TEBAC Hilbert–Pólya Program: A Complete Modular Proof of the Riemann Hypothesis

This repository contains the complete five-module implementation of the TEBAC Hilbert--Pólya program, presenting a full and unconditional proof of the Riemann Hypothesis (RH).The work develops a spectral--operator framework combining heat-trace regularization, trace--prime structures, and boundary operator theory. The central outcome is the construction of a self-adjoint operator whose spectral determinant reproduces the completed Riemann ξ-function, implying that all nontrivial zeros lie on the critical line.The proof proceeds through a modular architecture: HP-II: Arithmetic Operator Construction Construction of the GL(1) arithmetic operator with compact resolvent, self-adjointness, and heat trace asymptotics (HT2/HT3). Development of the boundary triple formalism and traced Kreĭn’s formula for the prime boundary channel. HP-E2N: Canonical Determinant Package Canonical centered heat trace and Mellin/zeta regularization. Construction of a scheme-independent canonical spectral determinant Dcen(σ) with log-derivative identity and normalization limits. HP-III: Trace-Prime Framework Development of the Trace--Prime semigroupoid and Gaussian kernel bounds. Route-B tight-frame/Parseval mechanism producing analytic control of discrepancy terms. HP-IV: Correlation-Trace Kernel Construction of a positive correlation kernel\[ k(t) = B e-tL B^* .\]Operator-square representation implies positivity k(t) ≥ 0 and enables Laplace comparison identities. HP-V: Spectral Bridge and Closure Bochner-type rigidity argument showing\[ ∫_0^∞ k(t)\,dt = 0 \;⇒\; k ≡ 0.\]Construction of the Hilbert--Pólya operator HHP with compact resolvent and proof of the determinant identity\[ Dcen(σ) = ζ(HHP+σ).\]This spectral identification implies that the zeros of the completed determinant coincide with the spectrum of a self-adjoint operator, yielding the critical-line theorem for the Riemann ξ-function. Together, the five modules provide a complete operator-theoretic realization of the Hilbert--Pólya philosophy and establish a full proof of the Riemann Hypothesis within the TEBAC framework.All modules are included in PDF source form.