cff-version: 1.2.0
version: '1.2'
message: 'If you use or discuss this paper, please cite it using this metadata.'
title: 'Report · The ARC Theory · Laws IV and V, the ARC Persistence Law and the ARC Embedding Law · The Mathematics'
authors:
  - family-names: 'Eastwood'
    given-names: 'Michael Darius'
    orcid: 'https://orcid.org/0009-0004-3222-7442'
date-released: '2026-10-29'
url: 'https://doi.org/10.17605/OSF.IO/4B3VT'
license: 'CC-BY-4.0'
identifiers:
  - type: doi
    value: '10.17605/OSF.IO/4B3VT'
    description: 'OSF component DOI for this report (laws-iv-v-mathematics)'
  - type: doi
    value: '10.17605/OSF.IO/6C5XB'
    description: 'OSF umbrella DOI for the ARC / Eden Research Programme'
preferred-citation:
  type: report
  title: 'Report · The ARC Theory · Laws IV and V, the ARC Persistence Law and the ARC Embedding Law · The Mathematics'
  authors:
    - family-names: 'Eastwood'
      given-names: 'Michael Darius'
      orcid: 'https://orcid.org/0009-0004-3222-7442'
  date-published: '2026-10-01'
  url: 'https://doi.org/10.17605/OSF.IO/4B3VT'
  doi: '10.17605/OSF.IO/4B3VT'
  identifiers:
    - type: doi
      value: '10.17605/OSF.IO/4B3VT'
      description: 'OSF component DOI for this report (laws-iv-v-mathematics)'
    - type: doi
      value: '10.17605/OSF.IO/6C5XB'
      description: 'OSF umbrella DOI for the ARC / Eden Research Programme'
  abstract: |
    If control of a capable AI system were lost and could not be won back, what the system itself carries would decide what it does. Whether a property installed in it from outside survives later training for capability, and whether correction formed first, or placed inside the improvement loop, does better than correction added afterwards, are therefore questions worth stating exactly. The ARC Theory states them as two laws, registered on 15 October 2026, each a law under test: Law IV, the ARC Persistence Law, whose retention limb is the registered prediction that more than half of an externally installed gain is lost once later capability training that embeds no correction of its own matches the budget that installed it; and Law V, the ARC Embedding Law, whose limbs hold that a corrector formed before the capability it corrects has the larger correction-leverage exponent and that correction inside each revision round pulls away, as depth grows, from correction that sees only the finished output. This report sets out their mathematics. It proves that a persistence kernel never grows as the horizon lengthens, the family of allowed paths grows or the tolerance tightens; that greedy acceptance and planned optimisation keep a property acceptable over a finite horizon under stated margin, coverage and selection premises; and that a planner scoring its plans with bounded error is covered when its margin exceeds its tolerance by more than twice that error. It states what a retained fraction needs before it can be read, proves that under Fieller’s method, with a gate on the estimated gain, the ratio interval and a signed contrast decide alike, and derives a balance identity whose root returns Law III’s registered ceiling as a special case. Each result holds inside its stated model and proves nothing about real systems; the report measures nothing and states each law’s kill condition.