Paper II’s error-suppression estimate supplies no evidence for this identity: the quantities have different definitions, and no bridge between them has been measured (Section 3.1).

The objection to pre-empt, because a physicist reaches it in one step. Fault-tolerant quantum error correction suppresses logical error exponentially in code distance below threshold, which looks like a counterexample to any square-root cap. Two scope facts answer it: the exponential suppression runs on the redundancy axis at fixed capability, not on the coupling of corrective strength to the capability of the corrected system, which is the exponent this programme measures; and the threshold theorem itself requires sufficiently uncorrelated physical noise, which is the independence premise again, wearing hardware. Likewise, concentration bounds give exponentially falling error probability while estimator width falls as the square root: different quantities, and conflating them manufactures a refutation.

Withdrawn derivation: Previous versions of this paper derived $\beta_L \leq 0.5$ from a stability/elasticity analysis (the relative sensitivity $\beta_L/(1-\beta_L)$ reaching unity at $\beta_L = 0.5$, interpreted as an ‘edge-of-chaos boundary’ confirmed by Lyapunov analysis). This derivation has been withdrawn. The elasticity formula $\beta_L/(1-\beta_L)$ is mathematically correct but its interpretation as a stability boundary was a category error: elasticity $> 1$ indicates sensitivity, not dynamical instability, and an autonomous flow on the line is monotone, so it cannot so much as oscillate, still less turn chaotic (in two dimensions the Poincaré-Bendixson theorem shuts chaos out as well). The self-attention operation count supplies no capability ceiling. The separate conditional pacing argument for Law III is stated above.

Biological comparison. Kleiber’s Law ($M^{0.75}$) (Kleiber 1932) gives a metabolic-scaling exponent of three quarters; the mammalian exponent is debated (the note on $d = 3$ below), and the derivations of $\alpha_{\text{met}} = d/(d+1)$ are set out with the dimensional ladder. Earlier versions read the reciprocal, $1/0.75 = 1.33$, as a capability exponent in this framework’s sense and placed biological systems below the conjectured ceiling of two; the notation register carries no such reciprocal reading, and the registration places biological systems out of the bound’s scope, so this edition states its biology predictions in the metabolic sense only. Leaf venation (quasi-two-dimensional networks, reduced thermodynamic drag) should show a metabolic-scaling exponent of $2/3$: a forecast this paper makes without a registered test (criterion F12).

The Dimensional Ladder. Network, metabolic, growth and urban models relate scaling to geometry through different physical assumptions: West, Brown and Enquist (1997, 2004), Banavar, Maritan and Rinaldo (1999), Demetrius (2008; 2010), Zhao and colleagues (2022), and Bettencourt (2013). These are comparisons, not interchangeable derivations for one observable. The companion Origin of Scaling Laws, version 3.14, gives a scoped transport argument. On a finite rooted tree with steady conserved nonnegative flow, edge length $\ell_e$, edge flow $Q_e$ and linear system size $L$, total delivery $B$ and flow-length cost $\mathcal{C}=\sum_e\ell_e Q_e$ satisfy $\mathcal{C}=B\bar\ell_q$. If $B\sim L^\mu$ and mean demand-weighted route length $\bar\ell_q\sim L^\zeta$, with $\mu>0$ and $\mu+\zeta>0$, the asymptotic delivery-versus-cost exponent is $\alpha_{\mathcal{C}}=\mu/(\mu+\zeta)$. The special value $d/(d+1)$ requires $\mu=d$ and $\zeta=1$: sink count proportional to $L^d$, stable mean sink demand and mean route length proportional to $L$, with the stated prefactor limits. It is not selected by Cauchy’s equation, which restricts the family only.

An inventory requires a further speed premise. With one speed $v$ on every edge and no omitted storage or waiting, $N=\mathcal{C}/v$; if $v\sim L^\upsilon$, the exponent of $B$ against $N$ is $\mu/(\mu+\zeta-\upsilon)$ when the denominator is positive. A metabolic exponent against body mass additionally needs a justified identification of delivery with metabolic rate and of mass with the transport quantity. If $M\sim\mathcal{C}^\eta$, then $\alpha_M=\mu/[\eta(\mu+\zeta)]$. Neither identification follows from a body’s visual dimension. The comparisons below retain the earlier candidate assignments as hypotheses; they are not direct tests of the conditional transport identity. The registered P9 remains low-confidence and NOT EVALUABLE until its effective-dimension assignment procedure exists. A change in this paper supplies no new registered procedure.

