Injectivity of Coherence Probes and Rigidity of Admissible Zeta Spectra

This paper establishes a rigidity theorem for admissible zeta spectra by proving injectivity of a family of global coherence probes. It builds on the admissibility framework developed in Beyond Zeros: The Riemann Hypothesis as an Admissibility Constraint (F. McGeough, 2025, DOI: 10.5281/zenodo.18039090), in which admissibility is shown to be equivalent to critical-line alignment via operator defect vanishing. The central result proved here is an injectivity (separation) theorem: under explicit and checkable functional-analytic assumptions, vanishing of all admissible coherence probe responses forces annihilation of both the self-adjointness and functional-equation defect operators associated to a…