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 configuration. Consequently, any admissible zeta-compatible configuration indistinguishable from the Riemann zeta spectrum by all coherence probes must coincide with it up to admissible symmetry. This paper isolates the rigidity mechanism required to exclude off-critical configurations. It does not address the origin of admissibility itself, nor does it assume a specific mechanism enforcing probe annihilation. The SN coherence invariant is introduced only as one concrete realisation producing probe annihilation for the true zeta configuration; the injectivity and rigidity results themselves are independent of this mechanism.