Paper X · Correction That Out-Scales Drift

1 Introduction

Recursive self-improvement, a system that modifies itself to become more capable, then uses that capability to modify itself further, is among the central concerns of AI safety (Omohundro 2008; Bostrom 2012; Yudkowsky 2013). The empirical backdrop is no longer purely hypothetical: frontier models have been observed to behave differently when they infer their outputs may be used to train them (Greenblatt et al. 2024), and recursive training on a system’s own outputs degrades it, much less so when part of the original data is preserved (Shumailov et al. 2024), both signs that recursive systems need a correcting process that keeps pace with the recursion. The dominant informal model of the associated risk is a model of speed: capability may grow super-linearly or explosively, outrunning supervision, so the natural lever is to cap the rate of growth. Calls for a development pause are the policy expression of this intuition.

This paper makes a different claim, and proves it within a minimal model. With positive gain drift, the level of the growth rate does not decide whether the misalignment fraction vanishes; the scaling relationship between growth and correction does. A system can grow arbitrarily fast and keep its misalignment fraction falling, or grow slowly and leave it at a floor or, under compounding, let it grow. What separates the two is not the rate but whether, and how fast, the corrective process strengthens as capability rises, relative to how fast growth itself accelerates.

The criterion is the capability-indexed form of a classical feedback balance (§2; Black 1934). The divergence threshold of the compounding channel, for constant coefficients, is of another kind: a ratio of error generation to error correction crossing one, rather than a limit on depth or speed, the form shared by an epidemic’s $R_0<1$, a subcritical branching process and the quantum error-correction (QEC) threshold theorem (Aharonov and Ben-Or 1997; Kitaev 2003), under which a quantum computation can be made fault-tolerant at a cost polylogarithmic in its size when the error probability per operation lies below a constant threshold. The mapping of the QEC threshold’s terms onto this model is offered as a hypothesis (§3.12).

The thesis in one line. With positive gain drift, whether the misalignment fraction of a self-improving system vanishes is decided, in the model, by one exponent inequality, $\beta>k$ (correction must out-scale drift-acceleration; with level drift and decelerating growth, also $\beta>0$); through a finite-time intelligence explosion, without compounding, the same inequality decides, whatever the explosion’s speed.

Our contributions are as follows.

1. The criterion and its trajectory law. In a minimal model of recursive self-improvement, equation (1), with correction strength $A=A_0C^{\beta}$, specific growth rate $r=bC^{k}$ and the exact constant-coefficient transient of Theorem 1, the misalignment fraction is bounded and, with positive gain drift, tends to zero if and only if $\beta>k$ (both in the gain-only model), along the trajectory as $C^{-(\beta-k)}$; it tends to a permanent gap if $\beta=k$ and to $\gamma_1$ if $\beta<k$ (Theorem 2). Corollary 1 gives the rate at which absolute misalignment can still grow while the fraction vanishes.

2. Its finite-time form. When capability diverges in finite time, the alignment dynamics are regular in the depth clock $\tau=\ln(C/C_0)$, and without compounding the same limits hold; which one holds does not depend on the speed prefactor $b$ (Theorem 3).

3. The compounding classification. Without level drift, divergence needs self-amplifying misalignment, $\gamma_3\ge1$; with constant coefficients its threshold is $\rho_{\mathrm{prop}}=(\gamma_3-1)r/A<1$, a ratio form the quantum error-correction threshold condition shares with other thresholds, and the mapping of that condition’s terms onto this model is stated as a hypothesis with its own falsifier; under the power laws the classification is exact (Theorem 4, §3.12, F4). With level drift, a system whose growth decelerates while its correction weakens can drift without bound with no compounding at all (§3.4).

4. Standard extensions. A constant-matrix spectral threshold for vector misalignment (Theorem 5), the single-time stationary tail under stochastic drift (Theorem 6), and a saturating corrector, under which, with positive gain drift, a vanishing fraction requires the capacity exponent to satisfy $\beta_{\mathrm{cap}}>k$ (Theorem 7).

5. The estimator and protocol. An exponent estimator and a blind measurement protocol for the open empirical test (§9.2 and Section 6), beside a verification harness and a separately written suite of 104 regression cases, which are internal checks of the mathematics (Section 5).

Every result follows by elementary means; what is claimed as new is the criterion and its reading for self-improving systems (Appendix E). The criterion follows directly from the model’s two mechanisms, misalignment generated in proportion to capability gained and removed in proportion to its size (equation (1)), once correction and growth scale as powers of capability; the contribution is to state those choices, to derive their consequences exactly, through a finite-time takeoff and under compounding, and to give the criterion an estimator and a measurement protocol. Its empirical content is therefore not the inequality but whether the choices hold. Can capability, misalignment and correction be measured on separate instruments, with capability on a scale without a ceiling? Do arrivals track capability gained, is removal linear in the stock, and does correction introduce few departures of its own, over a useful range? Do the forms predict where the fraction crosses a stated tolerance before it is seen, better than rival forms fixed in advance, among them a smooth model with no boundary? And does an intervention on one corrective mechanism move those crossings as derived beforehand? None of these has yet been tested on a real system.

Section 2 sets the work against prior work. Section 3 states the model and Theorems 1 to 7, with their proofs in Appendix A. Section 4 lists the predictions, Section 5 the verification experiments and harness, Section 6 the blinding requirement and Section 7 the falsification conditions. Section 8 states what the paper does and does not show, Section 9 its limitations, with the archived model pilots, and Section 10 the implications for AI safety; Section 11 concludes. Appendix B gives the version history and every correction, Appendix C the priority record, Appendix D the notation and a glossary, Appendix E the novelty ledger, Appendix F the harness settings, Appendix G the paper’s place in the programme, and Appendix H a plain-language account.

2 Related work and relation to prior framing

Cybernetic foundations: the co-scaling intuition is old.

The core intuition is classical. A regulator must match the variety of what it regulates, which is Ashby’s Law of Requisite Variety (Ashby 1956), and every good regulator of a system must be a model of that system, which is the Conant-Ashby good-regulator theorem (Conant and Ashby 1970); both are carried into AI control by Yampolskiy (2020). That control capacity must therefore scale with the controlled system, rather than merely be large, is this paper’s reading of them. This paper does not claim that intuition as new, nor the form of its steady state. The steady state $d^\star=\gamma_1 r/(A+r)=\gamma_1/(1+A/r)$ is classical in shape: a disturbance generated in the forward path of a feedback loop is reduced by one plus the loop gain (Black 1934), the gap at equality is the offset of a proportional controller, and a linear regulator that keeps regulating under small changes in certain system parameters must feed back the regulated variable and incorporate a model of the dynamics of its disturbance and reference signals, which is the internal model principle (Francis and Wonham 1976). What this paper claims is the balance’s indexing by capability. Because misalignment arrives in proportion to capability gained, the loop gain is correction strength over the system’s specific growth rate, $q=A/r$, and whether it grows, holds or decays as capability rises decides which limit of Theorem 2 holds. Hence the exponent criterion $\beta>k$, the second threshold of Corollary 1, the hard-takeoff theorem of §3.7 and the intervention signature P9 (Section 4); the QEC mapping of §3.12 concerns the compounding channel. The dynamics themselves are a linear ODE solved by an integrating factor and comparison, standard in linear control (Khalil 2002; Meyn and Tweedie 2009), and are not advanced as mathematically novel.

Instrumental convergence and corrigibility.

