Cauchy unification: an old mathematics as a scaling-law classifier
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A 200-year-old set of functional equations, tested as a cross-domain predictor of which scaling family fits which system.

Canonical path: /research/papers/paper-vii-cauchy-unification.html
Author: Michael Darius Eastwood
Research programme: https://doi.org/10.17605/OSF.IO/6C5XB

Cauchy unification: an old mathematics as a scaling-law classifier Michael Darius Eastwood · Independent AI alignment researcher Published 3 July 2026 Michael Darius Eastwood · Foundational theory · 3 July 2026 Michael Darius Eastwood, independent researcher, London: author of the ARC/Eden research programme; the embedded-correction alignment thesis is recorded in a source record dated 8 December 2024 (sent-side SHA-256 f0d1f38f). 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? Paper VII · Argues that Cauchy's functional equations constrain scaling laws to power law, exponential, or saturation families, and reports 19 of 25 empirical curve-fit domains confirming the predicted family under strict AICc model selection (p = 1.56 x 10^-5). OSF DOI 10.17605/OSF.IO/6C5XB. The question it asks 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.

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Machine-readable companion. Cite: Eastwood, M. D. (2026). "Cauchy unification: an old mathematics as a scaling-law classifier". /research/papers/paper-vii-cauchy-unification.html