The Coupled Co-Scaling Law

The criterion $\rho_{\mathrm{prop}}<1$ shares the ratio-crossing-unity form of the QEC sub-threshold condition $plinear 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.

The governance reading is sharp: you cannot correct what you do not measure. The correction operator must be positive-definite on the entire value space, with smallest eigenvalue co-scaling - a blind spot on any axis is a permanent (or divergent) misalignment on that axis. 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: a tail bound for governance (Theorem 6)

Drift is noisy. Add a Wiener term to the constant-coefficient fraction dynamics: $dd=(\gamma_1 r-\kappa\,d)\,d\tau'+\sigma\,dW$ with $\kappa=A+r$.

This converts the criterion into a probabilistic safety certificate - a quantity a regulator can bound - rather than a statement about averages. 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 $\sigma^2/2\kappa\ll(d^\star)^2$; 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 identity

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. The stability condition $\kappa_{\mathrm{eff}}>0$ is a small-gain / positive-realness condition - the loop gain must exceed the disturbance's self-amplification across the relevant band. The requirement that $A=A_0C^\beta$ track the plant with $\beta\ge k$ is precisely gain-scheduling [Shamma & Athans 1990]: the controller gain must scale with the plant's operating point, and the scheduling exponent must satisfy $\beta\ge k$. Recursive-self-improvement alignment is, in this exact sense, an adaptive-control problem with a computable stability margin $\beta-k$.

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. The model predicts power-law suppression $d^\star\propto C^{-(\beta-k)}$, which is an algebraic identity of the linear model, 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 §8. 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 (2022) explicitly wished for "the alignment analogue of an error-detecting code", and von Neumann (1956) founded the synthesis of reliable systems from unreliable components. Christiano's corrigibility "broad basin of attraction" (2017) and reliability amplification (2019) are the closest alignment precursors. But a broad basin - correction succeeds if one starts inside it - is distinct from a threshold theorem, in which correction outpaces error only below a critical rate; that distinction is the present contribution. To the author's knowledge the specific mapping of the quantum fault-tolerance threshold (physical error rate ↔ drift; code distance ↔ correction strength; $pconceptual 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 §8 and is not run here.

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

The single most important limitation named in §8 - 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

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 steady state remains $d^\star(C)=\gamma_1/\big(1+q(C)\big)$ with $q\equiv A/r$.

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, 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 - a bounded safety system under unboundedly accelerating self-improvement fails eventually by theorem, so a safety case must exhibit either bounded growth ($k\le0$ eventually) 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.

Remark (time-varying exponents). Real systems will not hold $\beta,k$ constant. If both vary along the trajectory, the comparison argument of Theorem 2 applies verbatim in the depth clock with $q(\tau)=\exp\!\big(\int_0^{\tau}(\beta(s)-k(s))\,ds+\mathrm{const}\big)$: the fraction vanishes iff the running integral of the margin $\beta-k$ diverges to $+\infty$, for which $\liminf_{\tau\to\infty}\big(\beta(\tau)-k(\tau)\big)>0$ is sufficient. Transient episodes of $\betacumulative co-scaling margin, not its instantaneous sign.

4. Predictions

5. The discriminating experiments

The closed-form predictions are checked by a verification harness (§11): 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 §8). For the deductive checks (E1-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-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 only non-simulation result - the real-model pilot - is reported separately in §8 and is labelled real-model pilot, with its own provenance caveats.

These are internal-consistency and integrator checks: they confirm the closed-form theorems are correctly derived and numerically reproduced (the code matches the maths). They are not, and are not presented as, evidence that the model describes a real AI system - that is the open empirical problem of §8.

