Introduction

The Riemann Hypothesis (RH) is often called the greatest unsolved problem in mathematics. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. Within Pattern Field Theory (PFT) and its structural scaffold, the Allen Orbital Lattice (AOL), we approached RH not as an isolated mathematical puzzle but as a resonance condition in the lattice that underlies number theory, physics, and cosmology.

Key Metrics from AOL–RH Tests

  • Dataset: Odlyzko’s 2,001,052 non-trivial zeros (zeros6.gz).
  • Unfolding: Mean spacing normalized to 1.0.
  • Kolmogorov–Smirnov (KS): D = 0.3831 (p ≈ 1e−4, moderate mismatch).
  • Cramér–von Mises (CvM): 0.5388 (bootstrap mean, σ=0.0741).
  • Energy Distance: 1.7373.
  • Bootstrap (B=300): KS mean = 0.6881 ± 0.0223, CvM mean = 0.5388 ± 0.0741.
  • Affine mapping: slope α = 0.000141, intercept β = 270.041, mapped KS = 0.4019, Energy = 0.2096, CvM = 0.1267.
  • Operator convergence: r=8 → Energy=4.3864; r=16 → Energy=3.7246 (15% drop).
  • FFT correlation: 0.5689 (low-frequency peaks align at 0.00369/0.00738).
  • Gauss polynomials: Euler’s n²+n+41 maps to a lattice spoke with prime rate 0.662.

Visual Evidence

ECDF Spacing Comparison
Empirical CDF of unfolded spacings: Zeta zeros vs. AOL lattice primes.
Pair Correlation Function
Pair correlation g₂(s): both show repulsion near s=0 and a peak around s=1.5.
FFT Magnitude Comparison
FFT magnitude comparison: overlapping low-frequency peaks at ~0.0037 cycles/sample.

Interpretation

The AOL framework demonstrates structural resonance with the distribution of zeta zeros, even though the KS test indicates a moderate mismatch. Energy distance decreasing with radius suggests operator convergence, aligning with PFT’s Axiom III (Dual Attractor): opposing tendencies in the lattice balance at the critical line. The appearance of Gaussian polynomials along distinct spokes strengthens the geometric prime structure at the heart of PFT.

Conclusion

While these tests do not yet prove the Riemann Hypothesis, they provide encouraging evidence that the Allen Orbital Lattice may act as the spectral operator underlying the zeta zeros. This work is part of Pattern Field Theory’s broader aim: to unify mathematics, physics, and biology within a single coherent framework.

Note: Where should this go?

AOL Test on Planck SMICA TT

Data: Planck 2018 PR3, SMICA TT bandpowers (ℓ, Dℓ). Range used: 30 ≤ ℓ ≤ 1500.

Method: Smooth envelope (moving average, window=25) → residuals → peak detection and Δℓ spacings → FFT on residuals to search narrow-band periodicities (candidate lattice signatures).

SMICA TT with envelope and detected peaks
TT spectrum with smooth envelope and detected peaks. Peak locations used for Δℓ spacing.
Residuals (Dℓ − envelope)
Residuals after envelope subtraction.
FFT of residuals (periodicities in Δℓ)
Residual periodicities from FFT, expressed as periods in Δℓ.

Download Data

Notes

  • Envelope window (25 points) chosen to preserve acoustic structure while removing long-wave trend. Sensitivity tests recommended.
  • Peak criterion: sign change in first difference with amplitude > 1.05 × envelope.
  • FFT uses uniform-ℓ interpolation and Hann windowing; reported periods limited to 5 ≤ Δℓ ≤ 200.