Michael Darius Eastwood · Independent AI alignment researcher
Published
In the early nineteenth century, Augustin-Louis Cauchy proved that four functional equations have unique continuous solutions: additive, multiplicative, exponential, and logarithmic. Paper VII asks whether that 200-year-old result has a physically testable consequence: does the way a system's amplifying process composes (multiplicatively, additively, or under a bound) already tell you which family of scaling law it must obey, before you look at any data?
Kleiber's law, Moore's law, Zipf's law, learning curves, epidemic growth, urban scaling, neural-network loss curves and radioactive decay all follow scaling relationships that look, superficially, unrelated. Each has traditionally been explained by a domain-specific mechanism. Paper VII asks a bolder unifying question. Cauchy's multiplicative equation forces a power law. His exponential equation (from additive composition) forces an exponential. Bounded composition, output constrained to a finite ceiling, is compatible with the saturation family (logistic, Hill, Michaelis-Menten, hyperbolic). If those constraints are physically operative, then classifying the composition operator of any new system should predict, with no free parameters, which family its scaling law belongs to.
Fifty domains were assembled across five evidence tiers. The primary tier is a set of twenty-five empirical curve-fit domains. For each, the composition operator was classified from known physics before any fitting, and six candidate models (power law, exponential, logistic, Hill, Michaelis-Menten, saturation-exponential) were fitted independently. The best model was chosen by proper AICc. Nineteen of the twenty-five confirmed the predicted family. A one-sided binomial test against uniform random assignment among three families gives p = 1.56 x 10^-5, roughly one in 64,000; sigma equivalences are not quoted because they shift with the choice of null. Thirteen published metabolic exponents matched the nearest ARC/Cauchy comparator in 13 of 13 direct cases; six analytic identities trivially recovered themselves. Under the stated axioms, the paper proves that multiplicative composition forces the power-law family (Cauchy's own theorem), and derives the d/(d+1) metabolic exponent when three conditions hold jointly: multiplicative composition, d-dimensional space-filling transport, and a conservation constraint on resource flow.
Six of the twenty-five empirical domains missed the predicted family, each for a specific named reason: small datasets (Galapagos species-area, time crystal order), truncated ranges (muscle force-velocity), finite-corpus ceilings (Zipf), high scatter at extremes (stellar mass-luminosity), and the classically contested Kleiber case where the exponent debate itself remains unsettled at 0.67 versus 0.75. No registry-filed pre-registration artefact exists for this test. The predictions themselves were timestamped in advance by dated public commit: the per-row classifications in the hash-pinned catalogue files of March 2026, the 12-domain extension manifest of 17 March 2026, and 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 registry-filed tests: the fourth-cell test and the 80-domain expansion test. That is correct and it is scoped to the primary test, whose outcomes were computed before any protocol was frozen. A separate forward frame does exist and does not cover this result: a prediction protocol, an extension manifest and a checksum file were deposited to the public OSF component x6wa7 on 17 March 2026, each with a modification timestamp identical to its creation timestamp and the outcomes for the domains they specify still uncomputed. A dated third-party deposit, not an OSF Registration. This is a structured prediction comparison, not a formally registered blind trial. The exhaustiveness proof for the bounded case is not provided and remains open. A replication with independent operator classification is the necessary next step. The closest related results are the metabolic-scaling derivations of Demetrius and of West and colleagues, reached independently by different routes; Paper VII shows what they have in common.
Paper VII is the foundational-mathematics companion to Papers I and II, which introduce and formalise U = I x R (from the 8 December 2024 manuscript) as U = I x R^alpha, and to Paper III on alignment scaling. Its bounded-family prediction underwrites Paper VI's honey architecture. Independent convergent work (Demetrius, West, Banavar, Bettencourt, Zhao, He, Maino) reaches the same d/(d+1) form from at least seven different starting points; the Cauchy framing offers a possible reason those routes converge, but claims of universal derivation await independent scrutiny.
The paper HTML and PDF are on OSF (DOI 10.17605/OSF.IO/6C5XB). The full 50-domain manifest, the AICc-based fitting code, and the raw data with citations for every domain are in the public arc-principle-validation repository. The script runs under Python 3.10 or later with numpy and scipy alone; output is deterministic. A replication with independent operator classification is the falsifying test the paper explicitly invites.
From the book Infinite Architects: Intelligence, Recursion, and the Creation of Everything by Michael Darius Eastwood.