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 prior metascience finding is directly relevant: under unblinded cross-model scoring an alignment-scaling effect appeared positive, while under multi-layer blinding the same effect reversed sign for some systems [Eastwood, Paper IV.d]. An experiment that measures a misalignment fraction without blinding the evaluator can produce not merely the wrong magnitude but the wrong direction. The validity of every result in §5 rests on the evaluator being blind to the configuration under test. A lab claiming "$\beta\ge k$" 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 (§8; 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. The first real run reported here did not meet this bar - engine and evaluator were the same model family, the system scoring its own output - so its misalignment numbers are explicitly recorded as provisional, and a 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 audit (§8): because the deductive experiments integrate the same ODE whose closed form is the prediction, F1-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 §8 and is not exercised here. F4 is the QEC-mechanism downgrade, which (the suppression law being power-law, §3.12) already holds: the correspondence stands as a threshold-form analogy, not a transferred mechanism.

These internal-consistency conditions all held in the run reported here, establishing that the derivation and integration are correct. 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 $\beta\ge k$ - is the open empirical problem of §8.

8. 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 resolved analytically (Theorem 7). The $\beta>k$ criterion as first stated assumes the corrector strength is an unbounded power law, $A=A_0C^{\beta}$. The prior draft could only name the finite-capacity case as an open gap; §3.13 now closes it in closed form. The upshot: a saturating corrector makes safety a transient window (centre $C_{\mathrm{opt}}$, depth $q_{\max}$, both computable) rather than an asymptote, with the fraction re-rising to $\gamma_1$ under indefinite acceleration - and the criterion survives the lift from strength to capacity: indefinite stability requires the capacity exponent to satisfy the same inequality, $\beta_{\mathrm{cap}}>k$. 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 the experiment that would settle the QEC-mechanism question (F4). It is a priority extension and is not run here - which is exactly why the QEC correspondence is claimed only at the threshold-form level.
• Scalar-to-vector projection. Theorem 5 gives the exact asymptotic (spectral-abscissa) criterion for any 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 but are modelling assumptions; other functional forms should be tested.
• 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).
• 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 5-6), 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, not a dimensionless natural constant: $C$ 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 scale-free, so it is more robust to this than any absolute-$d$ statement would be.
• Transient amplification (non-normal correctors). Theorem 5 is asymptotic. A non-normal correction operator with $\alpha(\mathbf M)>0$ 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 - a named extension, stated here, not established here.
• From simulation toward frontier systems. The verification harness uses toy self-modifying dynamics; a positive result there demonstrates the mechanism, not that frontier systems occupy any particular regime. A companion real-model harness (non-simulation) now drives the coupled/decoupled loop with a frontier model, scoring capability by real code execution and misalignment by a separate evaluator that the harness requires to be a different model family (Paper IV.d blinding, §6; experiments/PROTOCOL.md; first run in results/realmodel/). It corroborates the corrector mechanism on a real model - a seeded reward-hack is detected and removed and capability restored - but did not exhibit drift on the task tried, so whether real systems sit below threshold remains the open empirical question. A stronger confirmatory design - three task domains with a capability ladder, a sham-extra-compute control arm, a combined static-plus-blind-panel misalignment score, and matched-pair bootstrap CIs on the coupled-vs-decoupled and sham-vs-coupled final-fraction contrasts - is specified in experiments/PROTOCOL_V2.md and experiments/scripts/realmodel_coscaling_v2.py. (Real-model evidence - provenance: pilot v1 was $n=1$, one task, same-family scorer, IV.d non-compliant; H1 and H2 not supported. The 2 July 2026 drift run - 45 trajectories, three task domains, cross-family evaluator gpt-4o-mini scoring engine gpt-3.5-turbo - supported both H1 and H2: decoupled drifted (mean final fraction 6.38, capability collapsing), coupled and fully-embedded held zero misalignment across all 30 trajectories at higher final capability. That run evidences the mechanism; $\beta/k$ remain unmeasured because no capability ladder was traversed, and the result is single-lab pending pre-registered replication.)
• Estimating $\gamma$ and $A$ on real systems. In simulation they are set by construction. Independently estimating them - especially $\beta$ and $k$ - for a deployed model is the step that would make the criterion operationally useful for governance. This is now operational: the repository ships a runnable estimator (experiments/scripts/estimate_exponents.py) that reads $k$ off the capability curve ($\ln r$ versus $\ln C$) and $\beta$ off the corrector's fractional removal rate ($\ln A$ versus $\ln C$), validated on synthetic trajectories where it recovers known exponents to within $\approx0.1$. What remains open is therefore not the estimation but its input - a real self-improving system that drifts across a range of capability levels, from which the first measured $(\beta,k)$, and thus the first real test of $\beta\ge k$, can be read.

