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| Hovedforfatter: | |
|---|---|
| Format: | Recurso digital |
| Sprog: | engelsk |
| Udgivet: |
Zenodo
2026
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| Online adgang: | https://doi.org/10.5281/zenodo.19513301 |
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Indholdsfortegnelse:
- <p>Does the positional structure of anomalously large Riemann zero spacings carry arithmetic information that the GUE (Gaussian Unitary Ensemble) framework cannot encode? We answer: yes, with a 20.7-sigma separation from GUE and direct spectral evidence for Dyson's quasi-crystal conjecture (50 Bragg peaks, max SNR = 197). However, the specific arithmetic source and the exact identity of the coefficient B remain unknown.</p> <p>We define a normalized spacing weight w_n = delta_t_n * (1/2pi) * ln(t_n/2pi) and classify w_n as "anomalous at level c" if w_n > mu + c<em>sigma, where mu and sigma are sample statistics. Our numerical experiments show that the mean gap k(c) between consecutive anomalies satisfies a multi-scale exponential law: k(c) approx A * exp(B</em>c). At N = 10,000 zeros, we find A = 0.944, B = 1.688 +/- 0.062, with R^2 = 0.999 across four threshold levels, all exhibiting a coefficient of variation (CV) < 0.56 and bootstrap p < 10^-4.</p> <p>This law is absent in random matrix theory: a study of 500 GUE matrices (500 x 500) yields an anomaly CV of 3.228 versus 0.40 for Riemann zeros, representing a gap of 20.7-sigma in the opposite direction. Furthermore, the diffraction spectrum of anomaly positions yields an intensity excess of 3629% over the empirical random baseline. This spectral structure is stable across multiple thresholds and is consistent with the quasi-crystal conjecture.</p> <p>Whether the coefficient B converges to sqrt(3)---the fundamental constant of the number field Q[sqrt(3)] and the I_{2,1} Lorentzian lattice---remains the decisive open question. A comparison with 3,000 zeros of the L-function L(s, chi_3) remains inconclusive at current precision. No proof of the Riemann Hypothesis is claimed.</p>