The Regression Hypothesis, Priced: What Onsager Assumed, What the Assumption Buys, and Where It Stops Being True

Census pricing (row CC-91) of the regression hypothesis of Onsager's 'Reciprocal Relations in Irreversible Processes. I / II,' Phys. Rev. 37, 405 (1931); 38, 2265 (1931), member by member against a fixed bench (what is postulated, what is derived, what the notebook computes in a fully knowable test-bed, what it cannot reach). The reciprocal relations L_ij = L_ji are derived in the papers from assumed microscopic reversibility plus one stated statistical hypothesis -- that the regression of a spontaneous equilibrium fluctuation follows the same macroscopic law as the relaxation of a prepared state. Priced in linear Langevin dynamics, where the truth is known by construction: a prepared ensemble of 200,000 members relaxes on the macroscopic law 2e^(-lambda t) with worst deviation 0.0040, the spontaneous covariance matches the dynamics' own value (0.087204 against 0.087469), and the lagged correlation matrix is symmetric to the digits printed -- the hypothesis is a theorem of the assumed linear stochastic dynamics and its Gaussian equilibrium, neither derived in 1931. Entropy production is shown blind to the antisymmetric part of L (a non-reciprocal matrix returns the identical sigma = 20.220000), so the hypothesis carries the whole load; one step nonlinear, the stated law departs from the exact decay by 43.7-48.8%. Its 1950s linear-response justification keeps every fence and is credited to the later theory. EXTENSION. Vacuum catastrophe NOT SOLVED; quantum gravity NOT SOLVED.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-04
DOI
https://doi.org/10.5281/zenodo.23937743
Primary Topic
Advanced Thermodynamics and Statistical Mechanics
Type
preprint
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preprint

The Regression Hypothesis, Priced: What Onsager Assumed, What the Assumption Buys, and Where It Stops Being True

William Alexander Aldred
Zenodo (CERN European Organization for Nuclear Research)
Advanced Thermodynamics and Statistical Mechanics
preprint

The Regression Hypothesis, Priced: What Onsager Assumed, What the Assumption Buys, and Where It Stops Being True

William Alexander Aldred
preprint en

Abstract

Census pricing (row CC-91) of the regression hypothesis of Onsager's 'Reciprocal Relations in Irreversible Processes. I / II,' Phys. Rev. 37, 405 (1931); 38, 2265 (1931), member by member against a fixed bench (what is postulated, what is derived, what the notebook computes in a fully knowable test-bed, what it cannot reach). The reciprocal relations L_ij = L_ji are derived in the papers from assumed microscopic reversibility plus one stated statistical hypothesis -- that the regression of a spontaneous equilibrium fluctuation follows the same macroscopic law as the relaxation of a prepared state. Priced in linear Langevin dynamics, where the truth is known by construction: a prepared ensemble of 200,000 members relaxes on the macroscopic law 2e^(-lambda t) with worst deviation 0.0040, the spontaneous covariance matches the dynamics' own value (0.087204 against 0.087469), and the lagged correlation matrix is symmetric to the digits printed -- the hypothesis is a theorem of the assumed linear stochastic dynamics and its Gaussian equilibrium, neither derived in 1931. Entropy production is shown blind to the antisymmetric part of L (a non-reciprocal matrix returns the identical sigma = 20.220000), so the hypothesis carries the whole load; one step nonlinear, the stated law departs from the exact decay by 43.7-48.8%. Its 1950s linear-response justification keeps every fence and is credited to the later theory. EXTENSION. Vacuum catastrophe NOT SOLVED; quantum gravity NOT SOLVED.

Zenodo (CERN European Organization for Nuclear Research)
Advanced Thermodynamics and Statistical Mechanics
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The Regression Hypothesis, Priced: What Onsager Assumed, What the Assumption Buys, and Where It Stops Being True — William Alexander Aldred · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS