cff-version: 1.2.0
version: '3.12'
message: 'If you use or discuss this paper, please cite it using this metadata.'
title: 'Framework Paper · The ARC Theory · Law I, the ARC Principle · On the Origin of Scaling Laws: Functional Equations and the Scaling Forms Behind the Law'
authors:
  - family-names: 'Eastwood'
    given-names: 'Michael Darius'
    orcid: 'https://orcid.org/0009-0004-3222-7442'
date-released: '2026-10-10'
url: 'https://www.michaeldariuseastwood.com/research/papers/on-the-origin-of-scaling-laws.html'
license: 'CC-BY-4.0'
identifiers:
  - type: doi
    value: '10.17605/OSF.IO/XZY9U'
    description: 'OSF component DOI for this paper (on-the-origin-of-scaling-laws)'
  - type: doi
    value: '10.17605/OSF.IO/6C5XB'
    description: 'OSF umbrella DOI for the ARC / Eden Research Programme'
preferred-citation:
  type: article
  title: 'Framework Paper · The ARC Theory · Law I, the ARC Principle · On the Origin of Scaling Laws: Functional Equations and the Scaling Forms Behind the Law'
  authors:
    - family-names: 'Eastwood'
      given-names: 'Michael Darius'
      orcid: 'https://orcid.org/0009-0004-3222-7442'
  date-published: '2026-02-22'
  url: 'https://www.michaeldariuseastwood.com/research/papers/on-the-origin-of-scaling-laws.html'
  doi: '10.17605/OSF.IO/XZY9U'
  identifiers:
    - type: doi
      value: '10.17605/OSF.IO/XZY9U'
      description: 'OSF component DOI for this paper (on-the-origin-of-scaling-laws)'
    - type: doi
      value: '10.17605/OSF.IO/6C5XB'
      description: 'OSF umbrella DOI for the ARC / Eden Research Programme'
    - type: doi
      value: '10.17605/OSF.IO/P8CKQ'
      description: 'Theory-level predictions preregistered on OSF: DOI 10.17605/OSF.IO/P8CKQ, registered 8 September 2026 (version 1.100), updated 13 September 2026 (version 1.102), twenty-two propositions. Companion: Register · The ARC Theory · Operational Definitions, version 1.9.1, 5 October 2026.'
  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 They are withdrawn here and the corrected statement follows.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.

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