This paper establishes a rigidity and inevitability mechanism for the Riemann Hypothesis within a global coherence framework. Building on a previously introduced admissibility law, it shows that once admissible zero configurations are subject to sufficiently rich global coherence probes, no alternative configuration can remain indistinguishable from the true zeta spectrum. The central result is an injectivity theorem: vanishing of all admissible coherence probes forces annihilation of the associated operator defects. Combined with the admissibility–rigidity equivalence proved in earlier work, this yields critical-line alignment of all nontrivial zeros as a consequence of global consistency rather than local approximation. The approach reframes the Riemann Hypothesis as a rigidity theorem: the true zero configuration is not merely compatible with known constraints, but uniquely forced by them. The paper relies only on unconditional analytic inputs and avoids probabilistic or heuristic assumptions. For the admissibility framework on which this paper depends, see Beyond Zeros: The Riemann Hypothesis as an Admissibility Constraint (DOI: 10.5281/zenodo.18039090).Related structural results on injectivity of coherence probes are developed in a companion paper (DOI: 10.5281/zenodo.18111730).