Title: Paper VII · The ARC Theory · Law I, the ARC Principle · Cauchy Unification: the Classification Behind the Law’s Form Author: Michael Darius Eastwood Publication date: 2026-03-16 Version: v3.15 Revised: 2026-10-05 OSF DOI: 10.17605/OSF.IO/X6WA7 Canonical URL: https://www.michaeldariuseastwood.com/research/papers/paper-vii-cauchy-unification.html OSF version status ------------------ Site version: v3.3 OSF version: v3.0 Note: OSF PDF re-upload pending. The site's paper VII carries the v3.3 Tier A pass of 13 August 2026 (Section 3.2 model-set disclosure; every hit count qualified; four-family re-run referenced) plus v3.1/v3.2 corrections. The OSF component x6wa7 still hosts the v3.0 PDF of 24 March 2026. Any reader following the OSF DOI will read pre-correction figures. Re-upload the current PDF to OSF and bump this record. Registration ------------ 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 -------- Cauchy's four functional equations - additive, multiplicative, exponential, and logarithmic - are the only continuous solutions to their respective composition constraints. We argue that this 200-year-old mathematical result has a physically testable consequence: under the stated axioms, it forces a power law where composition is multiplicative and an exponential where depth adds while output multiplies; where output is bounded, the compatible forms are saturation curves, and a formal proof that they exhaust the bounded case is not provided here and remains an open problem. The family is determined by the composition operator of the underlying recursive amplification process. The novel contribution is not Cauchy's mathematics itself, but the claim that Cauchy-type functional equations have a physically testable consequence for scaling-law classification across domains. We present a structured prediction comparison across 50 real-world systems in five evidence tiers. Each empirical domain followed the same sequence: known physics fixed the composition operator before any fitting began, the operator class then set which scaling family to expect, published or standard reference data were loaded, and that expectation was finally put to the test by independent model fitting using AICc-based model selection. Primary result (computed on the six-model candidate set, which contained no logarithmic and no pure linear candidate (Section 3.2)): 19 out of 25 empirical curve-fit domains confirm the Cauchy-predicted family under strict AICc model selection. (The rerun of 11 August 2026, which added the logarithmic candidate, returns 18 of 25.) The primary significance statistic is a permutation test conditioned on both the predicted and the observed family marginals: $p = 7 \times 10^{-5}$. A one-sided binomial test against uniform $1/3$ chance gives $p = 1.56 \times 10^{-5}$ (roughly 1 in 64,000) and is retained as a sensitivity check. The result also holds on the strictly Cauchy-derived subset alone (the 18 multiplicative and additive domains, excluding the bounded family whose exhaustiveness remains an open problem): 14/18, permutation $p = 3.0 \times 10^{-4}$. Secondary results: a baseline-20 rerun yields 15/20 ($p = 1.67 \times 10^{-4}$); published metabolic exponents match the nearest ARC/Cauchy comparator in 13/13 direct cases; analytic identities confirm 6/6. These tiers are reported separately and are never combined into a single blended number. This result is strong enough to publish as a structured prediction comparison under a publicly stated rule rather than a per-domain advance timestamp, but not strong enough to claim as proven. The primary test is not an accepted registry submission, and no registry has accepted it as one. The predictions for this cohort are not timestamped in advance of their own fits: the classification rule reached the public repository at 23:33 UTC on 16 March 2026 and the fifty per-domain assignments were committed together with their results at 23:54 that night. The advance per-domain timestamp in this record belongs to the 12-domain extension manifest of 17 March 2026 and to the book's two prediction appendices in print on 2 January 2026, before the papers and the programme existed. What was not fixed in advance was the fitting protocol, and that gap is closed prospectively by the staged tests, the fourth-cell test and the 80-domain expansion test, both drafted and awaiting human submission; the fourth-cell test is preregistered by timestamp, first anchored on 12 August 2026 at Bitcoin block 962066. A separate, forward prediction frame does exist and is not claimed as covering this result. On 17 March 2026, between 02:06:51 and 02:07:05 UTC, a prediction protocol, an extension manifest and a checksum file were deposited to the public Open Science Framework component x6wa7. Each carries an OSF-assigned creation timestamp, each has a modification timestamp identical to its creation timestamp, and the outcomes for the domains they specify have not been computed. That is a dated third-party deposit rather than an OSF Registration, which is a distinct frozen and schema-bound object this component does not carry; the API reports registration: false and the page says so rather than blurring it. What the deposit establishes is the ordering: the predictions were public and unaltered before any result they predict existed. Independent replication is needed. Key findings (quotable) ----------------------- - Cauchy's four functional equations - additive, multiplicative, exponential, and logarithmic - are the only continuous solutions to their respective composition constraints. - We argue that this 200-year-old mathematical result has a physically testable consequence: under the stated axioms, it constrains scaling laws to one of three functional families (power law, exponential, or saturation curve). - The family is determined by the composition operator of the underlying recursive amplification process. - The novel contribution is not Cauchy's mathematics itself, but the claim that Cauchy-type functional equations have a physically testable consequence for scaling-law classification across domains. 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-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). Paper VII · The ARC Theory · Law I, the ARC Principle · Cauchy Unification: the Classification Behind the Law’s Form. OSF DOI 10.17605/OSF.IO/X6WA7. https://www.michaeldariuseastwood.com/research/papers/paper-vii-cauchy-unification.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). Provenance note --------------- Michael Darius Eastwood conceived and directs this research programme and is the author of this work. Across the programme, he has used more than six AI systems in parallel, under his own instructions, to stress-test his arguments, identify possible errors, and assist in preparing draft text from his own outlines. He determines what is adopted, revised or rejected and takes responsibility for the published content. These systems are tools, not authors. Generated from master --------------------- master_path: research/papers/paper-vii-cauchy-unification.html master_sha256: 45cc33bec9c4f23ddf1766228aba858e6eae8fcba9cc2ef60f068b3d0fd4c996 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-vii-cauchy-unification`.