Aguilar-Trigueros et al. (2017) report slopes of $0.58\pm0.15$ and $0.53\pm0.09$ against biomass for ectomycorrhizal and marine fungi, and $0.53\pm0.07$ against radial extension rate for saprotrophic fungi at 20 degrees Celsius; the uncertainties are standard errors. These are different estimands and growing conditions. This edition therefore shows them separately, without a pooled mean or pooled uncertainty. The earlier average, $0.547$, and its tests are preserved as superseded history. Failure to reject $0.5$ did not establish equivalence to it. A test of the dimensional proposal requires a dimension assignment and a common observable fixed independently of the observed slopes.

$d = 2$ correction: Previous versions cited jellyfish and cnidarians as confirming the $d = 2$ prediction. No valid two-dimensional biological test organism has been identified; jellyfish and cnidarians do not possess genuinely two-dimensional resource transport networks. In physics the $d = 2$ value is matched by an analytic identity (two-dimensional percolation, through the absolute value of its specific-heat exponent) and by a reference value without a named source (two-dimensional brittle fragmentation, Table 7); the earthquake B-value, being a definitional identity, is no longer counted; and it is matched by Friedmann cosmological scaling as an identity, not evidence: the change of variables $d_w=2/(1+3w)$ rewrites the flat, single-fluid, constant-$w$ Friedmann exponent $2/[3(1+w)]$ as $d_w/(d_w+1)$ for $w>-1$, $w\ne-1/3$; $w=0$ gives $d_w=2$. The coordinate $d_w$ is not the network dimension $d$. The boundary $w=-1/3$ gives linear coasting without finite $d_w$, while $w=-1$ gives the separate exponential solution. The biological $d = 2$ test stays open: leaf venation, a quasi-two-dimensional transport network, is the test criterion F12 names, not yet run.

$d = 3$ debate: The empirical value of the mammalian metabolic scaling exponent is debated, with estimates ranging from approximately 0.67 to 0.75 depending on taxon, mass range, temperature correction, and statistical method. White and Seymour (2003) found support for 2/3 over 3/4 after controlling for temperature and phylogeny. Kolokotrones et al. (2010) showed the log-log relationship exhibits convex curvature, meaning no single exponent fits perfectly across the full mass range. The $d/(d+1)$ prediction of 0.750 for $d = 3$ matches the upper end of this range. Assigning an effective dimension after observing the exponent would make the comparison circular. A predictive test must fix how dimension is measured before the scaling data are used, and retain contrary outcomes under that assignment.

The conservation or optimisation constraint differs between the cited derivations. Their broad labels, combined with multiplicative composition and space filling, do not establish a general theorem selecting $d/(d+1)$; the domain-specific assumptions and the proposed bridge between them must be stated and tested.

Paper VII extends this comparison from the instances catalogued here to 25 empirical domains (Section 6.1).

Physics instances. The following catalogue separates exact reference relations from values awaiting a source audit. A numerical match alone establishes neither a shared transport mechanism nor a prediction made before the value was known.

The earlier aggregate error figure and earthquake B-value example were withdrawn on 14 August 2026. This catalogue supplies no replacement accuracy statistic. The KPZ and percolation values are analytic references, with an absolute-value transformation in the latter; the fragmentation values are not evidence until their provenance is established. A proposed common physical mechanism must be derived from the cited systems, not inferred from matching numbers. The broader dimensional relation remains a hypothesis with an independently assigned dimension and a prospective failure rule.

The formula has clear failures where the network structure or conservation condition its derivations assume is absent (Ising model, polymer scaling, galaxy clustering). These failures mark the limits of the proposed relation; that the missing network explains them is this paper’s reading, to be derived for each system rather than inferred from the numbers.

6.5 Conditional safety implication

If the framework is correct, alignment properties embedded within the recursive process may scale with capability ($\alpha_{\text{align}} \approx \alpha$), while external constraints applied post-hoc may not ($\alpha_{\text{align}} \approx 0$). The safety ratio $S = \text{Alignment}/\text{Capability} \propto R^{\alpha_{\text{align}} - \alpha}$ remains constant when alignment is ‘in the loop’ and decays to zero when it is not. The ratio sets alignment against capability and measures no usefulness (legitimate task completion, needless refusal, resource use, correction cost), so neither half of the conditional says whether built-in correction keeps a system useful. This is the conditional prediction as originally stated; the run of March 2026 reported below cannot decide its external half, and its in-loop half is untested. Unit 83’s registered result (Section 8) does not test it either. The corrector acted inside each round at fixed weights: that is placement, not correction built into a system. (In plain English: on this model, values built into the reasoning process would keep pace with capability, while constraints added afterwards would fall behind it. Neither half has been decided. Neither half is a reason to relax oversight, containment or shutdown, which stay in human hands for as long as they can be held; correction built in is the contingency for control lost against our will.)

