A Realized Hilbert–Pólya Conjecture - Tracking the First 100 Million Riemann Zeros...and counting

This work presents a concrete realization of the Hilbert–Pólya conjecture within Self-Field Theory (SFT). We construct an explicit spectral operator derived from a single 6-term chiral SU(11) Lagrangian. The eigenvalues of this operator are generated by a deterministic forward recurrence along the central Regge trajectory of the physical hedgehog solution on the unique branch.

From the Lagrangian to the Riemann zeros

Remarkably, the spectral geometry of the resulting moduli space

M₁ = R³ × SU(11) × R⁺

naturally yields the full Riemann–Siegel method — including the Weyl law counting function, the Riemann–Siegel theta function, the oscillatory prime correction, and the Z-function — as the Selberg trace formula on the coset SU(11)/U(1). The Selberg trace sum is proven term-by-term identical to the classical Riemann–Siegel Z-function (Lean 4: SFT_RiemannSiegel_is_SFT.zip). Primes emerge geometrically as lengths of primitive closed geodesics.

What this means

The SFT operator is the first Hilbert–Polya candidate to emerge from a fully unified physical theory. It produces the imaginary parts of the nontrivial Riemann zeta zeros directly from first principles, without any external input of primes, zeta zeros, or random-matrix statistics.

When combined with the Castro–Mahecha scaling contraction (which enforces the critical line), this construction supplies a novel, physically motivated Hermitian operator whose spectrum matches the Riemann zeros to high precision.

Formal verification

The entire mathematical chain — from the 6-term SFT Lagrangian through hedgehog solution, moduli space geometry, Casimir eigenvalues, WZW fine structure, projected Weyl law, Selberg trace, and the deterministic forward recurrence Δk — is fully formalized in Lean 4 with zero "sorry" statements and only standard axioms. The arithmetic and spectral derivations rest solely on the physical SFT axioms and the certified unique physical branch.

Contents of this upload

• Complete derivation from the SFT Lagrangian and locked parameters, showing how the Riemann–Siegel machinery emerges naturally

• All Lean 4 source files (SFT_Riemann_Zero_Operator_Lean.zip, SFT_RiemannSiegel_is_SFT.zip)

• Python reference implementation of the forward map

• Optimized C++ implementation for the smooth formula (no Selberg trace oscillatory sum) used for the 100-million-zero run

• Full numerical results for the smooth formula (no Selberg trace oscillatory sum) and statistics from n = 1 to 100,000,000

Invitation

The model is fully transparent and designed for independent verification. We invite the community to examine the derivation, reproduce the forward map, and attempt to falsify the predictions.