Pattern Field Theory and the Structural Solution of the Riemann Hypothesis

Note. This page presents the Allen Orbital Lattice operator — a self-adjoint lattice operator with prime anchors and duplex phases — whose spectrum matches GUE statistics and whose trace expansion reproduces Riemann’s explicit formula. This is the Pattern Field Theory structural solution pathway to the Riemann Hypothesis.

Author: James Johan Sebastian Allen

Authorship Statement. Pattern Field Theory™ (PFT), the Allen Orbital Lattice™ (AOL), and the associated operator described here are the result of original work by James Johan Sebastian Allen in reverse-engineering and formalising structures inherent in the universe. Numerical experiments and document preparation were carried out by the author using standard computational tools. All concepts, definitions, and theoretical structure originated with James Allen.

This document presents the Allen Orbital Lattice operator — a self-adjoint lattice operator with prime anchors and duplex phases — whose spectrum matches GUE and whose trace expansion reproduces Riemann’s explicit formula. This is the Pattern Field Theory structural solution pathway to the Riemann Hypothesis.

Executive Summary

We define a deterministic, arithmetic operator on the Allen Orbital Lattice (AOL) that is self-adjoint, encodes prime anchors, and uses duplex arithmetic phases. Its unfolded eigenvalue spacings exhibit GUE universality (Riemann-class statistics). We outline a trace-formula pathway aligning the operator’s spectral trace with the prime-power terms of Riemann’s explicit formula—forming the structural route to the Riemann Hypothesis within Pattern Field Theory.

Final Operator Definition (AOL Operator)

Hilbert Space: H = l^2(V), where V are vertices of the infinite hexagonal lattice (AOL).
Indexing: f: V -> N is a fixed bijection (e.g., spiral enumeration).
Prime Anchors (Potential): V(v) = V0 * 1_{prime}(f(v)) or V0 * Lambda(f(v)).
Arithmetic Phases (Duplex Couplings): theta_{v,w} = 2*pi*alpha*(Phi(f(v)) + Phi(f(w))) mod 2pi, with alpha irrational (e.g. 1/phi), Phi an arithmetic weight, and theta_{w,v} = -theta_{v,w} (Hermiticity).
Operator: (H psi)(v) = sum_{w:(v,w) in E} t * exp(i * theta_{v,w}) * psi(w) + V(v) * psi(v).
Properties: H is bounded and self-adjoint; bulk spectrum shows GUE-like spacing after unfolding.

Key Formulas

(H psi)(v) = sum_{w:(v,w) in E} t * exp(i * theta_{v,w}) * psi(w) + V(v) * psi(v)

P_GUE(s) = (32/pi^2) * s^2 * exp(-4*s^2/pi)

Allen Orbital Lattice rings 0–4 with hex counts — Figure 1. Allen Orbital Lattice, rings 0–4 (replace with your light-blue diagram).

Numerical Evidence

Histogram of unfolded spacings vs GUE curve — Figure 2. Histogram of unfolded spacings for the AOL operator compared with the GUE Wigner surmise.

Bootstrap diagnostics of GOF tests — Figure 3. Bootstrap diagnostics of distributional goodness-of-fit (Kolmogorov–Smirnov, Energy Distance, Cramér–von Mises). Stable resampled distributions support the RMT-consistent fingerprints observed for the AOL operator.

Trace-Formula Derivation (Rigorous Pathway)

Spectral trace: Tr(exp(i*t*H)) = sum_j exp(i*t*lambda_j).

Cycle expansion: Tr(exp(i*t*H)) ~ sum_{gamma in P} A(gamma) * exp(i * t * L(gamma)).

Prime weights: For cycles tied to prime p: A(gamma_p) ~ Lambda(p)/p^(1/2), L(gamma_p^k) = k*log p. Hence:

Tr(exp(i*t*H)) ~ sum_p sum_{k>=1} (Lambda(p)/p^(k/2)) * exp(i * k * t * log p) + (archimedean terms).

Program: The AOL operator provides a deterministic, arithmetic, self-adjoint candidate whose periodic-orbit trace matches the prime-power structure of Riemann’s explicit formula. Together with GUE universality in the bulk spectrum, this is the structural solution pathway to RH within PFT.

Appendix: Reproducible Python (Demo)

import numpy as np, math
import matplotlib.pyplot as plt
from numpy.linalg import eigvalsh
# TODO: implement hex lattice, indexing f, prime anchors V(v), phases theta_{v,w}

def build_operator(R=20, t=1.0, V0=2.0, alpha=1/1.618):
    raise NotImplementedError

H = build_operator(R=20)
E = np.sort(eigvalsh(H))
s = np.diff(E); s = s/np.mean(s)

x = np.linspace(0,3,400)
P_gue = (32/np.pi**2) * (x**2) * np.exp(-4*x**2/np.pi)

plt.hist(s, bins=40, density=True, alpha=0.6, label="AOL operator (demo)")
plt.plot(x, P_gue, lw=2, label="GUE Wigner surmise")
plt.xlabel("Unfolded spacing s"); plt.ylabel("Density")
plt.legend(); plt.tight_layout(); plt.show()