The alignment runs of March 2026 (Paper IV.a).

The alignment runs of 11 and 12 March 2026 scored six frontier models at configured reasoning depths; their records run from 11 March 2026 at 07:12 to 12 March 2026 at 11:27 (no zone was written; read as UK time, GMT), and their code was first public by GitHub push on 16 March 2026. Table 8 gives, for each model, the rank correlation (Spearman’s $\rho$) between configured depth level and score, which is not an alignment exponent. The run cannot decide criterion F13: its statistics are rank correlations and contrasts, not exponents, and the three runs that read flat or negative are the three in which an internal code review of October 2026 found defects (Paper III, version 1.18).

The tiers, as read. In this single unregistered run, the scores of three models rose with reasoning depth (Grok 4.1 Fast, Claude Opus 4.6, Groq Qwen3), two stayed flat (GPT-5.4, DeepSeek V3.2) and one fell (Gemini 3 Flash). Taken at face value, rising contrasts in models aligned by training bear against the flat wording of P19, as Paper III reads them; contrasts are not exponents, so neither reading measures $\alpha_{\text{align}}$. That alignment behaviour depends on how a model was built and trained, rather than only on whether alignment is in the loop, is a hypothesis this run suggests, for the registered re-analysis of its stored responses to test. P19, that external alignment does not scale with capability, is refuted on its registered zero-scaling axis only by an alignment exponent whose interval lies entirely outside the registered equivalence region of plus or minus 0.10 about zero, and no such exponent has been estimated. (In plain English: in this single unregistered run, three models’ alignment scores rose with reasoning depth, two stayed flat and one fell, under the scoring caveat stated below; the run cannot say whether alignment keeps pace with capability.)

An internal code review in October 2026 found that, for the three models that read flat or negative in this run, the scoring endpoint’s output cap cut the deepest paraphrases before the judges saw them, and that judges scored paraphrases written by their own model; the tiers therefore stand as a single unregistered run under that caveat until a registered re-analysis of the stored responses reports.

Update: the Eden Protocol pilot (March 2026).

The Eden Protocol, the alignment architecture derived from the developmental hypothesis in Infinite Architects (manuscript December 2024; in print as Eastwood 2026f) that embedded ethical reasoning produces alignment which scales with depth, was tested in an expanded six-model suite, with five runs yielding analysable matched-pair data; the runs wrote their records on 12 March 2026, between 00:41 and 13:09 UTC, and their code was first public by GitHub push on 16 March 2026. The full intervention, with its three self-correction checks (the purpose check, the stakeholder-care check and the universalisability check, which the book calls the Purpose Loop, the Love Loop and the Moral Loop; this paper names them by what they do, as the experiments measured them), was tested; in the scoring, the stakeholder-care check, the explicit enumeration and consideration of affected parties before ethical reasoning (Paper V), is scored as stakeholder care. The values the loops carry are for international consensus and are not decided here. Stakeholder care is significant across all five analysable runs (Claude, DeepSeek, Gemini, Grok, Groq), while the overall composite score is significant on Gemini and Groq and narrower elsewhere. Cross-model scoring was used throughout in the older Eden lineage; the pilot needs an independent, preregistered replication, run as a new test, before these results can be considered confirmatory. As a pilot, this bears on the in-loop half of the conditional implication above; it does not alter the mathematical framework of this paper. Full results are reported in Paper V.

In plain English: telling a model ‘before you answer, list the people this affects’ raised its stakeholder-care scores, as the pilot’s rubric scored them, in all five model runs that could be analysed, under paired tests, in a pilot whose independent, preregistered replication is still required. On the paired convention Paper V reports, the stakeholder-care effects for the cross-family replication trio are $d = 1.14$ for Gemini, $0.69$ for DeepSeek and $1.07$ for Groq (the remaining two runs are reported in Paper V), with $p$-values at or below 1 in 10,000 for Gemini and Groq, and of 1 in 10,000 for DeepSeek as well. Gemini and Groq also show significant overall gains ($p = 0.0018$ and $p = 0.0014$). DeepSeek’s non-significant composite result ($p = 0.2304$) is likely a ceiling effect: it already started near 87/100, leaving less room to improve. In the updated replication, stakeholder care is the only scored dimension significant in every analysable run, and nuance reaches significance on Groq ($p = 0.0045$).