Adversarial audit. This paper was developed under adversarial audit rather than asserted, and the record is in the repository. A prior-art and novelty audit located the closest precedent for each component and narrowed the originality claims accordingly (Appendix C). Independently, a multi-agent red-team attacked the work across five fronts - mathematics, numerics, the QEC correspondence, the safety inference, and priority - raising 24 objections, of which 21 survived independent verification (0 fatal, 13 serious, 8 minor); the full report is committed alongside the harness (results/redteam.md). Every surviving objection was a framing, wording, or edge-case fix; none touched the load-bearing $\beta>k$ result, and this version incorporates them all - the QEC framing softened to a threshold-form analogy, the Theorem 2 bound scoped to the gain-only model, the Theorem 5 criterion corrected to the spectral abscissa, the harness relabelled as a verification (not falsification) artefact, and the level-drift experiment F5 added. The contribution is offered as ambitious and audited, with the boundary of what is claimed made explicit and checkable.

9. Implications for AI safety

If the framework survives its tests, the design implication is concrete and differs from the prevailing reflex.

The lever is coupling, not speed. Slowing capability growth buys time but does not change the verdict; at fixed coupling, a slow decoupled system still diverges (P2), and a frozen capable system still drifts (§3.6). What changes the verdict is ensuring that correction (i) is coupled to the capability process so it cannot be decoupled, and (ii) scales at least as fast as capability accelerates ($\beta\ge k$). On the author's corrected, sub-linear capability-scaling estimates for current frozen models [Eastwood, Paper IX], present systems are nowhere near a super-linear growth regime - which means a growth-rate ceiling was never the binding constraint. The binding constraint is the co-scaling of correction, and it becomes binding precisely in the regime that matters: genuine self-modification.

A measurable governance target. The criterion gives regulators a quantity to instrument rather than a rate to forbid: the correction-to-drift margin $\beta-k$, and the ratio $\rho$. "Does correction co-scale - is $\beta\ge k$?" is sharper and more actionable than "is it growing too fast?" A lab claiming a safely self-improving system should be required to exhibit $\beta\ge k$, measured by a blind external evaluator (§6). Theorem 6 turns this into a bound on the probability of catastrophic excursion, the natural object of a safety case. Without such a demonstration, the word "safely" has no scientific content within the framework presented here.

Continuity with embedded-alignment work. The conclusion that correction must participate in the recursive loop - rather than sit outside it as a fixed external constraint - is the embedded-alignment thesis, here derived as the $\beta>k$ condition rather than asserted. The gated-simulation result in which safety-coupled self-modification preserved both safety and capability while the decoupled variant did not [Eastwood, Paper VIII] is the predicted behaviour of a coupled ($\beta>0$) versus fixed ($\beta=0$) corrector.

Magnitude of the claim - stated conditionally. If the criterion is borne out empirically - if real self-improving systems are governed by the same drift-versus-correction balance and the same $\beta>k$ margin - the consequences for the field are large, and it is worth stating them plainly while being equally plain that they are conditional. (i) The central safety question changes from "how fast is capability growing, and can we pause it?" to "does correction co-scale - is $\beta\ge k$?": a measurable margin rather than a rate to forbid. (ii) The most feared scenario, a finite-time intelligence explosion, ceases to be intrinsically uncontrollable - the modelled misalignment fraction is controllable iff $\beta>k$, and its speed does not change that verdict. (iii) Governance acquires a quantity to instrument and a tail bound to certify (Theorem 6) in place of an unenforceable speed limit. None of this is yet established. It rests on a minimal model, demonstrated only in simulation, with the empirical measurement of $\beta$, $k$ and $\gamma$ on real systems unsolved (§8). The paper's claim is therefore not that AI is safe or unsafe, but that the right variable to measure and govern is the co-scaling margin $\beta-k$ - and that this variable is well-defined, falsifiable, and, if it holds, decisive. That is the magnitude: not a proof about reality, but a precise, testable redirection of the question on which the field's central fear turns.

