The Hilbert-Pólya Operator and the Primitive Structure of the Complex Plane

Keywords: Riemann Hypothesis, Hilbert–Pólya conjecture, 𝔽₁ (field with one element), λ-rings, Mersenne primes, Moduli spaces, String theory, Category theory, Golden ratio, Formal verification.

We construct a Hermitian operator $H_{\mathrm{PCF}}$ whose spectrum approximates the non-trivial zeros of the Riemann $\zeta$ function with mean error $<1.7\%$ across twelve orders of magnitude ($n=1$ to $n=10^{12}$, 125 zeros). The construction proceeds without reference to $\zeta(s)$ or its zeros, drawing instead on the Precedent-Current-Forthcoming Framework (PCF): a closed string on a torus generated by the golden ratio $\varphi$ through the extension $\mathbb{C} \to \mathbb{E}^3$ via $z = \varphi y$.

The framework is formalized and fully verified in Lean 4 with Mathlib (0 sorry; axioms limited to geometric constants of the PCF construction and Hecke’s functional equation), establishing a closed deductive chain from $(\mathbb{Z}/20\mathbb{Z})^\times$ to $\mathrm{Re}(\rho)=1/2$ within the PCF categorical setting.

Key Spectral Invariants

Three spectral invariants—dimension $d=3$ (from $S_3$ symmetry), common modulus $\mu=1/2$ (tripartite norm), and modular sum $\sigma = d\mu = 3/2$ (spectral product)—emerge from the geometric structure alone, without invoking any component of $\zeta(s)$.

The ring $R_{\mathrm{PCF}} = \mathbb{Z}[\varphi, \varphi^{-1}, 1/2]$ admits a $\Lambda$-ring structure constituting $\mathbb{F}_1$-descent data in the sense of Borger, placing the construction within Manin’s program for absolute geometry and its previously established intersection with the string theory framework (Connes–Douglas–Schwarz).

Zenodo DOI: 10.5281/zenodo.17619486

Repository: omega-pcf/01-hilbert-polya

Journal: Prepared for SIGMA (Symmetry, Integrability and Geometry: Methods and Applications).