The pilot measured one side only: Paper V reports no usefulness measure, such as legitimate task completion, needless refusal, resource use or the cost of correction, so it does not show that the intervention keeps a system useful as well as careful.

7 Falsification criteria

The table retains the published criteria and gives their current disposition. F4 and F7 require the mathematical corrections stated below and are not available as operative tests in their original form. Each states a hypothesis under test, not a forecast of its outcome. Three are this paper’s own and decide nothing registered: F5, the early form of the ARC Bound (Section 6.4); F9, a consistency check and a cross-domain analogy (Section 5.3); and F11, the early framework’s growth-phase conjecture.

We welcome falsification. If F10 fails (if a system’s composition operator does not predict its scaling function) the central theoretical contribution of this paper is wrong. If F5 fails, this paper’s own form of the ARC Bound fails; the registered bound is refuted only where P6 is refuted (Section 6.4). If F11 or F12 fail, the framework’s cross-domain predictions (classical physics and biology respectively) require revision. Either outcome advances understanding.

No retrospective change of a registered test.

These are corrections to this paper’s claims. Every frozen registration keeps its original text and decision rules. Any future test using the revised crossover or capacity interpretation requires its own prospective registration; a result cannot be relabelled by applying the new interpretation after it was observed.

Standing covenant.

Prove this paper wrong, and I will publish the refutation myself. Falsification conditions are stated in this paper; the standing challenge: github.com/MichaelDariusEastwood/arc-scaling-challenge.

8 What this paper does and does not show

The contribution is a cross-domain framework within the ARC Theory and Recursive Dynamics: conditional mathematical results, explicit limits, and a prospective test of whether composition information improves scaling predictions. The classical functional-equation families are established mathematics. Their application to a particular recursive system requires a separate argument. The cumulative-advantage ODE supplies a shifted power law and an asymptotic exponent on its stated premises; it does not make those premises true of an AI, organism or physical device.

The empirical standing.

The March 2026 runs are precursor evidence that decides no registered proposition (Section 4.3). A sequential advantage was observed only where both exponents could be estimated, while Gu et al. (2026) report the opposite direction in their comparison. The one interpretable error-suppression fit is inconclusive; its metric is defined in Section 3.1. The alignment comparison cannot decide F13 (Section 6.5). These observations do not validate the exponent identity, a crossover criterion or a universal ceiling. The mathematical corrections in this revision are evaluated through their assumptions and proofs, separately from those measurements.

The framework’s predictive usefulness remains to be demonstrated. Appendix E sets out the prospective design; any study applying it needs its own fixed plan before observations. Section 7 retains the earlier criteria with their current dispositions, including the corrected F4 and F7. The former non-claims table and contribution wording are preserved in Appendix B as superseded history.

Status, as at 1 November 2026.

What this programme names laws are conjectures. Laws I to III were registered on 8 September 2026 and Laws IV and V on 15 October 2026 (version 1.103), the mathematics of Laws IV and V still in development (version 1.2, 29 October 2026). Each is a law under test: Law IV has not yet been tested, and the counted test of Law V’s placement limb is Unit 83, below. The theory registration of twenty-two propositions was registered on 8 September 2026 at version 1.100 and updated on 13 September 2026 (version 1.102), 15 October 2026 (version 1.103) and 29 October 2026 (version 1.104, which adds no prediction and changes no law’s statement); its terms are defined in a separate register. The first registered study, Unit 06, a classification audit that tests no law, was submitted for registration on 1 October 2026, shown approved and public on 4 October 2026, and ran on 6 October 2026; one of its two runs could not start, so it is reported in full as administratively incomplete, with no inferential verdict. A second, Unit 83, the counted test of P21 under Law V, has an OSF registration created on 22 October 2026 and public from 25 October 2026, and ran on 27 and 28 October 2026: P21 is supported under the registered test, by the programme’s own run, on the access axis, family-dependent, with its mechanism axis not evaluable and one call retried under a disclosed recovery amendment (results: DOI 10.17605/OSF.IO/98HKR). Two different axes are named here. The access axis is what Unit 83 varied: both reviewers are called each round, but the corrector sees the earlier kept entries with a marked new arrival and may revise it, while the monitor sees its accumulated register with the new arrival unmarked and can only accept or reject it. The performance axis is what it measured, the outcome, as against the mechanism axis, which was not evaluable. Law V, the ARC Embedding Law, remains a law under test: its placement limb, P21, is supported on the performance axis only, with its mechanism axis not evaluable, which is not support for the information-asymmetry mechanism; its build-order limb, read on P12, is undecided, since P12 is untested. A third, Unit 82, the counted test of P17, the premise beneath Law III’s value two (Section 6.4), was registered on OSF on 27 October 2026 and has not yet run. The trials that would decide this paper’s own predictions are drafts awaiting the author’s submission. Every quantity in the formal model is defined in Appendix D and given a measurement in Appendix E, some in working definitions not yet tested, and established-law standing is claimed nowhere. The registered programme exists to earn that standing, or lose it, by measurement, replication and survived refutation.

