The Hilbert-Pólya Operator and the Primitive Structure of the Complex Plane: Between 𝔽₁, String Theory, and Ancient Geometry

We construct a Hermitian operator H_PCF whose spectrum approximates the non-trivial zeros of the Riemann ζ function with mean error <1.7% across twelve orders of magnitude (n=1 to n=10¹², 125 zeros). The construction proceeds without reference to ζ(s) or its zeros, drawing instead on the Precedent-Current-Forthcoming Framework (PCF): a closed string on a torus generated by the golden ratio φ through the extension ℂ -> E³ via z = φ 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 (ℤ/20ℤ)* to Re(ρ)=1/2 within the PCF categorical setting. Three spectral invariants—dimension d=3 (from S₃ symmetry), common modulus μ=1/2 (tripartite norm), and modular sum σ = dμ = 3/2 (spectral product)—emerge from the geometric structure alone, without invoking any component of ζ(s). The ring R_PCF = ℤ[φ, φ^-1, 1/2] admits a Λ-ring structure constituting 𝔽₁-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).