What question did this study set out to answer?

The aim is to present a complete and unconditional proof of the Riemann Hypothesis using the TEBAC Hilbert–Pólya program.

March 12, 2026Open Access

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

Key Points

The aim is to present a complete and unconditional proof of the Riemann Hypothesis using the TEBAC Hilbert–Pólya program.
Developed a five-module implementation of the TEBAC cycle.
Utilized heat-trace regularization and boundary operator theory.
Constructed a self-adjoint operator and canonical spectral determinant.
The proof shows that all nontrivial zeros of the Riemann zeta function lie on the critical line.
The constructed operator’s spectral determinant matches the completed Riemann xi-function.
Established a positive correlation kernel that leads to important Laplace comparison identities.

Abstract

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 -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 D₂₄₍ () 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^*. -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. of the Hilbert--Pólya operator H₇₏ with compact resolvent and proof of the determinant identity\ D₂₄₍ () = _ (H₇₏+). 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 -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.

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

Key Points

Abstract

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