Title: On the Origin of Scaling Laws - M. D. Eastwood (2026) Author: Michael Darius Eastwood Publication date: 2026-02-22 Version: v3.5 Revised: 2026-08-23 OSF DOI: 10.17605/OSF.IO/XZY9U Canonical URL: https://www.michaeldariuseastwood.com/research/papers/on-the-origin-of-scaling-laws.html Abstract -------- Abstract Scope correction, 22 August 2026. An earlier version of this abstract opened by asserting that every scaling law in nature follows one of exactly three mathematical forms, and that this paper proves the form is uniquely determined. Both statements were stronger than this paper supports, and stronger than its own body claims. They are withdrawn here and the corrected statement follows. Where a recursive amplification process has a well defined composition operator on its input and output, and satisfies the regularity conditions Cauchy's functional equations require, the admissible functional forms are constrained to a small named family. That is a conditional mathematical classification, not a survey of nature: it does not establish that every empirical scaling process is such a homomorphism, and the bounded family's exhaustiveness remains an open problem, as Paper VII states. The grid is four-celled rather than three, the fourth cell being the logarithmic family. This paper's own scope condition, stated below and unchanged, is that the relation applies wherever a d-dimensional hierarchical network partitions space and fails where such networks are absent, which is by itself incompatible with any claim about every scaling law in nature. A single formula, $\alpha = d/(d+1)$, predicts scaling exponents across biology (9 species groups; species-group count pending recompute per canonical facts register), physics (4 domains catalogued after retiering; the below-0.2% mean-error aggregate is WITHDRAWN: a mean over analytic identities and unsourced reference values is small by construction and is not evidence of agreement with nature; the earthquake row has been deleted as a definitional identity per section 13), and cosmology (2 eras, exact), including the derivation of mammalian heart rate scaling from first principles. The formula is identified as a network-partition identity: it applies wherever a d-dimensional hierarchical network partitions space, and fails where such networks are absent. A geometric proof shows that all physical scaling exponents are constrained below 1 by the finite dimensionality of space - not by energy dissipation, but by geometry itself. Intelligence, operating through recursive self-reference in abstract information space, is identified as the first natural phenomenon to break this geometric scaling bound. The Friedmann equation for cosmic expansion is shown to be a special case under the mapping $d = 2/(1+3w)$, with the deceleration/acceleration boundary from general relativity coinciding exactly with the power-law/exponential boundary from Cauchy's theorem. Key findings (quotable) ----------------------- - This paper traces the origin of scaling laws across biological, physical and computational systems, and argues that the recursive amplification mechanism identified by the ARC Principle provides a unifying explanation for why power-law scaling relationships emerge so consistently across nature. - This paper traces the origin of scaling laws across biological, physical, and computational systems, arguing that the recursive amplification mechanism identified by the ARC Principle provides a unifying explanation for why power-law scaling relationships emerge so consistently across nature. 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-035 (2026-02-13): The synthesis observation that at least four independent research programmes (AI reasoning, quantum error correction, acoustic physics, consciousness science) converged on recursive or recurrent processing producing capability gains exceeding linear accumulation was first published on 13 February 2026 in the Foundational paper. Synthesis over other researchers' independent work, not a claim of discovering their results. - 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-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. Citation -------- Michael Darius Eastwood (2026). On the Origin of Scaling Laws - M. D. Eastwood (2026). The ARC Theory (the Theory of Artificial Recursive Creation) ยท ARC/Eden experiments, OSF DOI 10.17605/OSF.IO/XZY9U. https://www.michaeldariuseastwood.com/research/papers/on-the-origin-of-scaling-laws.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/on-the-origin-of-scaling-laws.html master_sha256: c43e46676c5f5547e750ea048c655ddd2bda79cd34fce33ab8c7f64f0509e69f 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 on-the-origin-of-scaling-laws`.