Riemann Hypothesis as a Conditional Spectral Theorem: A Self-Adjoint Dirac-Type Operator with Full Riemann Spectral Structure

Source abstract

We construct a self-adjoint Dirac-type operator Dχ on L²(R; C²) with a confining cosh-potential, whose zeta-regularised Fredholm determinant ξχ(s) = det_reg(Dχ − (s − 1/2)I) reproduces all known analytic properties of the completed Riemann zeta function ξ(s) = (1/2) s(s − 1) π^{−s/2} Γ(s/2) ζ(s): entire function, functional equation ξχ(s) = ξχ(1 − s), order of growth, zero density matching the Riemann–von Mangoldt formula, and all zeros lying on the critical line Re(s) = 1/2. We prove that the Riemann Hypothesis follows as a conditional theorem: if ξχ(s) = ξ(s) (the Identification Lemma), then all non-trivial zeros of ζ(s) lie on Re(s) = 1/2. The Identification Lemma is the unique open step. We discuss three concrete pathways to close it: Euler product (trace formula), special-value matching, and trivial-zero verification. Numerical evidence confirms correct spectral density and scale.