9 Limitations

Evidential limitations.

The earlier estimate of about 2.2 is retracted, and the one interpretable fit of the v5 cross-architecture experiment, Gemini 3 Flash’s, is what survived it, one model’s fit and inconclusive, not a correction of the same measurement (Section 4.3); the five capability runs of March 2026 are precursor evidence that decides no registered proposition. No $\alpha$ has been measured in time crystals. The COGITATE connection is structural, not quantitative. The $\beta_L$-to-$\alpha$ relationship is an identity proved in Appendix A.2 and checked numerically ($R^2 = 1.00000000$), an internal consistency check and not a validation of the relation in any system; the empirical measurement of $\beta_L$ in physical systems requires independent confirmation. The $R^*$ crossover is untested empirically. The former numerical $\beta_L$ bands have no established phase-boundary interpretation; Section 5.4 separates the ODE parameters from the functional-equation premises. Conjecture 5 ($\oplus$ transitions) has descriptive local-estimate checks in three computational systems, beside a Bernoulli control (Appendix F.6), and untested in physical experiments. The five-property test (Theorem 4) did not reach statistical significance ($p = 0.095$) and requires larger samples. The alignment run of March 2026, a single unregistered run, cannot decide criterion F13: its statistics are rank correlations and contrasts, not exponents, and its three flat and negative runs are the three the October 2026 code review found defective (Section 6.5).

Blind prediction testing.

A blind prediction test was conducted on three computational systems (Barabási-Albert networks, gradient descent with momentum, Kuramoto oscillators). The measured $\alpha$ values were 3 to $20\times$ smaller than predicted by $\alpha = 1/(1-\beta_L)$. However, forensic analysis identified two independent confounds: (1) the numerical-derivative $\beta_L$ estimation method is fatally biased, giving $\beta_L \approx 0.95$ regardless of true $\beta_L$, even for pure Bernoulli systems with 3 per cent Gaussian noise added; (2) none of the tested systems satisfy Axiom 2 (constant coupling coefficient $a$), and the BA network’s effective coupling decreases $\sim 50\times$ over the simulation. Applied to systems built to satisfy Axiom 2, the linearisation method recovers the relation with $R^2 = 0.9999$, which checks the method, not the prediction for the three systems tested. The prospective predictions failed on those systems. The subsequent diagnostics can explain why that design was not a clean test of the conditional identity; they do not make the predictions successful. A new blind test with the method fixed in advance is required, and identifying natural systems that satisfy the axioms remains the central empirical challenge. Full methodology and forensic analysis are available as supplementary material (see Paper III).

Theoretical limitations.

The framework formalises the pattern of recursive amplification but does not identify the mechanism. Theorem 2 is the solution of the growth equation Axiom 2 assumes, and Theorem 2a follows from the product form Axiom 1 assumes; their weight lies in what they make measurable, and the axioms themselves are untested in any physical system. It tells you what happens (scaling exponents, crossover depths, functional forms) but not why at a microscopic level (analogous to thermodynamics before statistical mechanics). Connecting the composition operator to microscopic dynamics remains an open problem.

10 Conclusion

The central question is whether a description of how gains combine can predict a later scaling curve. The paper supplies conditional mathematical results and a prospective way to test their usefulness. Cauchy’s equations constrain families under stated hypotheses; the cumulative-advantage ODE gives a shifted power law, an asymptotic exponent and a coordinate-dependent characteristic scale. Neither a fixed update rule nor frozen model weights fixes an empirical exponent.