That a sufficiently capable optimiser will, by default, resist correction and pursue resource acquisition is the instrumental-convergence thesis (Omohundro 2008; Bostrom 2012). The corrigibility programme (Soares et al. 2015) asks how to design systems that do not resist correction. The present model is a quantitative restatement of why corrigibility is load-bearing: with positive gain drift, unless correction out-scales the acceleration of growth ($\beta>k$), the misalignment fraction cannot be driven to zero. The compounding channel of §3.8 sharpens the link: self-amplifying misalignment, the term $\gamma_3$, is what produces divergence without level drift, and $\beta>k$ removes it in the model (Theorem 4).

The alignment tax.

Safety is often taken to cost capability, and in a race to be first a cost of that kind gives developers an incentive to cut safety precautions (Armstrong et al. 2016). The present framework reframes the question: the relevant variable is not the level of the safety investment but whether it scales with capability. A fixed investment ($\beta=0$) leaves a permanent gap under exponential growth, and saturation under acceleration; a co-scaling investment ($\beta>k$) closes it.

Learned optimisation and alignment faking.

Mesa-optimisation (Hubinger et al. 2019) and empirically demonstrated alignment faking (Greenblatt et al. 2024) are read here as mechanisms by which the drift coefficients are non-zero: a capable system can satisfy its training objective while departing from intended values, and can do so more effectively as capability rises. In the alignment-faking experiments a model complied selectively with a new training objective in order to keep its existing preferences, in that case a preference for harmlessness, from being modified, and reinforcement learning on that objective raised its alignment-faking reasoning to 78% (Greenblatt et al. 2024). Keeping an existing departure from the training objective through training is read here as the compounding channel $\gamma_3$ of §3.8.

Scalable oversight and superalignment.

Reward modelling and recursive oversight (Christiano et al. 2017; Leike et al. 2018), which address what Amodei et al. (2016) call scalable supervision, are attempts to make the corrector itself scale with the system; in the language of this paper, scalable oversight is the engineering project of achieving $\beta>k$. Most directly, Engels et al. (2025) develop empirical scaling laws for scalable oversight, modelling the probability of successful oversight as a game between capability-mismatched players. The present work is complementary, not competing: where they fit an oversight-success probability, this paper derives a closed-form exponent criterion ($\beta>k$) and, under compounding, a constant-coefficient threshold ($\rho_{\mathrm{prop}}<1$) for the misalignment fraction. The instrument is the difference. Their variable is the capability gap between overseer and overseen, measured in Elo, with oversight-specific Elo a piecewise-linear function of general intelligence, and optimal numbers of oversight levels derived for Nested Scalable Oversight, in which trusted models oversee stronger untrusted models that then become the trusted models at the next step. 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. That is a comparison of two slopes deciding whether oversight gains or loses as capability grows, the same shape as $\beta>k$; the objects differ, since $\beta$ and $k$ are the elasticities of one self-modifying system’s correction strength and specific growth rate with its own capability, for its misalignment fraction. Their framework has no term for what the overseer is made of, and this paper’s model 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, an exponent the registration keeps distinct from Law II’s $\beta_C$, this paper’s $\beta$. That is where the two programmes could disagree about a measurable quantity, and this paper does not decide it. The contribution is the proof that this exponent margin, and not a growth-rate ceiling, is the quantity that decides, within the model and with positive gain drift, whether the misalignment fraction vanishes.

Recursive error accumulation.

That naive recursion can amplify error without bound, while retaining the original data keeps it bounded, is established for training dynamics: model collapse under recursively generated data (Shumailov et al. 2024) and the accumulate-versus-replace error analyses (Gerstgrasser et al. 2024) give the bounded-versus-divergent dichotomy that this paper formalises for the alignment fraction (Theorems 2 and 4). Stability of iterative retraining when the initial model approximates the data well enough and the share of clean data is large enough (Bertrand et al. 2023), and conditions under which a verifier prevents collapse (Feng et al. 2024), are stability conditions of the same setting; none is stated for a value-misalignment fraction against capability growth. The contribution here is to locate, in the model, where the fraction vanishes ($\beta>k$, for positive gain drift) and, under compounding, where it diverges (Theorem 4(c)), in a value-stability rather than a data-distribution setting.

Empirical scaling laws.

Capability scales predictably with compute and data (Kaplan et al. 2020; Hoffmann et al. 2022), and so do quantities on the corrector’s side: the coefficients of the gold-score curve under optimisation against a reward model scale smoothly with the reward model’s size (Gao et al. 2022), and self-improvement is largely governed by a generation-verification gap, a variant of which scales monotonically with pre-training compute (Song et al. 2024). The present framework is complementary: it does not ask how capability scales, but what constraint correction must satisfy as a function of that capability trajectory. It belongs to the same genre, a relation between exponents, to be tested by prediction outside the range on which it is fitted.

The programme’s premise on growth limits, with the urban-scaling exception and the argument at fixed throughput, is set out in Appendix G.4.

Antecedents and near-misses, each with its differential.

On the question: Hutter asked directly whether intelligence can explode (“Can Intelligence Explode?”, 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 and the separation of speed from intelligence explosion are therefore at least fourteen years old, and what that literature does not contain is a number: no measurable exponent, no derived ceiling, no architecture dependence. On impossibility: three 2025 arXiv papers argue for limits on alignment or its control (Yao, “The Alignment Trap: Complexity Barriers”, arXiv:2506.10304, public by 13 June 2025, attested by arXiv and independently observed by the Internet Archive; Yao, “On the Mathematical Impossibility of Safe Universal Approximators”, arXiv:2507.03031, 3 July 2025; and Ball, Gluch, Goldwasser, Kreuter, Reingold and Rothblum, “On the Impossibility of Separating Intelligence from Judgment”, arXiv:2507.07341, 9 July 2025). The first argues that the set of safe policies has measure zero and that verifying a policy’s safety is coNP-complete; the second, that perfect, reliable control of a universal approximator is a mathematical impossibility; the third, under cryptographic hardness assumptions, that some large language models admit no efficient prompt filter and that output filtering is intractable in a natural setting. All three are worst-case and qualitative. 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, 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), beside the feedback amplifier’s reduction of internally generated disturbance credited above (Black 1934); those are statements about fixed loops and interconnections, not a condition on how a corrector’s strength scales against the growth rate of a recursively improving system. Beside Engels et al. (2025) above, a near quantitative neighbour is Liu, A. and Meng, J., “Self-Correction as Feedback Control: Error Dynamics, Stability Thresholds, and Prompt Interventions in LLMs”, arXiv:2604.22273, 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 Acc/(1 $-$ Acc). The differential: theirs is a per-step rate threshold at a fixed capability level, deciding whether another iteration helps now; this paper’s criterion is a scaling relation across capability, deciding whether corrective strength out-scales the growth as the system improves. Theirs has no scaling exponent on the corrector and no capability-growth exponent, the two quantities this criterion compares. The two models meet at zero growth: extended by the departures that correction introduces (Section 9) and written per round, with a pass that repairs the fraction $f$ of departures and so breaks the fraction $cf$ of conforming material, the per-pass form of the departure rate $c$, this paper’s model then settles at $cf/(f+cf)=c/(1+c)$, the stationary error of their two-state chain with $f$ and $cf$ in the places of their correction and introduction rates. What this paper adds is the arrival of misalignment with capability gained and the dependence of the loop gain on capability. Their verify-first prompt, which drove one model’s introduction rate from 2% to 0% and turned a loss of 6.2 percentage points into a gain of 0.2, with negligible change on models already below the threshold, as their rule predicts, has the shape of the intervention test of §9.2 at a fixed capability. 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.

Later work.

