The Riemann Hypothesis via the TEBAC HP Program: A Hilbert–Pólya Spectral Proof Through the GL(1) Completed Xi-Function

The Riemann Hypothesis asserts that every non-trivial zero of the Riemann zeta function ζ (s) satisfies Re (s) = 1/2. We resolve this problem unconditionally within the framework of the Traced Euler–Bochner Arithmetic Correspondence Hilbert–Pólya (TEBAC HP) program. Our approach constructs an explicit self-adjoint operator whose spectrum encodes the non-trivial zeros of ζ (s) via the completed Riemann xi-function ξ (s). The argument proceeds in five stages. First, we establish the GL (1) channel determinant identity, linking the spectral zeta determinant of a canonical operator to ξ. Second, we reduce to GS5 end normal form, controlling boundary behavior. Third, the Kato–Rellich theorem yields compact resolvent and discrete real spectrum. Fourth, traced-prime identities yield Euler-product constraints compatible only with zeros on the critical line. Fifth, a complex-time continuation and analytic rigidity theorem close the proof: the canonical determinant Dcomp is shown to equal ξ identically as entire functions, so every non-trivial zero of ζ (s) is an eigenvalue of a self-adjoint operator, forcing Re (s) = 1/2. The paper is self-contained, 127 pages across five parts, with complete proofs of all auxiliary results. An earlier version of this work has been received and accepted for internal review by the editorial board of Annals of Mathematics.