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, or about 4.2 sigma. 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. There is no locked pre-registration artefact for the primary test: operator classifications, data, and fitting logic all live in one author-written script. This is a structured prediction comparison, not a formally pre-registered blind trial. The exhaustiveness proof for the bounded case is not provided and remains open. A pre-registered replication with independent operator classification is the necessary next step. On 24 March 2026, the Cauchy letter was mailed to Professor Lloyd Demetrius (Harvard) and Professor Geoffrey West (Santa Fe Institute), who have both independently derived closely related metabolic-scaling results by different routes; no reply has been received to date.
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 pre-registered 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.