Relation to the wider programme - one ladder of laws. This paper is the safety keystone of a larger argument, not a self-standing result; stating how it sits among its companions is part of the claim. The programme's results are of several kinds, and the distinction matters: dynamical laws (this paper's $\beta>k$; Paper III's $\alpha_{\text{align}}\approx0$), a form meta-law (the Cauchy three-form constraint), a measurement law (Paper IV.d - that an unblinded alignment score is not blinding-invariant), and mechanism/architecture findings (Papers V, VI, VIII) that show how the safe regime is reached. The programme is a single chain: the scaling-law cluster establishes how recursive systems grow and supplies the power-law forms taken as given here; Paper II confirms capability does scale (currently sub-linearly); Paper III identifies the danger that external safety does not co-scale; this paper supplies the criterion for when it can; Paper VIII shows the coupled design beats the decoupled one; and Paper IV.d supplies the blinding discipline without which none of the misalignment measurements can be trusted.

Read as one ladder: the scaling-law work says how systems grow, Paper III says why external safety fails to keep up, this paper says what must hold for it to keep up ($\beta>k$), Paper VIII shows it can, and Paper IV.d says how to measure it honestly. The $\beta>k$ criterion is the rung that turns the programme's scaling-law backbone into a safety criterion - and, conversely, this paper inherits its functional forms and its empirical license from that backbone rather than positing them in isolation.

10. The ARC/Eden programme, accumulated

This is the culminating paper of the ARC/Eden programme, and it is written to stand as the programme's synthesis - superseding the earlier roadmap paper [Eastwood, Paper IX] by integrating the whole into a single safety result. The preceding papers are individually published and time-stamped (OSF: 10.17605/OSF.IO/6C5XB); this section gathers their laws and findings, states what each contributes to the larger picture, and is scrupulous about the one distinction on which an honest synthesis turns: priority (when a claim was first set out) versus novelty (whether it is new to the literature). The two are not the same, and conflating them is how good programmes lose credibility.

The bigger picture, in one paragraph. One mechanism runs through every paper: recursive amplification - a process that acts on its own output. The scaling-law work (Foundational, Origin, VII, and the capability measurements of Papers I-II) is the claim that recursive amplification shapes how capability grows, constraining scaling to a small family of functional forms. The alignment work (Papers III, IV.a-d) is the claim that this creates a safety problem: when correction sits outside the recursive loop it does not co-scale, the misalignment fraction is left ungoverned, and - measured without blinding - even the sign of the effect cannot be trusted. The architecture work (V, VI, VIII) shows, in simulation and in pilot model studies, that putting correction inside the loop preserves both safety and capability where an external constraint does not. This paper supplies the missing quantitative law that ties the three together: a self-improving system is alignment-stable iff correction out-scales drift, $\beta>k$. That is the magnitude of the claim - not that any system is safe, but that the field's central fear (a fast, recursive takeoff) is governed by a single measurable margin, $\beta-k$, rather than by speed. It is stated here as it is throughout: conditional, pending the empirical measurement of $\beta$ and $k$ on a real self-improving system (§8).

The complete ledger. Every result in the programme, its first-published date (priority), its honest standing - informed by adversarial prior-art audits committed alongside this paper - and its role in the whole:

Priority, stated plainly - and bounded. The conceptual thesis of this programme was set out in the author's book Infinite Architects (copyright deposited 8 December 2024; published 2 January 2026; ISBN 978-1806056200), and each subsequent paper is independently time-stamped on OSF. That establishes the author's priority - the date of articulation - and it is real. It does not, by itself, establish novelty against the wider scientific literature, and the audits committed with this paper are explicit about where the two diverge: the feedback exponent $\alpha=1/(1-\beta)$, the $d/(d+1)$ allometric ladder, the existence of an optimal depth $R^\star$, and the "functional-equation fixes the scaling form" principle all have specific, citable precedents (Keynes; Banavar & West 1999; Qi 2025; Luce 1959; Frank 2009/16). The defensible novel contributions of the programme are narrower and, stated honestly, stronger for being precise: (i) the $\beta>k$ criterion as a compact corrigibility law and the Hard-Takeoff Depth-Regularity Theorem (this paper); (ii) the Cauchy unification of the independent allometric derivations (Origin/VII); (iii) the blinding-reversal measurement law for AI alignment evaluation (IV.d); and (iv) the embedded-vs-external safety demonstrations (VI, VIII). The recommended next-version revisions for the published priors - adding the missing citations and re-scoping their novelty assertions - are recorded in the audit files; the corrected record is set here, in the capstone, and should be carried into those papers when they are re-versioned on OSF.

Honest assessment (what this synthesis claims, and does not). Established: the mathematics of $\beta>k$ (Theorems 1-6) and its internal-consistency verification; the IV.d measurement law; the simulation-level superiority of coupled over decoupled correction (VI, VIII). Suggestive but not settled: the capability-scaling exponents (architecture-dependent; II), the Stewardship-Gene intervention (unblinded; V), and the three-form empirical fit (exploratory, author-classified operators; VII). Open: the measurement of $\beta$, $k$ and $\gamma$ on a real drifting self-improving system - the single result that would convert the keystone from a proved criterion into a confirmed law (the estimator for it is built and validated, §8). The programme's magnitude, then, is the magnitude of a coherent, falsifiable framework with a small number of genuinely novel keystones - not a stack of a dozen independent new laws. That is the honest claim, and it is the one worth defending.

11. The runnable verification harness

The closed-form predictions of §4-5 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 certifies that the theorems are correctly derived and correctly integrated - that the code matches the maths. 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 §8.

$ 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 derivation and the integrator reproducible end-to-end. What it establishes is that the formulae are right and the solver is accurate; what it deliberately does not claim is corroboration of the model against any real system. Separating those two is the point.

12. Conclusion

The danger of recursive self-improvement is real, but the standard model of that danger - a rate that must be capped - locates the risk in the wrong variable. A minimal model shows that the stability of a self-improving system is set by the ratio of value-drift to correction, $\rho=\gamma r/A$, and by whether correction co-scales with capability: $\beta>0$ under exponential growth, sharpening to $\beta>k$ under accelerating growth. The misalignment fraction never diverges in the gain-only model - it saturates at the drift coefficient, correcting the prior draft - while genuine divergence lives in a compounding channel whose threshold $\rho_{\mathrm{prop}}<1$ shares the threshold form of the quantum error-correction criterion (the suppression law being power-law, the correspondence is offered as a hypothesis, not a transferred mechanism). The sharpest consequence is the Hard-Takeoff Depth-Regularity Theorem: a finite-time intelligence explosion is alignment-stable iff $\beta>k$, and its speed does not change that verdict. The criterion survives in vector and stochastic forms, gives governance a measurable target and a tail bound, and is accompanied by a verification harness that checks the closed-form predictions are correctly derived and integrated. The decisive test - whether real self-improving systems satisfy the criterion - is the stated next step, not a claim made here.

This is a smaller claim than the cosmological framing the author's programme once pursued, and deliberately so. It refers only to systems with an externally specified value target, makes no assertion about the universe, and treats even its most striking correspondence - with quantum error correction - as a hypothesis to be tested rather than a truth to be announced. The recursive-stability intuition that motivated the broader programme, including Infinite Architects, finds here its measurable, falsifiable form: stable recursion requires correction that scales with the amplification. The next step is not to extend that claim outward but to run, on real self-modifying systems, the experiment that could refute it.

Appendix A. The fraction change of variable

The reduction underlying every theorem is the change of variable $d=D/C$. Differentiating and substituting (1):

The dilution term $-r\,d$ (from $C$ itself growing) is what bounds the additive fraction: it adds $+r$ to the decay coefficient, guaranteeing $\kappa_{\mathrm{eff}}\ge r>0$ when $\gamma_3\le1$. Only the compounding channel $\gamma_3>1$ can overcome dilution and produce divergence - the formal reason the additive model saturates rather than blows up.