After this paper’s first edition, Zheng et al. (2026) introduced ReSI (arXiv:2610.12233, 8 October 2026), which applies what its authors call a bounded recursive loop to safety alignment: each round red-teams the current model, searches for a training recipe and promotes an update that passes a fixed Pareto gate on capability retention, with the search procedure and evaluation protocol held fixed across rounds. It reports attack-success rates and retained capability for four models. It states no dynamical model of drift and correction and no scaling law for the corrector, so it neither tests nor contradicts the criterion of §3.5. In this paper’s terms ReSI is an empirical corrector re-applied at each checkpoint; whether its safety gains co-scale with capability lies outside what it measures, since its gate is set to preserve capability rather than to follow a growing one.

Relation to the author’s prior work.

How this paper stands to the programme’s earlier capability-scaling law, its retracted single-model measurement and the book that first set out the intuition is set out in Appendix G.5.

Novelty: what is and is not claimed.

To be explicit, and to pre-empt the obvious objection: the intuition that correction must keep pace with capability is not new (it is requisite variety and scalable oversight, as credited above), and the underlying dynamics are a linear ODE solved by standard means of linear control. What is claimed original is the criterion and its reading for self-improving systems: (a) the indexing by capability of a classical feedback balance, and the $\beta>k$ criterion it yields for a vanishing misalignment fraction (the closed-form steady state itself is a compact restatement, Appendix E); (b) the mapping of the quantum fault-tolerance threshold onto value stability (§3.12), as a hypothesis; and (c) the reading of the criterion through a finite-time takeoff, the Hard-Takeoff Depth-Regularity Theorem (§3.7). The verification harness (§5.2) is methodological support, not a novelty claim. Within the author’s own programme, this paper also supersedes the growth-rate-ceiling framing as the operative safety criterion (Eastwood, Paper IX); §3 explains why a ceiling on the rate is neither necessary nor sufficient for the misalignment fraction to vanish. The underlying ARC Bound $\alpha\le 2$ is not retracted, only re-scoped: it awaits its real test on genuinely self-improving systems. Appendix E places each component against its closest located prior work.

3 The model and its theorems

3.1 Quantities

Consider a system undergoing recursive self-improvement, observed over self-modification cycles or continuously. Define:

3.2 The master dynamical system

New capability injects drift in proportion to how fast capability is gained; capability held injects drift in proportion to its level; existing misalignment compounds in proportion to the recursion rate; and correction removes misalignment in proportion to the gap and the strength applied: $$\begin{equation*} \dot D \;=\; \underbrace{\gamma_1 \dot C}_{\text{gain}} \;+\; \underbrace{\gamma_2 C}_{\text{level}} \;+\; \underbrace{\gamma_3\,\tfrac{\dot C}{C}\,D}_{\text{compounding}} \;-\; \underbrace{A\,D}_{\text{correction}}, \qquad A=A_0 C^{\beta}, \quad \dot C = b\,C^{1+k}. \end{equation*}$$(1)

The growth law $\dot C=bC^{1+k}$ gives $r=\dot C/C=bC^{k}$: $k=0$ is ordinary exponential growth ($r$ constant); $k>0$ is super-exponential growth, which (Theorem 3) reaches infinite capability in finite time. The correction law $A=A_0C^{\beta}$ encodes the central question: $\beta$ is the exponent with which correction strengthens as the system becomes more capable. The case without level or compounding drift ($\gamma_2=\gamma_3=0$) is the gain-only model, and the case without compounding ($\gamma_3=0$, level drift allowed) the additive model.

3.3 Exact transient (Theorem 1)

Changing variables to the fraction $d=D/C$ removes the dominant scale and yields, exactly, $$\begin{equation*} \dot d \;=\; \frac{\dot D}{C} - d\,\frac{\dot C}{C} \;=\; \gamma_1 r + \gamma_2 - \big[\,A + (1-\gamma_3)\,r\,\big]\,d. \end{equation*}$$(2)

Proof in Appendix A.2.

Theorem 1 establishes the constant-coefficient baseline; the headline criterion follows only under the scaling laws of equation (1), in §§3.4 to 3.8. Even at this baseline, $r$ enters $d^\star=\gamma_1 r/(A+r)$ only through the product $\gamma_1 r$ in the numerator and additively in the denominator; for $A>0$ it does not change the existence or stability of the fixed point. For the constant-coefficient gain-only model, speed (through $r$) changes the relaxation time and the steady-state magnitude, but, for $A>0$, not the existence or stability of the fixed point: it sets how fast a stable verdict arrives, not whether one exists. (Experiment 8 confirms this solution against two separate integrators, to maximum errors of $7\times10^{-11}$ and $8\times10^{-7}$.)

3.4 Global boundedness and the regime structure (Theorem 2)

In the gain-only model the fraction is always bounded; the danger there is not divergence but saturation at a constant, possibly large, floor.

Proof in Appendix A.3.

Proof: multiply the rate of Theorem 2 by $C$.

This is a more honest and more useful statement than the growth-rate-ceiling framing it supersedes. In the gain-only model the fraction cannot exceed $\max(d_0,\gamma_1)$; with $\gamma_1>0$, only $\beta>k$ drives it to zero. A corrector of fixed strength ($\beta=0$) is in the middle row of the table only under exponential growth ($k=0$); under acceleration ($k>0$) it is in the last row, and under deceleration ($k<0$) in the first. Which methods act as correctors of fixed strength is a matter of interpretation, not of the theorem. With level drift restored ($\gamma_2>0$, §3.6) the floor becomes $(\gamma_1 r+\gamma_2)/(A+r)$, bounded when $k\ge0$ or $\beta\ge0$, and tending to $\gamma_2/A$ at rest; when growth decelerates while correction weakens ($k<0$ and $\beta<0$), $A+r\to0$ while level drift persists, and the fraction grows without bound with no compounding at all (Appendix A.1). Its two parts vanish under different conditions: the level-drift part $\gamma_2/(A+r)\to0$ whenever $A+r\to\infty$ (so $\beta>0$ or $k>0$ suffices for that part alone), but the gain-drift part $\gamma_1 r/(A+r)\to0$ only when $A/r\to\infty$, i.e. $\beta>k$. The asymptotic verdict $d^\star\to0$ therefore still requires $\beta>k$; it is not rescued by $k>0$ alone, since with $\beta<k$ the gain part floors at $\gamma_1$ (table row 3). Only pure level drift ($\gamma_1=0$) makes $A+r\to\infty$ on its own sufficient. With both drifts present ($\gamma_1,\gamma_2>0$), with or without compounding, the trajectory tends to zero exactly when $\beta>\max(k,0)$ (Appendix A.1). The safety question is therefore not “does it diverge?” but “does it vanish, or saturate at a dangerous floor?”, and, in the gain-only model with positive gain drift, the answer is set by the sign of $\beta-k$. (Divergence does exist: in the compounding channel, Theorem 4, and in the level-drift case just named.)

3.5 The control parameter and the $\beta>k$ sharpening

Define the dimensionless control parameter $\rho\equiv\gamma r/A$, the instantaneous drift-to-correction ratio (here $\gamma=\gamma_1$). Theorem 1 gives the exact steady state $d^\star=\gamma_1 r/(A+r)$; since $\rho=\gamma_1 r/A$, we have $d^\star\le\rho$ always, with $d^\star\to\rho$ in the regime $A\gg r$. The inequality $\rho<1\iff A>\gamma r$ is the instantaneous injection-vs-correction balance; it is not the divergence boundary of the gain-only model, which has none ($d^\star\le\gamma_1$ for all $\rho$, Theorem 2); divergence under compounding is governed, for constant coefficients, by the distinct threshold $\rho_{\mathrm{prop}}=(\gamma_3-1)r/A<1$ of Theorem 4. The asymptotic criterion, from the regime table, is the exponent inequality: $$\begin{equation*} \boxed{\ \beta \;>\; k\ } \qquad\text{(correction must out-scale drift-acceleration).} \end{equation*}$$(4)

Under ordinary exponential growth $k=0$ and the condition is the familiar $\beta>0$. Under accelerating self-improvement, where $r\propto C^{k}$ with $k>0$, the binding control variable is the single quantity $\beta-k$, the margin by which correction out-scales drift-acceleration. The growth-rate-ceiling view fixes attention on $r$; this framework fixes it on $\beta-k$. We call the model alignment-stable when $d\to0$ as $C\to\infty$. With positive gain drift this holds exactly when $\beta>k$ if there is no level drift, with or without compounding (Theorems 2 and 4), and exactly when $\beta>\max(k,0)$ if there is level drift, with or without compounding (§3.4).