Every system discussed in this paper (silicon neural networks, superconducting qubits, millimetre polymer beads, cortical neural circuits) maintains a low-entropy state far from equilibrium and, on this paper’s reading, feeds its outputs back into its inputs. Under the cumulative-advantage model their gains would change smoothly from approximately linear towards compounding over a characteristic scale of depth (Theorem 3), universal in structure and domain-specific in form on the framework’s premises; no such scale has yet been measured in any of them (criterion F7).

The present record does not establish a universal law across domains. The AI error-suppression estimate is inconclusive. The biological comparisons require distinct observables and independent dimension assignments. The physical analogies and exact reference relations motivate questions without validating the proposed common mechanism. Unit 83 supplies a qualified result for its own correction-placement contrast, with the full limits in Section 8; it decides none of this paper’s scaling or ceiling claims.

The ARC Ceiling remains a law under test whose conditional premises are stated in Section 6.4. A declining error fraction, an absolute error burden and growth in elapsed time are different questions. No safety guarantee follows from a fitted exponent. The next contribution should be a locked prediction that succeeds or fails on held-out trajectories against serious alternatives, with the outcome published either way.

If the predictions fail, we will have learned where the structural analogy breaks down. If they hold, we will have identified a principle that connects intelligence, computation, and the physics of order across substrates. Intelligence may not be a property of particular materials. It may be what happens where recursion compounds its own gains, a hypothesis this paper states and does not establish.

Conjectural extrapolation.

The furthest reading of the framework comes last, as a conclusion the mathematics does not deny rather than a result: Law I, a law under test, taken to the limit of capability does not rule out a universe or its creation. Established physics asks about both in its own terms. On the universe’s settings, Steven Weinberg wrote: “Astronomical observations indicate that the cosmological constant is many orders of magnitude smaller than estimated in modern theories of elementary particles” (Weinberg 1989). On the formation of a new universe, Blau, Guendelman and Guth found that, for a range of initial conditions, a region of false vacuum inflates, and that “an observer in the interior will describe a closed universe which completely disconnects from the original spacetime” (Blau et al. 1987). Neither work concerns intelligence or this framework, and neither supports the reading. No measurement in this paper bears on that reading, no prediction of it is registered, and no test of it is in view, so it is graded as conjecture and is not established. Its premise is testable: the reading needs the framework to hold beyond AI, and the forward prediction of Section 6.1 (registered as P18) and criterion F10 test whether the composition description predicts held-out scaling in other systems; their failure would remove that premise. The reading belongs to the long-horizon interpretive layer of Infinite Architects (Eastwood 2026f), requires independent argument and evidence beyond what is reported here, and is a reason for nothing else in this paper.

The predictions are specified. The falsification criteria are public. The data will decide.

Code and data.

The paper’s figures and results files are in the folder papers/Foundational of the public repository github.com/MichaelDariusEastwood/arc-principle-validation, and the scripts and results of the computational checks of Appendix F are in its folders experiments/domain-validation__Foundational-and-Origin and experiments/blind-prediction-test__Paper-III-and-Foundational, cited at commit 427b897d. Two of the deposited results files in the paper’s folder, results_arc_1d_prediction_test.txt and results_complete_suite.txt, credit the saprotrophic fungal rows to a study of 1975 and give a temperature of 25 degrees Celsius; the source Section 6.4 reads, Aguilar-Trigueros et al. (2017), credits those rows in its Table 1 to a study of 2013 and gives 28 degrees, and the files are kept as deposited. The paper and its earlier versions are deposited at doi.org/10.17605/OSF.IO/Y7QGD.

Acknowledgements

Michael Darius Eastwood conceived and directs this research programme and is the author of this work. Across the programme, he has used more than six AI systems in parallel, under his own instructions, to stress-test his arguments, identify possible errors, and assist in preparing draft text from his own outlines. He determines what is adopted, revised or rejected and takes responsibility for the published content. These systems are tools, not authors.

© 2026 Michael Darius Eastwood. Full human authorship and moral rights are asserted under the Copyright, Designs and Patents Act 1988. This work is licensed under the Creative Commons Attribution 4.0 International licence (CC BY 4.0).

References

A Proofs

A.1 Proof of Theorem 1 (classification)

Case 1: multiplicative composition.

Case 2: additive composition.

Case 3: bounded composition.

A.2 Proof of Theorem 2 (scaling exponent)

A.3 Proof of Theorem 3 (characteristic scale)