The zeros of the Riemann zeta function on the critical lineencode the oscillatory structure of the primes through theexplicit formula. The Montgomery-Odlyzko law describes theirspacing statistics via the Gaussian Unitary Ensemble (GUE) ofrandom matrix theory -- but GUE is blind to the arithmeticorigin of the zeros. We ask a sharper question: among thespacings that deviate most from the mean, do their positionscarry arithmetic structure that GUE cannot replicate? We introduce a probability measure pₙ = wₙ/Z on zerospacings and establish four numerical results on N=3000 zeros (verified at N=1000 in two independent environments). (1) Multi-scale exponential law: the mean gap k (c) betweenanomalies at threshold c satisfies k (c) ~ 0. 834 * exp (1. 730 * c), R² = 0. 998, across five levels (c in 1. 0, 1. 5, 2. 0, 2. 5, 3. 0), eachwith CV = 1e4 is required for convergence. (2) GUE matrix comparison: 500 independent GUE matrices (500x500) yield CVGUE = 3. 23 versus CVRiem = 0. 40 -- adifference of 20. 7 sigma in the OPPOSITE direction. Thisstructure lies strictly outside the GUE framework. (3) Shannon entropy: Sₕat (T) = S (T) / log N (T) -> 1 asT -> infinity, consistent with RH. (4) Zero detection: local maxima of |zeta' (1/2+it) /zeta (1/2+it) |²locate all 29 zeros in 14, 100 with 100% accuracy and zerofalse positives. No proof of the Riemann Hypothesis is claimed. The analyticidentity of B ~= 1. 730 is the central open problem. ----------------------------------------------------------------MSC 2020: 11M06 (primary) ; 11M26, 94A17, 65B99 (secondary) Keywords: Riemann zeta function; non-trivial zeros;anomalous spacing; multi-scale exponential law; Gram law;random matrix theory; GUE comparison; zero detection================================================================
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Y.Y.N. Li (Sun,) studied this question.
www.synapsesocial.com/papers/69dc887f3afacbeac03ea5b1 — DOI: https://doi.org/10.5281/zenodo.19513301
Y.Y.N. Li
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