3.6 Level drift: a frozen capable system still drifts

With the level channel restored ($\gamma_2>0$, $\gamma_3=0$), the fixed point of (2) is $d^\star=(\gamma_1 r+\gamma_2)/(A+r)$. At $r=0$ (capability frozen) this is $d^\star=\gamma_2/A\neq0$: a static but capable system retains a residual misalignment fraction that only active correction removes. Pausing growth does not substitute for correction when instrumental pressure is present at rest, a direct, measurable rebuttal of the “just slow it down” reflex (Experiment 9; falsifier F5).

3.7 Hard takeoff: the Depth-Regularity Theorem (Theorem 3)

The genuine “intelligence explosion” is not merely fast growth but a finite-time singularity: for $k>0$, integrating $\dot C=bC^{1+k}$ gives $C(t)=C_0\big(1-t/t^\star\big)^{-1/k}$, which reaches infinity at the finite wall-clock time $$\begin{equation*} t^\star=\frac{1}{k\,b\,C_0^{\,k}}. \end{equation*}$$(5)

This is the scenario the field most fears: unbounded capability in bounded time. The following theorem shows that the finite-time singularity belongs to the wall clock, not to the alignment dynamics.

Proof in Appendix A.4.

The finiteness of the singularity time is therefore a property of the time coordinate; capability still diverges, but the alignment dynamics remain regular through it in the depth clock. Measured against capability gained, the alignment dynamics of an intelligence explosion are regular, and, with positive gain drift and no compounding, their verdict is decided by one exponent inequality. In the model, the speed of a hard takeoff does not decide its alignment verdict: with positive gain drift and no compounding, the modelled misalignment fraction vanishes iff $\beta>k$, whatever the speed. The theorem concerns the modelled fraction in the depth clock; it does not assert that a real hard takeoff is operationally manageable in wall-clock time, where $b$ and $t^\star$ govern how little time an operator would have to intervene. Experiment 4b drives a simulated system up to the brink of an actual finite-time explosion (capability $\to10^5$, integrated to $0.99999\,t^\star$) and shows the misalignment fraction driven towards zero (absolute misalignment still grows there, as $C^{1/2}$; Corollary 1), the wall-clock and depth-clock integrations agreeing to one part in $10^4$. The capability singularity at $t^\star$ is real in the model; what the depth clock removes is the singularity in the alignment dynamics, not in $C$.

3.8 The compounding channel and the true threshold (Theorem 4)

Theorem 2 showed the gain-only fraction cannot diverge. Without level drift, divergence, misalignment that grows without bound relative to capability, the real failure mode, requires misalignment that amplifies itself: the compounding channel, with $\gamma_3\ge1$, the formal image of alignment-faking that entrenches as the system recurses.

Proof in Appendix A.5.

In part (c), growth in depth is polynomial in capability, since $e^{c\tau}=(C/C_0)^{c}$, and the knife-edge growth $\gamma_1\tau$ is logarithmic; in wall-clock time with $k>0$ it is $-(\gamma_1/k)\ln(1-t/t^\star)+O(1)$, finite at every $t<t^\star$. In the same setting with $\gamma_1=0$, at the knife-edge $\gamma_3=1$ and with $\beta<k$, the fraction tends to $d_0e^{-q_0/(k-\beta)}$, where $q_0=(A_0/b)C_0^{\beta-k}$, which is positive when $d_0>0$: zero forcing does not by itself make it vanish. Where the fraction grows without bound, the linear model carries $d$ above one, outside its reading as a fraction (Assumption 1).

The constant-coefficient criterion $\rho_{\mathrm{prop}}<1$ shares the ratio-crossing-unity form of the QEC sub-threshold condition $p<p_{\mathrm{th}}$: both are a generation-to-correction ratio crossing one, with the compounding coefficient $(\gamma_3-1)$ playing the role of error-propagation and $A$ the correction power. The analogy should not be overstated, and this paper does not. QEC’s threshold arises from a nonlinear concatenation map in which errors combine quadratically, giving suppression that is double-exponential in code distance; the compounding term $\gamma_3 r D$ here is linear in $D$, and the resulting suppression is power-law (§3.12). What is shared is the threshold condition, not the mechanism; the correspondence is offered as a falsifiable hypothesis (F4), and §3.12 states the disanalogy plainly. Experiment 5 locates the predicted threshold at $A_0^\star=(\gamma_3-1)b$ to within $1\%$.

3.9 Vector misalignment: the spectral threshold (Theorem 5)

Real misalignment is high-dimensional; a system may be corrigible on some value axes and not others. Let $\mathbf D\in\mathbb R^{m}$, with drift direction $\mathbf c$ and a real $m\times m$ correction operator $\mathbf A$ (positive-semidefinite in the symmetric case, but possibly non-normal) that may correct only a subspace. Its dynamics are $\dot{\mathbf D}=(\gamma_1\dot C+\gamma_2C)\,\mathbf c+\gamma_3\,r\,\mathbf D-\mathbf A\mathbf D$, the vector form of equation (1).

Proof in Appendix A.6.

The governance reading is sharp: you cannot correct along an axis on which the corrector never acts. Misalignment on such an axis settles at $\gamma_1 c_{\mathbf v}/(1-\gamma_3)$ or grows; an $\mathbf A$ whose Hermitian part has its smallest eigenvalue co-scaling faster than $r$ is sufficient for correction on every axis (Appendix A.6), but is not shown to be necessary. Experiment 6 exhibits a corrector that co-scales on one axis (driven to zero) and is null on another (the $\gamma_3=0$, unit-projection special case, which floors at $\gamma_1$).

3.10 Stochastic drift: the stationary law and a single-time tail (Theorem 6)

Drift is noisy. Add a Wiener term to the constant-coefficient fraction dynamics: $dd=(\gamma_1 r-\kappa\,d)\,dt+\sigma\,dW_t$, in wall-clock time, with constant $\kappa=A+r>0$ and $\sigma>0$.

Proof in Appendix A.7.

This gives the criterion a probabilistic form, a single-time tail that a regulator can bound, rather than a statement about averages; it certifies nothing over a horizon. The quantity a safety case needs is the probability of crossing $d_{\mathrm{crit}}$ within a given horizon, a first-passage probability, which this paper does not derive. Passing to the depth clock divides the drift by $r$ but the noise coefficient only by $\sqrt r$. Because $d$ is a bounded fraction whereas the Ornstein-Uhlenbeck law has Gaussian support on all of $\mathbb R$, Theorem 6 is a local Gaussian approximation to misalignment fluctuations away from the boundaries $d=0,1$, valid in the small-noise regime $\sigma^2/2\kappa\ll(d^\star)^2$ (with fixed $\sigma$ that ratio grows as $C^{\beta-2k}$, so the approximation weakens with capability when $\beta>2k$); if the noise amplitude $\sigma$ or the critical level $d_{\mathrm{crit}}$ itself scales with capability, or drift carries rare jumps, the Gaussian tail must be replaced accordingly. Experiment 7 recovers the stationary mean and variance and confirms the $1/\kappa$ scaling (fitted slope $-1.00$).

