Pattern Field Theory
Audit Note — Run005 Repeat (September 26, 2025)

This page documents a clean repeat of Run005, using the same dataset (2M Riemann zeros, von Mangoldt unfolding) but with protocol corrections: unfolding fixed to mean 1.0, no overwriting, consistent KS/GUE reporting.

Purpose: to confirm that Run005’s GUE match (KS D ≈ 0.0125, Brody β ≈ 1.00) holds under a reproducible and auditable pipeline.

run005 — Repeat (GUE Alignment, 2M zeros)

Objective: re-run and audit run005 with fixed unfolding, figure regeneration, and a clear methods trail. Result: ECDF vs GUE with residuals within ±0.01; KS \(D = 0.0125\); Brody \(\beta \approx 1.00\).

Executive summary
  • Dataset: first \( N = 2{,}000{,}000 \) non-trivial zeros (Odlyzko).
  • Unfolding: von Mangoldt zero-density \( \bar{N}(T) \), unit mean spacing (unfolded s).
  • Fit: GUE spacing model; ECDF residuals bounded in ±0.01 band for \( s \le 3.5 \).
  • Statistics: KS \(D = 0.0125\); Brody \(\beta \approx 1.00\) (repulsion), GOE rejected.

Figures

ECDF of unfolded spacings vs GUE CDF
ECDF of unfolded spacings (\(N=2\,\mathrm{M}\)) overlaid with GUE CDF. Mean(s)=1 by construction; visual agreement is close across \(s\in[0,3.5]\).

Residual Plot

ECDF minus GUE residuals (run005 repeat)
Empirical CDF − GUE CDF (unfolded spacings, \( s \le 3.5 \)). Oscillations remain within ±0.01, consistent with the KS statistic \( D = 0.0125 \). Bounded fluctuations with no systematic drift.

Methods (audit)

  1. Data: First \(2\text{M}\) zeros (Odlyzko). Use imaginary parts \(\gamma_n\), compute gaps \( \Delta \gamma_n = \gamma_{n+1}-\gamma_n \).
  2. Unfolding: Map gaps to unit mean via the local zero density \( \rho(T) \approx \frac{1}{2\pi}\log\!\Big(\frac{T}{2\pi}\Big) \); unfolded spacings \( s_n = \rho(\tilde{T}_n)\,\Delta \gamma_n \).
  3. ECDF & KS: Compute empirical CDF \(F_N(s)\); compare to GUE CDF \(F_\mathrm{GUE}(s)\); record \( D = \sup_s |F_N(s)-F_\mathrm{GUE}(s)| \).
  4. Brody fit: Fit \(\beta\) from Brody distribution to measure level repulsion; \(\beta\to1\) ≈ GUE-like.
  5. Residual band: Plot \(F_N - F_\mathrm{GUE}\) for \(s\le 3.5\) with ±0.01 guide lines.
Protocol note: All runs in this repeat use the same unfolding and plotting ranges; this resolves earlier contradictions between “raw mean” and “unfolded mean = 1.0”.

Results

RunZeros (N)UnfoldingKS DBrody βResidual bandNotes
run005-repeat 2,000,000 von Mangoldt 0.0125 ≈ 1.00 ±0.01 (s ≤ 3.5) GUE ≫ GOE; no drift in residuals

Minimal reproducibility (Python sketch)

# Pseudocode for unfolding + ECDF + KS (replace ... with actual I/O)
import numpy as np
from math import log, pi
from scipy import stats

# zeros: array of gamma_n (imag parts), length N+1
g = np.loadtxt("zeros_first_2000000.txt")  # placeholder
d = np.diff(g)                             # raw gaps

# local density at midpoints (rough)
T = 0.5*(g[1:] + g[:-1])
rho = (1.0/(2*pi))*np.log(T/(2*pi))
s = rho * d                                # unfolded spacings (mean ~1)

# GUE reference CDF via Wigner surmise (approx) or tabulated CDF
def gue_pdf(s): return (32/(pi**2))*(s**2)*np.exp(-(4/pi)*(s**2))
def gue_cdf(s): # numeric integrate or pre-tabulate
    x = np.linspace(0, s, 4000)
    return np.trapz(gue_pdf(x), x)

# ECDF & KS
s = s[(s>0) & (s<= 5)]
s.sort()
F_emp = np.arange(1, len(s)+1)/len(s)
F_ref = np.array([gue_cdf(x) for x in s])
D = np.max(np.abs(F_emp - F_ref))
print("KS D =", D)

Use your production scripts for exact unfolding and CDF reference; this sketch is only to illustrate the pipeline.

Notes

  • Why GUE? RH-compatible statistics predict GUE-like repulsion. GOE/Poisson fail to match the near-origin behavior.
  • Residuals: The ±0.01 band is a visual diagnostic; KS reports the sup norm across all s.
  • Limitations: Empirical evidence ≠ proof; no explicit Hermitian \(H\) is constructed here.

Author: James Johan Sebastian Allen · Pattern Field Theory (AOL lattice)