Purpose
Cascade
The cascade is the density of zeros / eigenvalues vs spacing: Φ(s) = local density of normalised level spacings interior coordinate = spacing s ∈ [0, ∞) conserved cascade = Wigner-Dyson distribution (GUE class) cascade tail = bulk-region spacings far from rigidity edges injected energy = finite-N corrections, model-specific deviations
Operation (Law II, no parametric distributional assumption)
1. Generate M independent GUE realisations at size N. For each, compute eigenvalues, unfold to mean spacing 1, and extract the bulk-region nearest-neighbour spacings. 2. Pool all GUE bulk spacings → empirical EDF of the universal class (Wigner-Dyson). This is the conserved-cascade reference. 3. Bootstrap a within-trial CVM threshold from the GUE pool at controlled Pfa = 0.05. 4. Read Odlyzko's first 100 Riemann zeta zeros, unfold them with the exact Riemann-Siegel mean density formula, and form the spacing sample. 5. Per-realisation CVM admissibility: - each individual GUE realisation tested against the GUE pool (sanity floor — pass rate should be near 1−Pfa) - the Riemann unfolded-spacing sample tested against the GUE pool (the Hilbert-Pólya verdict at this resolution) 6. Cross-realisation consensus on a per-class summary statistic (the small-s exponent φ of the spacing density: Wigner-Dyson predicts φ = 2; Poisson would give φ = 0). Read φ by binning the spacings — no parametric fit, just a slope of log P(s) vs log s on the small-s window.
The framework's verdict is structural, not parametric. There is no χ² critical value, no Gaussian assumption, no fixed threshold for "in same universality class" — only the empirical reference EDF and the within-trial CVM gate.
Pass criterion
The framework should classify Odlyzko's real zeros as Wigner-Dyson: the Riemann CVM distance to the GUE pool should fall AT OR BELOW the threshold; the small-s exponent φ should fall in cross-realisation consensus with the GUE φ.
Run wall: ~5–10 s (no C kernels needed).
Methodology
The cascade is described above. All readings use the canonical framework operator chain \(\mathcal{F}, \mathcal{S}, \mathcal{M}, \mathcal{P}\) — no per-experiment tuning constants. Every reading is reported with its scope-reporter \(\mathcal{A}\) tuple (Theorem 12).
Results
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Riemann zeros vs GUE — Hilbert-Pólya empirical anchor
Cascade : density of zeros / eigenvalues vs spacing
Conserved cascade : Wigner-Dyson universal class
Operation : empirical EDF + within-trial CVM (Law II)
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VERDICT
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Riemann (Odlyzko, n=101 zeros) → WIGNER-DYSON
Hilbert-Pólya pass : True
Empirical EDF reference
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GUE realisations : 24 matrices at N=200
Pool spacings : 3384
CVM threshold (Pfa=0.05) : 0.0040
GUE pass rate vs own pool : 1.000 (target ≥ 0.90)
Riemann admissibility (within-trial CVM, no parametric form)
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Riemann unfolded spacings : 100
Threshold @ that window L : 0.0056
Riemann CVM to GUE pool : 0.0018
Admissible (CVM ≤ thresh) : True
Small-s exponent φ (Wigner-Dyson predicts φ = 2)
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GUE consensus : φ = +1.101 σ_cross = 0.094 (M=24)
Riemann : φ = +1.930
z(Riemann vs GUE consensus) = +8.78
Discipline (Law II)
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• NO Gaussian assumption on spacings.
• NO χ² critical value used.
• NO fixed threshold for 'in Wigner-Dyson class'.
• Threshold bootstrapped from GUE pool itself; CVM is
a non-parametric two-sample distance.
Wall time : 0.27 sCode
Implementation: domains/pure-math/experiments/riemann_zeros_vs_gue.py