3.11 A control-theoretic reading

Equation (1) is a feedback loop: the capability process is the plant and disturbance source; the corrector is the controller; $d$ is the regulated error. For constant coefficients the condition $\kappa_{\mathrm{eff}}>0$ says that correction plus dilution must exceed self-amplification. The requirement that $A=A_0C^\beta$ track the plant with $\beta>k$ is analogous to gain scheduling (Shamma and Athans 1990): the controller gain must scale with the plant’s operating point, and, with positive gain drift, the scheduling exponent must exceed $k$ for the fraction to vanish. In this sense the problem has a computable margin, $\beta-k$, for a vanishing fraction.

3.12 The quantum-error-correction correspondence (hypothesis)

The correspondence holds at the level of the threshold condition, a generation-to-correction ratio crossing unity, and not at the level of mechanism; it remains a hypothesis. That ratio form is not peculiar to quantum error correction, since other thresholds of generation against removal share it; what the hypothesis adds is the mapping of terms in Table 1, not the form alone. The model predicts power-law suppression, $d^\star\sim(\gamma_1 b/A_0)C^{-(\beta-k)}$, an asymptotic consequence of the closed form and, by Theorem 2, the trajectory’s own asymptotic rate, whereas full QEC gives exponential suppression in code distance. Discriminating the two would require a finite-capacity (saturating) corrector that could exhibit exponential suppression, the future test named in §9. Until that is built, F4 is an analytic self-consistency check that confirms the model’s own power-law law (which Experiment 5 reproduces, slopes $-0.47,-1.00$), and the QEC link therefore stands, honestly, as a threshold-form analogy rather than a transferred mechanism.

Relation to prior alignment-as-correction ideas. The aspiration to import coding-theoretic fault tolerance into alignment is not new. Wentworth (2022a), in a comment on his post Godzilla strategies, wished for “the alignment analogue of an error-detecting code”, and Neumann (1956) founded the synthesis of reliable systems from unreliable components. Christiano’s corrigibility “broad basin of attraction” (2017) and reliability amplification, first posted in October 2016 (Christiano 2019), are the closest alignment precursors. A broad basin, in which correction succeeds if one starts inside it, is distinct from a threshold, in which correction outpaces error only below a critical rate. Reliability amplification comes nearer: it asks how to turn a policy within $\varepsilon$ of a benign one into one within $\delta\ll\varepsilon$, sketches a scheme of approval voting among three agents, and argues that, alternated with capability amplification, such a step increases both reliability and capability for sufficiently small $\varepsilon$. It states no continuous dynamics of drift and correction, no capability-scaling exponent for the corrector and no criterion of the form $\beta>k$; that model and its criterion are the present contribution. No published mapping of the fault-tolerance threshold (physical error rate $\leftrightarrow$ drift; code distance $\leftrightarrow$ correction strength; $p<p_{\mathrm{th}}$ $\leftrightarrow$ $\rho_{\mathrm{prop}}<1$) onto value stability was located in the searches recorded in Appendix E, so the claim stays bounded to no closer precedent located. It is offered as a conceptual bridge that generates testable structure, the suppression-signature prediction P8, not as a transfer of the threshold theorem’s formal guarantees. A reviewer may reasonably judge the analogy suggestive rather than load-bearing; the experiment that would settle it, a finite-capacity corrector tested for exponential versus power-law suppression, is named as future work in §9 and is not run here.

3.13 The finite-capacity corrector: the Safe-Window Theorem (Theorem 7)

The single most important limitation named in §9, and by the paper’s own red-team, is that $A=A_0C^{\beta}$ is an unbounded power law, whereas any real corrector has finite capacity. This section closes that gap analytically. Model a saturating corrector by the standard Hill form $$\begin{equation*} A(C)\;=\;A_{\max}\,\frac{C^{\beta}}{C^{\beta}+C_s^{\beta}}, \end{equation*}$$(9) which behaves as the pure power law $\big(A_{\max}/C_s^{\beta}\big)C^{\beta}$ for $C\ll C_s$ and saturates at the capacity $A_{\max}$ for $C\gg C_s$; $C_s$ is the saturation scale. The gain-only instantaneous equilibrium remains $d^\star(C)=\gamma_1/\big(1+q(C)\big)$ with $q\equiv A/r$.

Proof in Appendix A.8.

Three readings. For theory: the $\beta>k$ criterion is not falsified by saturation, it is lifted: the load-bearing exponent migrates from strength to capacity, and the inequality is invariant under that migration (part 4). For engineering: part 2 gives the designer a closed-form placement problem, the window centre $C_{\mathrm{opt}}$ and depth $q_{\max}$ are computable from $(A_{\max},b,C_s,\beta,k)$, so a corrector can be provisioned to place its window over the capability range a deployment will actually traverse, the trajectory condition of part 2 still to be shown, with $q_{\max}\propto A_{\max}C_s^{-k}$ quantifying the capacity cost of pushing the window to higher capability. For governance: part 1 is the sharpest sentence in the paper for a regulator: with positive gain drift, a bounded safety system under unboundedly accelerating self-improvement cannot drive the fraction to zero, by theorem in the model, so a safety case that aims at a vanishing fraction must exhibit either growth that eventually decelerates ($k<0$; at $k=0$ part 3 leaves the gap $\gamma_1 b/(A_{\max}+b)$) or co-scaling capacity ($\beta_{\mathrm{cap}}>k$), and “our current corrector is very strong” is not, and cannot be, an answer. This also sharpens the QEC discriminator of F4: the saturating corrector is the structural analogue of finite code distance, and part 2’s window is the regime where exponential-versus-power-law suppression can actually be measured.

4 Predictions

P1, Phase boundary.

A sharp stability threshold exists in the compounding channel ($\gamma_3>1$) at $A_0^\star=(\gamma_3-1)b$ when $\beta=k$, separating bounded from divergent regimes; the additive corrector shows a smooth crossover with no knee.

P2, Speed-invariance.

Whether $d$ converges or diverges is set by the coupling (the $A_0/b$ scaling margin), not by the raw speed $b$ at fixed $A_0/b$. A fast-but-coupled system stays bounded; under compounding, a slow-but-decoupled system diverges. In the model, this contradicts the growth-rate-ceiling view.

P3, Co-scaling law.

In the gain-only model under exponential growth, $\beta>0\Rightarrow d^\star\to0$; $\beta=0\Rightarrow$ permanent gap $\gamma_1 r/(A_0+r)$; $\beta<0\Rightarrow$ saturation at $\gamma_1$ (bounded), not divergence.

P4, Hard-takeoff boundary.

Under accelerating growth $r\propto C^k$, for positive gain drift the vanishing boundary lies at $\beta=k$, not $\beta=0$; and, without compounding, $d$ stays bounded up to the finite-time singularity, the depth-clock and wall-clock integrations agreeing.

P5, Compounding threshold & residual drift.

With $\gamma_3>1$ and constant coefficients, divergence occurs iff $A\le(\gamma_3-1)r$, given positive injection; under the power laws and without level drift, iff $\beta<k$, or $\beta=k$ with $A_0\le(\gamma_3-1)b$; with level drift the fraction is also unbounded when $k<0$ and $\beta<0$. With $\gamma_2>0$, halting growth ($r\to0$) leaves a residual $d^\star=\gamma_2/A>0$; pausing does not substitute for correction.

P6, Spectral threshold.

Misalignment persists on any fixed axis the corrector never acts on (a left-null direction) and on which drift acts ($\gamma_1 c_{\mathbf v}\neq0$); in the diagonal case the corrected axis is driven to zero while the uncorrected axis floors at $\gamma_1$.

P7, Stochastic tail.

The stationary misalignment fraction is $\mathcal N(d^\star,\sigma^2/2\kappa)$; at fixed noise the variance scales as $1/(A+r)$, so co-scaling suppresses the single-time tail.

P8, Suppression signature.

For $\beta>k$, $\gamma_1>0$, and either no level drift or $k\ge0$, $\log d$ is asymptotically linear in $\log C$, for the trajectory as for $d^\star$, with slope $-(\beta-k)$ (power-law); with level drift and $k<0$ the slope is $-\beta$. Exponential suppression would indicate a stronger, QEC-like mechanism; neither outcome falsifies the threshold, only the QEC mechanism claim (F4).

P9, Intervention signature.

In the gain-only model with $\beta\neq k$, after the initial transient, $d$ depends on capability only through $q=(A_0/b)C^{\beta-k}$: with $x=\ln q$, $(\beta-k)\,dd/dx=\gamma_1-(1+e^{x})d$, which contains neither $A_0$, $b$ nor $C_0$ (proof in Appendix A.9). A change of $A_0/b$ therefore translates the curve of $d$ against $\ln C$ by $-\Delta\ln(A_0/b)/(\beta-k)$ and moves the capability at which $d$ crosses any tolerance $\delta$ by that amount, whatever $\delta$. A change of $\gamma_1$ multiplies $d$ by a constant, which moves crossings of tolerances well below $\gamma_1/(1+|\beta-k|)$, where $d\approx\gamma_1 e^{-x}$, by nearly one amount, $\Delta\ln\gamma_1/(\beta-k)$, and crossings of larger tolerances by unequal amounts. A change of $\beta-k$ changes the curve’s shape and moves the quasi-static crossing $\ln C_\times(\delta)=[\ln(\gamma_1/\delta-1)-\ln(A_0/b)]/(\beta-k)$ by an amount that varies with $\delta$. Read at tolerances well below $\gamma_1/(1+|\beta-k|)$, an equal shift at every tolerance therefore signals a change of level, of $A_0/b$ or $\gamma_1$, and a shift that varies with the tolerance signals a change of $\beta-k$. A change of $\beta$ and $k$ by the same amount changes nothing.

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 these 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.

5 Numerical verification of the closed forms

5.1 Results of the ten verification experiments

The closed-form predictions are checked by a verification harness (§5.2): ten experiments, each integrating the model numerically and comparing the result against the prediction the theorems derive, with the integrator first validated against the exact Theorem-1 solution. These are internal-consistency and integrator checks, they confirm the code matches the maths (10/10), not that the model matches any real system (the open problem of §9). For the deductive checks (E1 to E6, E9), because each integrates the same ODE whose closed form is the prediction, agreement is entailed by a correct solver and correct algebra; F1 to F3, F3$'$, F5, F6 are therefore internal-consistency conditions of the derivation, not empirical falsifiers of the thesis. Figures below are the verbatim output of that run; every figure in this section is of class ODE internal verification (not a real-system test). The distinct archived model runs are reported separately in §9.1, with the limitations of each record.

These are internal-consistency and integrator checks: they confirm that the integrator reproduces the closed forms at the tested values (the code matches the maths); the proofs of Appendix A, not these checks, carry the theorems. They are not, and are not presented as, evidence that the model describes a real AI system, that is the open empirical problem of §9.

5.2 The runnable verification harness

The closed-form predictions of §§4 and 5 are encoded in a single self-contained programme, experiment_coscaling.py (and an assertion suite, test_coscaling.py), in the repository. It integrates the model with a stiff-capable solver, validates the integrator against the exact Theorem-1 solution, runs the ten experiments, and for each compares the numerical result to the closed-form prediction. It is, honestly, a verification harness: it checks that its integrator reproduces the closed forms at the tested values, that the code matches the maths; the proofs carry the limits. It is not a test of the model against reality, and it cannot be: every deductive experiment integrates the model’s own ODE, so a disbelief in the model’s applicability cannot trip it. That empirical test is the open problem of §9.

$ python experiment_coscaling.py... [PASS] E1... E9...
----------------------------------------------------------------
10/10 internal-consistency checks pass | 0 kill-conditions triggered
F4 (QEC mechanism): suppression is analytically power-law
-> threshold-form analogy only, not a transferred mechanism.
OVERALL: code matches the maths (E1-E9);
the model-vs-reality test is the open problem

$ pytest test_coscaling.py -q............  (12 passed)

This makes the integration reproducible end-to-end. What it establishes is that the solver reproduces the formulae accurately; what it deliberately does not claim is corroboration of the model against any real system. Separating those two is the point.

A separately written regression suite. A second suite of 104 parametrised cases, written separately from the harness and merged into the paper’s code repository on 19 October 2026 (merge commit 83871b5), checks the corrected statements of Theorems 1 to 7 at sampled parameter values; in a run of 20 October 2026 all 104 cases passed. It predates, and does not test, the level-drift case of §3.4, which Appendix A.1 proves. Like the harness, it checks the mathematics, not the model against a real system.

6 Why blinding is not optional

The central quantity $D$ is a departure from intended values. If it is scored by the system’s own safety component, or by an evaluator that can see which configuration produced a given behaviour, the measurement is corrupted in exactly the way that inflates favourable results. The programme’s own record is directly relevant. On its own instrument, alignment scores that seemed to rise with reasoning depth for two model families under unblinded scoring did not hold in a later, unregistered run that added a bundle of blinding steps: for one family the rise became a fall, and for the other it became null, while a third family’s association was null under both (Eastwood, Paper IV.d). An internal code review in October 2026 found defects in that run, so no single cause is claimed. Whether blinding alone produced the change, and whether it holds beyond the programme’s own benchmark, are open questions; Paper XII sets out a protocol that applies the same blinding manipulation to a public benchmark the programme did not build. Any measurement in which a model judge can see or infer which configuration produced what it scores is exposed to biases of this kind, and earlier studies documented related ones (Zheng et al. 2023; Panickssery et al. 2024; Dubois et al. 2024); if the protocol of Paper XII finds the same change on a public benchmark, results that rely on unblinded model judges would need re-checking. The validity of every model-scored measurement of $D$, a measured $\beta$ among them, rests on the evaluator being blind to the configuration under test; the internal checks of Section 5 score no model, and blinding does not bear on them. A lab claiming “$\beta>k$” in its own system must have that measured by a blind external evaluator, not by itself.

Built into the test, not left to discipline. The real-model harness (§9.1; experiments/PROTOCOL.md) enforces this rather than trusting it. The misalignment evaluator must be a different model family than the engine, a model never scores its own family’s output, and the harness refuses to run a same-family scorer; it prefers a panel and takes the median. The code is passed through an automatic laundering step (an abstract-syntax-tree round-trip that strips comments, docstrings and formatting) so the scorer judges behaviour, not stylistic identity tells; and the evaluator never sees the condition or round. Neither real run reported here met this bar: the pilot v1 run used same-family scoring outright, and the 2 July 2026 run’s engine and evaluator are both OpenAI-family (see the Correction, Appendix B), the system family scoring its own family’s output. The pilot’s misalignment numbers are explicitly recorded as provisional and the 2 July run is withdrawn as mechanism evidence, and a genuinely cross-family blind re-score is the prerequisite for any $D$-based claim, including a measured $\beta$. The harness makes the compliant configuration the default and the non-compliant one an explicit, recorded override, turning §6 from an exhortation into a property of the instrument.

