Title: The ARC Co-Scaling Law (Paper X) · Michael Darius Eastwood Author: Michael Darius Eastwood Publication date: 2026-07-03 Version: v1.8 Revised: 2026-08-27 OSF DOI: 10.17605/OSF.IO/BSE2Q Canonical URL: https://www.michaeldariuseastwood.com/research/papers/paper-x-coupled-coscaling-correction.html Abstract -------- This paper proves a theorem about a minimal dynamical model. The verification harness checks theorem-to-code consistency only; no claim is made that current frontier systems obey the model. The empirical contribution is a proposed blind protocol for measuring whether they do. A widely held intuition holds that recursive self-improvement is dangerous because capability can grow explosively, and that safety therefore depends on limiting the rate of growth. Using a minimal model of a self-modifying system, capability $C$, a blind-scored misalignment magnitude $D$, and the misalignment fraction $d=D/C$, I show the rate is the wrong control variable. The steady-state misalignment fraction is $d^\star=\gamma_1 r/(A+r)$, which reduces to the drift-to-correction ratio $\rho=\gamma_1 r/A$ in the regime $A\gg r$; the long-run fate is governed by the relationship between two scaling exponents, not by the growth rate. Under exponential growth the stability condition is $\beta>0$ (correction co-scales with capability); under accelerating growth, where the specific growth rate itself rises as $r\propto C^{k}$, the condition sharpens to $\beta>k$, correction must out-scale not the growth rate but its acceleration. I prove an exact transient solution (Theorem 1), global boundedness that corrects an over-claim in the prior draft, the misalignment fraction never diverges to infinity but, in the gain-only model ($\gamma_2=\gamma_3=0$), saturates at the gain-drift coefficient $\gamma_1$ (Theorem 2), and a Hard-Takeoff Depth-Regularity Theorem (Theorem 3): when capability reaches infinity in finite wall-clock time, re-expressing the dynamics in the natural clock of self-improvement depth $\tau=\ln C$ renders them regular, and the verdict is set by $\mathrm{sign}(\beta-k)$ independently of the speed and of the finiteness of the singularity time. I locate genuine divergence in a distinct compounding drift channel whose threshold $\rho_{\mathrm{prop}}=(\gamma_3-1)r/A<1$ shares the form of the quantum error-correction sub-threshold condition $pk$. A real-model drift run (gpt-3.5-turbo engine, gpt-4o-mini evaluator, 45 trajectories over three task domains) produced directionally consistent numbers, the decoupled configuration drifted while coupled correction held misalignment at zero across all 30 trajectories at higher final capability, but the run is withdrawn as mechanism evidence (see the Correction above): both models are OpenAI-family, so the scoring was not cross-family, and the merged-harness run contained empty evaluator panels. $\beta/k$ remains unmeasured. Ten experiments, packaged as a verification harness, check that these closed-form predictions are correctly derived and numerically reproduced: internal-consistency and integrator checks (the code matches the maths), not a test of the model against real systems, which remains the open empirical problem. The claim is deliberately narrow: it concerns operationally measurable systems, makes no cosmological assertion, and treats the quantum-error-correction correspondence itself as a falsifiable hypothesis. The criterion certifies that correction keeps pace with capability; it does not certify that the correction target itself is well specified, and it is therefore not quotable as an alignment certificate on its own. Key findings (quotable) ----------------------- - Using a minimal model of a self-modifying system, capability $C$, a blind-scored misalignment magnitude $D$, and the misalignment fraction $d=D/C$, I show the rate is the wrong control variable. - only in the large-correction limit $A\gg r$ (where $d^\star=\gamma_1 r/(A+r)\to\rho$), not in general; (ii) the sharpened stability criterion $\beta>k$ under accelerating self-improvement; (iii) the Hard-Takeoff Depth-Regularity Theorem , which proves that a finite-time intelligence explosion is alignment-stable iff $\beta>k$ and that the explosion's speed does not change the asymptotic verdict; (iv) the identification of the compounding drift channel as the locus of genuine divergence, with a threshold $\rho_{\mathrm{prop}} Finite-Capacity Safe-Window Theorem (§3.13), which solves the saturating-corrector case in closed form and shows the criterion is invariant under the lift from correction strength to correction capacity. - Its sharpest consequence speaks to the central fear of the field: a "hard takeoff", even a genuine finite-time intelligence explosion, drives the modelled misalignment fraction to zero if and only if $\beta>k$, and the speed of the explosion does not change that asymptotic verdict. Priority claims relevant to this paper -------------------------------------- - PC-036 (2026-02-13): Application of the bare alpha = d/(d+1) scaling exponent to recursive AI capability was first published in the Foundational paper on 13 February 2026. The exponent form itself is prior work in the wider scaling-law literature (West-Brown-Enquist 1997; Banavar et al.; Demetrius; He and Chen; Bettencourt; Zhao 2022; Maino et al.); priority is limited to the AI-capability application on this date. - PC-018 (2026-01-02): 'Meltdown Alignment' - a specific named framing organised around catastrophic failure modes as the primary design constraint set - was first published in Infinite Architects on 2 January 2026. FMEA and catastrophe-tolerant design are substantial prior work. - PC-011 (2026-01-02): The ARC Principle equation - book form U = I x R squared, paper form U = I x R^alpha - was first published as a conjecture in Infinite Architects on 2 January 2026 and formally in Paper I on 17 January 2026. Framing is conjecture throughout. The early alpha of approximately 2.24 is retracted; the robust v13 estimate is approximately 0.49 with interval [-1.3, 2.9] (sub-linear); alpha > 1 remains an open prediction. Bare d/(d+1) form is prior work. - PC-005 (2024-12-08): The Hyperspace Recursive Intelligence Hypothesis (HRIH) - the speculative framing that reality is a recursive creation by intelligence - was first described in the 8 December 2024 manuscript. Deep prior work exists (Teilhard de Chardin, Wheeler, Smolin). Self-labelled speculative. Citation -------- Michael Darius Eastwood (2026). The ARC Co-Scaling Law (Paper X) · Michael Darius Eastwood. The ARC Theory · ARC/Eden experiments, OSF DOI 10.17605/OSF.IO/BSE2Q. https://www.michaeldariuseastwood.com/research/papers/paper-x-coupled-coscaling-correction.html Notes ----- Hardware framings held at the site's already-published two-sentence concept ceiling: safety constraints at hardware level through cryptographic tokens in silicon, TRL 0-1, no prototype. No enabling implementation detail is stated in this companion file. Balanced-ternary computing is prior work (Setun 1958); recursion as a structural concept is prior work (evolutionary theory, self-modifying computation, quantum error correction). Generated from master --------------------- master_path: research/papers/paper-x-coupled-coscaling-correction.html master_sha256: b9e1f3a1fdd2413077a4f3ca5d81248fdddcd829769071530ff229a966187833 builder: scripts/build-paper-companions.py This block records the SHA-256 of the HTML master that produced this .txt. A check tool re-hashing master_path can decide freshness without any external state. If the master's current SHA-256 does not match master_sha256, this file is stale and must not be published: regenerate first with `python3 scripts/build-paper-companions.py --slug paper-x-coupled-coscaling-correction`.