The Level That Was Never Returned: Kolmogorov's Similarity Hypotheses, Priced
Year: 2026
Type: preprint
Abstract: A ruler can be true in every ratio and still never tell you how long anything is. Kolmogorov's 1941 theory of turbulence is that ruler, and it is also — I credit this at full strength in this paper — the most successful scaling theory in classical physics. From two declared inputs, the viscosity ν and the mean dissipation ε, its hypotheses organise the entire small-scale structure of high-Reynolds-number turbulence: the microscales, the inertial-range spectrum E(k) ∝ ε^(2/3) k^(−5/3), the scale separation (L/η) = Re^(3/4), which my bench verifies reads back at ratio 1.000000 across Re = 10⁴–10⁸. What I evaluate is what the hypotheses are: premises their author named as hypotheses — local isotropy, and unique determination by (ν, ε) — which return every exponent and no level. The spectrum's constant, C in E(k) = C ε^(2/3) k^(−5/3), is fitted: ≈1.5 in the standard record, 1.68 ± 0.15 in the direct-simulation record my census carries — a 12% band (bench ratio 1.120000) that no argument from the hypotheses has ever computed. And the exponent family itself, ζ_p = p/3, fails against the record at every order except one: deviations +0.029333 at p = 2, −0.053333 at p = 4, −0.220000 at p = 6, −0.466667 at p = 8 on the bench's declared record inputs — the sole agreement, p = 3, belonging not to the hypotheses but to the Kármán–Howarth derivation of 1938, three years before them. Grade for this census entry, printed at full size: REASON FOUND / STILL TURNING — explicitly not a kill and not a falsification. The program headline stands unmoved: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.
Author: Aldred, William Alexander
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225346266