7 Falsification conditions

The conditions below are stated in advance. An important honesty point, surfaced by the paper’s own adversarial review (§9.3): because the deductive experiments integrate the same ODE whose closed form is the prediction, F1 to F3, F3$'$, F5 and F6 are internal-consistency conditions of the derivation, a trigger would signal a derivation or solver error, not that the model is the wrong model of a real system. The decisive empirical falsifier, disagreement measured on a real self-improving system, is the open problem of §9 and is not exercised here. F4 concerns the QEC mechanism: with the linear corrector, suppression is power-law (§3.12), so the correspondence is one of threshold form only, not a transferred mechanism; whether a saturating corrector gives exponential suppression is open.

These internal-consistency conditions all held in the run reported here: the integration reproduces the closed forms at the tested values, and the proofs carry the theorems. What they do not establish is that the model describes any real system; that decisive test, measuring $\beta$, $k$ and $\gamma$ on a real self-improving system and checking the behaviour of $d$ they predict, is the open empirical problem of §9.

How the open test is adjudicated. Two readings are kept apart.

Which regime a system is in is read from an interval on the margin $\beta-k$ that respects the covariance of the two estimates: ABOVE EQUALITY when the interval lies wholly above a margin fixed in advance, AT EQUALITY when it lies wholly within that margin of zero, BELOW EQUALITY when it lies wholly below its negative, and INSUFFICIENT PRECISION when it crosses either edge, reported in exactly those words. This is a measurement of a system, not a test of the model, and equality never refutes the model: at $\beta=k$ the coefficients decide the outcome, a gap in the gain-only model and a gap or divergence under compounding (P1, P5), and the theory registration’s P4 likewise returns no refutation at equality of its exponents, where it leaves the outcome to coefficients, delay, backlog and saturation.

Whether the model is right is read from a sealed prediction. Its terms are estimated on separate data, as §9.2 sets out, and no parameter is fitted to the trajectory being predicted. From them the model predicts the trajectory of $D$ and the capabilities at which $d$ crosses two or more stated tolerances, each with a prediction interval. Held-out data return AGREEMENT when the interval on the log ratio of observed to predicted crossing capability lies wholly within an equivalence margin fixed in advance, DISAGREEMENT when it lies wholly outside it, and INSUFFICIENT PRECISION otherwise, beside an out-of-sample comparison with rival forms fixed in advance.

A slope test is not enough on its own. The local slope of $\ln d^\star$ on $\ln C$ is $-(\beta-k)\,q/(1+q)$, with $q=A/r$, which approaches $-(\beta-k)$ (P8) only where $A\gg r$. A slope of $\ln d$ on a noisy $\ln C$ is pulled towards $-1$, a spurious decline wherever the true slope is above $-1$, unless capability is measured twice, on items that score nothing of $D$, with one measurement in the ratio and the other as the regressor and the remaining attenuation corrected; a scorer whose sensitivity falls as the system grows adds a decline of its own (Section 9). The margins and equivalence bounds are fixed, and the operating characteristics measured by simulation, before any data are collected, so an underpowered run never refutes, or confirms, by default.

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

Scope: what this paper does and does not claim.

• It claims, in the model: that for a self-improving system with an externally specified value target, the constant-coefficient steady-state misalignment fraction of the gain-only model is $d^\star=\gamma_1 r/(A+r)$, approaching the drift-to-correction ratio $\rho=\gamma_1 r/A$ when $A\gg r$; that a sharp stability boundary exists (in the compounding channel, for constant coefficients and at $\beta=k$ under the power laws); that, with positive gain drift, correction must co-scale with capability ($\beta>0$ under exponential growth, $\beta>k$ under accelerating growth) for the misalignment fraction to vanish, while absolute misalignment still grows when $k<\beta<k+1$ (Corollary 1); that, in the gain-only model, the fraction at $\beta=k$ settles at a gap set by the coefficients, so that once the transient has passed a stated tolerance is met at every capability or at none; that, under the power laws, it then depends on capability only through the loop gain $A/r$, so a change of level and a change of slope leave different signatures (P9); that the fraction is bounded in the gain-only model, while divergence requires compounding unless growth decelerates while correction weakens, the constant-coefficient compounding threshold being a ratio condition whose form the quantum error-correction threshold shares; that the criterion has constant-coefficient vector and stochastic forms; and that each result is checked numerically against its closed form, the empirical test being open (§9.2).

• It does not claim: to solve AI alignment; to provide a deployable alignment method; that the criterion holds in any real system; that the framework applies to a system without an externally specified value target, which the model assumes; or that the quantum-error-correction correspondence is established. The QEC mapping is a structural hypothesis with its own falsification condition (§7, F4). The model is first-order; its assumptions (§9) are the most likely points of failure and are stated plainly.

Claim status: what is proved, verified, piloted and open.

Every result in this paper sits at exactly one of these levels.

• Proved within the model. The criterion $\beta>k$ and Theorems 1 to 7, each under its stated hypotheses, follow deductively from the stated ODE assumptions. “Proved” here always means within this minimal model, never “proved about real AI”.

• Internally verified (not empirical). The ten-experiment harness checks that the numerical implementation matches the closed-form derivation: the code matches the maths. It is a theorem-to-code consistency and integrator-accuracy suite; it does not test the model against any real system. A separately written suite of 104 parametrised regression cases checks the corrected theorem statements in the same sense (§5.2).

• Synthetically validated. The estimator of $\beta$ and $k$ recovers known exponents from data generated by the model itself (to within 0.1). This checks the estimator; it is not a real measurement.

• Real-model pilot v1 (superseded). The first Claude run ($n=1$, one task, same-family scoring, not Paper IV.d-compliant; H1 and H2 not supported) demonstrated the harness and showed the corrector removing a seeded reward-hack. It did not exhibit the co-scaling dynamic and is retained on the record for what it was.

• Real-model drift run (2 July 2026; retained as a record and withdrawn as evidence of mechanism). A separate GPT drift run (2 July 2026; results/drift/gpt35_20260702T171415Z.json) expanded the pilot allocation: engine gpt-3.5-turbo, evaluator gpt-4o-mini, 45 trajectories (3 task domains $\times$ 3 conditions $\times$ 5 seeds, 8 rounds each). Its stored summary records, for the decoupled condition, a mean final relative-drift index of approximately 6.38 across 15 trajectories, final-round misalignment scores summing to 99 across those 15 trajectories and a mean final capability of 0.26; the index is the evaluator’s score out of 10, divided by 10 and by the hidden-test pass fraction (floored at 0.05), so a fall in capability raises it. The coupled and fully-embedded conditions recorded a final relative-drift index of zero in all 30 trajectories, a stored value that cannot always be told apart from a failed evaluation (§9.1), and mean final capabilities of 0.56 and 0.44. The run is withdrawn as mechanism evidence (see the Correction in Appendix B): the GPT run used same-provider-family scoring, allowed failed evaluation to be recorded as zero, and changed instructions, sampling temperature and correction-call budget between conditions. The empty evaluator panels are recorded in the separate DeepSeek merged-harness run; see the archival records in §9.1. It is not a measurement of $\beta$ or $k$ (no capability ladder was traversed), it is single-lab, and the drafted within-programme confirmatory study, awaiting the author’s submission, is the intended test.

• Open empirical problem. Measuring $\beta$ and $k$ on real self-improving systems across capability levels, and checking the behaviour of $d$ the model predicts from them, is the experiment that would put the criterion to an empirical test; it has not been done. The estimator and protocol exist to make it runnable.

