Michael Darius Eastwood · Independent AI alignment researcher
Published
A mouse's heart beats roughly 600 times per minute. An elephant's beats about 28. Plotted on a log scale the relationship is a straight line with slope minus one quarter. Not approximately. Exactly. The same specific fractions (one half, two thirds, three quarters) appear in cosmology, shattering rocks, and urban infrastructure. This paper argues those recurrences are Cauchy's 1821 theorem applied to recursive composition.
Every scaling law in nature is one of exactly three shapes. Why only three, and why do the same specific fractions (one half, two thirds, three quarters) reappear across matter dominated cosmology, mammalian metabolism, and rock fragmentation? The paper argues this cannot be coincidence, and points at a two hundred year old theorem that fixes the answer.
Cauchy's 1821 classification says that a continuous function preserving multiplication must be a power law, and a continuous function turning addition into multiplication must be an exponential. Combined with the ARC Principle's cumulative advantage dynamics, this pins the scaling form to the composition operator: multiplicative gives a power law, additive gives an exponential, bounded gives a saturation curve. Within the power law regime, the paper derives the network partition identity alpha = d / (d + 1), which applies wherever a d dimensional hierarchical network partitions space. The Friedmann equation for cosmic expansion is shown to be a special case under the mapping d = 2 / (1 + 3w), and the deceleration and acceleration boundary from general relativity is shown to coincide with the power law and exponential boundary from Cauchy's theorem. All physical scaling exponents are argued to sit below 1 because space has finite dimensionality; intelligence, operating in abstract information space, is identified as the first natural phenomenon to break that geometric ceiling.
The paper's cross domain confirmations include cosmology (two eras, exact) and physics (several domains with low error). Its biological confirmations are the load bearing claim in this synthesis, and here the honest position is that the metabolic scaling headline figures are under recompute and unreconciled. The paper's abstract, its own figure caption, and downstream programme summaries disagree on which specific percentage and how many species groups are being cited, so no specific percentage is quoted here. Independent work exists on the same d / (d + 1) form (Demetrius and Tuszynski derived it from quantum oscillator coupling), which supports the structure but does not settle the empirical fit. A subsequent revision also reclassified the two dimensional biological test case from confirmed to unconfirmed pending a valid test organism.
This paper is the cross domain evidence companion to the Foundational Paper. Paper VII (the Cauchy Unification) formalises the theorem within the ARC framework. Paper II supplies the AI scaling empirical results (the earlier unblinded single-model fit of alpha approximately 2.24 was retracted, corrected to approximately 0.49 under blinding for the best fitting current frozen model). Paper X supersedes the fixed-exponent framing of U = I × R^alpha as the operative safety criterion and locates the current operative quantitative claim in a beta greater than k co scaling stability condition; the equation and the ARC Bound remain live hypotheses. The equation family U = I × R appears in the 8 December 2024 manuscript and is formalised as U = I × R^alpha in Papers I and II.
The paper is on OSF at DOI 10.17605/OSF.IO/6C5XB. Replication code lives at github.com/MichaelDariusEastwood/arc-principle-validation, in the domain validation directory. The Cauchy classification is a two hundred year old theorem; the reader can verify it in any standard functional equations text. The alpha = d / (d + 1) formula is derivable from three stated conditions (multiplicative composition, space filling geometry, and a conservation or optimisation constraint), not from space filling alone. The load bearing biological figures should be treated with caution until the recompute closes.
From the book Infinite Architects: Intelligence, Recursion, and the Creation of Everything by Michael Darius Eastwood.