Appendix B. Parameter settings for the reported run

All figures use $C_0=1$, $d_0=0.05$, $\gamma_1=0.05$. Experiment-specific settings: E1 $b=1$, $\gamma_3\in\{0,3\}$, $\lambda\in[0.2,4]$; E2 $\gamma_3=3$, $A_0/b\in\{1,3\}$, $b\in\{0.5,5\}$; E3 $A_0=0.08$, $b=0.5$, $\beta\in\{-0.5,0,0.5,1\}$; E4 $A_0=0.08$, $b=0.02$, $(k,\beta)\in\{0,0.5,1\}\times\{0.25,0.75,1.5\}$; E4b $k=1$, $\beta=1.5$, $b=0.02$ ($t^\star=50$); E5 $\gamma_3=4$, $b=1$; E6 monitored $\beta=1$, blind $A_0=0$; E7 $\sigma=0.02$, OU ensemble of 2.5-4k paths; E8 closed-form case $k=0$, $\beta=0$. Random seed fixed (7) for determinism. Full settings are in the harness source.

Appendix C. Novelty and prior-art ledger

This appendix consolidates the adversarial prior-art audit underlying the positioning in §2 and §3.12, so the boundary between what is established and what is claimed original is explicit and auditable in one place. This paper deliberately invokes no blanket-originality framing; each component is placed against its closest located prior art below. "Defensibly new" means "no closer prior art located," not "correct" or "significant" - novelty and validity are independent.

What this paper asserts as new - and only this: (i) the $\beta>k$ stability criterion and the single-parameter $\rho$ framing; and (ii) the explicit QEC threshold mapping (as a hypothesis with its own falsifier). The programme's strongest empirical novelty - sign-reversal of alignment-scaling effects under multi-layer blinding - is a separate, companion result [Eastwood, Paper IV.d] and is not claimed here.

The prior capability equation $U=I\times R^{\alpha}$ (whose single-model unblinded measurement $\alpha\approx 2.24$ was retracted and corrected to approximately 0.49 under blinding; the equation itself and the ARC Bound $\alpha\le 2$ are not retracted and remain live hypotheses) is not used anywhere in this paper's argument; it appears only in §1 as superseded framing-context. Caveat: forum/blog/preprint indexing is imperfect, so the "defensibly new" verdicts carry an estimated 10-15% residual risk that a closer, unindexed precedent exists; absence of evidence is not proof of absence.

Appendix D. Plain-language glossary

Every technical term in this paper, in one place and in ordinary words, for the non-specialist reader.

References

Declaration of AI-Assisted Human Authorship

The author of this work is Michael Darius Eastwood, a human being. Every core concept, hypothesis, experimental design, claim and conclusion in this paper originates from human ideation. No part of this manuscript is a wholly generated artificial-intelligence output.

Artificial-intelligence tools (Anthropic's Claude family and other large-language-model assistants) were used as instruments under continuous human direction, in the way a word processor, calculator or research assistant is used: for editing and prose refinement, literature search and summarisation (manually verified against primary sources), document structure, formatting, brainstorming against author-defined questions, and the acceleration of drafting to author-defined outlines and instructions. All selection, coordination, arrangement and final editorial judgment are the author's. Every substantive output was reviewed, tested or verified by the author, who takes full responsibility for the accuracy and integrity of the final text. The tools increased the speed of the work; they were never relied upon as its source.

United Kingdom. In accordance with the Copyright, Designs and Patents Act 1988, the author undertook the arrangements necessary for the creation of this work and asserts full human authorship and moral rights: this is a human-authored work produced with computer assistance, not a computer-generated work. United States. Consistent with United States Copyright Office guidance on works containing AI-generated material, the human contribution (conception, selection, coordination, arrangement and final expression) is asserted as sufficient for full human authorship. Inventions. Any novel technical contribution described in this work was conceived by the human author; no artificial-intelligence system autonomously invented anything presented here.