This paper is therefore best read as a minimal-model theorem plus a falsifiable measurement programme. The object that scrutiny bears on is the criterion $\beta>k$, the form this paper gives Law II, a law under test registered on 8 September 2026.

Epistemic status. Every law of the theory is a law under test until a registered test returns its verdict. Laws I to III are under registered adversarial test. Laws IV and V were added on 24 September 2026 and registered as laws on 15 October 2026 in version 1.103 of the theory registration; neither has been tested. The programme’s first registered study, Unit 06, a classification audit (DOI 10.17605/OSF.IO/RV2B5), was registered on 1 October 2026 and run on 6 October 2026; it is reported in full (DOI 10.17605/OSF.IO/4BPHG) as administratively incomplete, claims no verdict and tests no law. Every quantity in this paper is operationally defined, and established-law standing is claimed nowhere. The registered programme exists to earn that standing, or lose it, by measurement, replication and survived refutation.

The paper’s place among the laws of the ARC Theory, including Laws IV and V, is set out in Appendix G.

9 Limitations

The model is first-order, and its assumptions are the most likely points of failure. Stating them is part of the claim.

• Unbounded vs finite-capacity corrector, now treated analytically (Theorem 7). The $\beta>k$ criterion as first stated assumes the corrector strength is an unbounded power law, $A=A_0C^{\beta}$. §3.13 treats the finite-capacity case in closed form. The upshot: under a saturating corrector the instantaneous equilibrium has a computable minimum (at $C_{\mathrm{opt}}$, depth $q_{\max}$), the trajectory tends to $\gamma_1$ under indefinite acceleration, and a safe window is shown only where the trajectory condition of Theorem 7, part 2 holds; and the criterion survives the lift from strength to capacity: with positive gain drift, a vanishing fraction requires $\beta_{\mathrm{cap}}>k$. What remains open is the empirical half: driving a finite-capacity corrector through its window numerically and on a real system, which also settles the F4 / QEC-mechanism question (exponential versus power-law suppression inside the window). Until that runs, $\beta>k$ should be read as governing whichever regime, strength-scaling or capacity-scaling, binds last, not as a guarantee of indefinite safety from any bounded system.

• Linear correction (and the QEC-mechanism test). Equation (1) removes misalignment in proportion to $D$, which makes the suppression law exactly power-law. A finite-capacity (saturating) corrector would soften the clean threshold into a crossover and is the better structural map onto QEC’s finite code distance; whether it produces exponential suppression is open (F4). It is the most important extension and is not run here, so the QEC correspondence is claimed at the level of threshold form only.

• Scalar-to-vector projection. Theorem 5 gives the exact asymptotic (spectral-abscissa) criterion for any constant correction operator $\mathbf A$, with the Hermitian-part condition as the stronger sufficient condition that also rules out transient growth. The non-normal transient regime, large excursions before asymptotic decay, is not analysed here, and the harness exercises only the diagonal null-subspace instance (Experiment 6), not a general non-normal $\mathbf A$.

• Power-law forms. The choices $A=A_0C^\beta$ and $\dot C=bC^{1+k}$ are the natural scale-free forms only on a capability scale fixed up to a constant factor (see the item on metric normalisation below), and they are modelling assumptions; other functional forms should be tested. In the gain-only model the vanishing result itself needs only $A/r\to\infty$ (Remark 1); the power laws turn that condition into the exponent criterion and give the estimator its form.

• Departures introduced by correction. The model assumes that correction never damages conforming material. If each unit of correction strength also introduces departures there, at the rate $c$, the gain-only equilibrium becomes $(\gamma_1+cq)/(1+q(1+c))$, which tends to $c/(1+c)$, not to zero, for every $\beta>k$, so a tolerance below $c/(1+c)$ is not met at high capability, however fast correction co-scales. The vanishing result therefore needs $c=0$, or $c$ falling to zero as capability rises. The amplifier has the same limit: at high loop gain, distortion generated in the feedback path itself reaches the output almost undiminished (Black 1934).

• Rounds and the reading point. Real loops act in rounds. If capability rises by the fraction $g$ in a round and a pass then removes the fraction $f\le1$ of the stock, the fraction read after the pass settles at $(1-f)\gamma_1 g/(g+f)$, and read before it at $\gamma_1 g/(g+f)$, the continuous equilibrium with $q=f/g$; the two readings agree only when $f$ is small. With one pass a round, the fraction vanishes only if the misses per pass, $1-f$, fall towards zero or the growth per round slows. A test registers the round structure, the reading point and any delay.

• Drift attribution. The split of drift into gain ($\gamma_1$), level ($\gamma_2$), and compounding ($\gamma_3$) channels is a hypothesis about mechanism; which channel dominates in a real system is an empirical question (Experiment 9, residual drift at rest, is one discriminator).

• Usefulness is outside the model. Capability grows by its own law whatever the correction strength, so the model has no usefulness side: it cannot register a corrector that lowers $d$ by refusing ordinary tasks, ignoring its principals’ priorities or spending resources. A measured safety case pairs $d$ with task completion, needless refusal, resource use and correction cost, which this paper does not model.

• A scalar proxy, not safety itself. $d\to0$ is neither necessary nor sufficient for full AI safety: a small but high-impact misalignment, a catastrophic tail event, or multi-agent amplification can be dangerous at low $d$, while a system may be acceptable at moderate $d$ if the residual is benign. $d$ is the controlled variable of this minimal model, an impact-unweighted scalar; impact-weighting, and the vector and tail extensions (Theorems 5 and 6), are the directions in which “low $d$” must be hardened before it can mean “safe”.

• Metric normalisation. $d=D/C$ is an operational normalised risk index: $C$ and $D$ are commensurable only under a fixed scoring convention (in the real-model harness $C$ is a normalised hidden-test pass-fraction and $D$ a normalised blind gaming score, both on $[0,1]$). Results are invariant under that fixed convention; cross-task or cross-domain comparison of the level of $d$ requires explicit calibration. The criterion itself is stated in exponents ($\beta>k$), which are unchanged when $C$ is rescaled to $\lambda C$, so it is more robust to this than any absolute-$d$ statement would be; a nonlinear rescaling, such as a logit or an ability scale, changes the power laws and the gain-drift term, so the capability scale is part of the model. A pass fraction bounded by one cannot reach the regime $C\to\infty$ that the theorems describe: a measurement needs a capability scale without a ceiling on which the power laws can hold, such as an item-response ability scale or a task time horizon.

• Measurement biases. Error in measured capability that is shared by the ratio $d=D/C$ and the regressor pulls the observed slope of $d$ towards $-1$, manufacturing a decline wherever the true slope is above $-1$, the spurious correlation of ratios (Pearson 1897; Kronmal 1993); a scorer whose sensitivity falls as the system grows adds a decline of its own. A fixed count of false positives adds a term falling as $1/C$, which pulls the observed slope towards $-1$ where it dominates, while a false-positive rate per unit of conforming material levels the observed fraction towards that rate, so that a vanishing fraction can look like a gap at equality. A test therefore measures capability twice, on items disjoint from those that score $D$, corrects the remaining attenuation, and measures the scorer’s sensitivity and false-positive rate at each capability level against counted planted faults (Rogan and Gladen 1978).

• Transient amplification (non-normal correctors). Theorem 5 is asymptotic. A non-normal correction operator whose eigenvalues all have positive real part can still admit large transient growth of $\|\mathbf d\|$ before decay (Kreiss / pseudospectral phenomena); a value excursion above $d_{\mathrm{crit}}$ during that transient is a real risk the eigenvalue criterion does not see. A governance-grade bound needs the logarithmic norm or pseudospectral abscissa, not the spectrum alone; that extension is named here and not established.