Title: The ARC Equation: the Law of Conversion (Paper I) Author: Michael Darius Eastwood Publication date: 2026-01-17 Version: v1.10 Revised: 2026-08-23 OSF DOI: 10.17605/OSF.IO/B6N27 Canonical URL: https://www.michaeldariuseastwood.com/research/papers/paper-i-arc-principle.html Abstract -------- This paper formalises and preliminarily tests the ARC Principle (Artificial Recursive Creation), first proposed in Infinite Architects (Eastwood, 2026): that capability in intelligent systems scales super-linearly with recursive depth. The principle is expressed mathematically as $U = I \times R^{\alpha}$, where effective capability ($U$) scales with base intelligence ($I$) multiplied by recursive depth ($R$) raised to an empirically estimated exponent $\alpha$. Analysis of publicly available test-time compute data from reasoning models reveals a critical distinction between two forms of recursion. Parallel recursion (majority voting across independent samples) yields sub-linear scaling with $\alpha \approx 0.1$ to $0.3$. Sequential recursion (chain-of-thought reasoning where each step builds on previous steps) yields super-linear scaling with $\alpha \approx 1.3$ in this paper's early two-point estimate; the later six-model cross-architecture measurement (Paper II) finds the sequential exponent architecture-dependent, with best fit $\alpha_{\text{seq}} \approx 0.49$ and an interval spanning sub-linear and super-linear alike, so the super-linearity claim is unresolved rather than established. This preliminary finding, if validated by further research, suggests that the form of recursion determines whether intelligence compounds or merely accumulates. We propose that $\alpha = 2$ is a stability ceiling, not an impossibility limit: it is the proposed stable limit, the most a recursively self-correcting system can grow and stay correctable, a scaling limit and not a speed limit. Systems can exceed $\alpha = 2$; they do not remain stably self-correcting above it. A durably stable system above 2 would refute the bound; a transient excursion is the predicted supercritical regime, not the refutation. Stable is measured, not asserted: correction out-scaling drift, $\beta > k$ from Law II, with the lower bound of $\beta - k$ above zero across a window fixed before the run. Key findings (quotable) ----------------------- - This paper formalises and preliminarily tests the ARC Principle (Artificial Recursive Creation), first proposed in Infinite Architects (Eastwood, 2026): that capability in intelligent systems scales super-linearly with recursive depth. - The principle is expressed mathematically as $U = I \times R^{\alpha}$, where effective capability ($U$) scales with base intelligence ($I$) multiplied by recursive depth ($R$) raised to an empirically determined power $\alpha$. - Analysis of publicly available test-time compute data from reasoning models reveals a critical distinction between two forms of recursion. - Parallel recursion (majority voting across independent samples) yields sub-linear scaling with $\alpha \approx 0.1$ to $0.3$. - "U = I x R^alpha, where effective capability (U) scales with base intelligence (I) multiplied by recursive depth (R)" - published as a conjecture; alpha open to measurement. Priority claims relevant to this paper -------------------------------------- - PC-024 (2026-01-17): The ARC / Eden research programme was placed on the DOI record on 17 January 2026 under OSF DOI 10.17605/OSF.IO/6C5XB, coinciding with Paper I. - PC-003 (2024-12-08): The ARC Principle - Artificial Recursive Creation, as a named cross-domain framework, was first described in the 8 December 2024 manuscript and formalised in Paper I on 17 January 2026. Recursion as a structural concept is prior work; priority is over Eastwood's specific naming and cross-domain framing under this label. - 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-001 (2024-12-08): The 8 December 2024 self-emailed manuscript is the earliest timestamped anchor for the programme (Google-server timestamp 02:45:18 UTC; Message-ID CAGPsKAnp-DLDBOeBAh4t9XsqJBKdoe07uORGF86meoSMFnin7w; approx. 221,236 words (five-attachment scope); SHA-256 prefix f0d1f38f). Existence-by-date, not DKIM-authenticated authorship. - PC-010 (2026-01-02): The Infinite Architects paperback (ISBN 978-1806056200, ISBN-10 1806056208) was first published on 2 January 2026. The book presents the ARC Principle, the Eden Protocol, the semiconductor Chokepoint framing, the HARI Treaty proposal, the Three Ethical Loops and the 37 named concepts. Book's featured equation is U = I x R squared; the research papers use U = I x R^alpha with the exponent open to measurement. Citation -------- Michael Darius Eastwood (2026). The ARC Equation: the Law of Conversion (Paper I). The ARC Theory (the Theory of Artificial Recursive Creation) ยท ARC/Eden experiments, OSF DOI 10.17605/OSF.IO/B6N27. https://www.michaeldariuseastwood.com/research/papers/paper-i-arc-principle.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). 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