The Primacohedron and the Hilbert–Pólya Program: A Conditional Spectral Proof of the Riemann Hypothesis

This paper formulates an axiomatic spectral-geometric framework, called the Primacohedron, and proves a conditional theorem: if the Primacohedron axioms hold and the associated zeta spectral operator exists with the stated self-adjointness and spectral correspondenceproperties, then the Riemann Hypothesis follows. The argument is deliberately conditional. It does not claim an unconditional proof of the Riemann Hypothesis, which remains an open Millennium Prize Problem. Rather, it gives a precise Hilbert–Pólya-type route: a self-adjoint operator whose spectrum is exactly the set of imaginary parts of non-trivial zeros forces all corresponding zeros to lie on the critical line. The contribution is to organize the proposed Primacohedron framework into geometric, spectral, arithmetic, and analytic axioms, to derive the conditional Riemann Hypothesis theorem from these axioms, and to identify exactly what remains to be proved in order to convert the conditional implication into an unconditional proof. Connections with the explicit formula, random matrix statistics, quantum chaos, adeles, noncommutative geometry, and the Langlands program are discussed.