Spectral Admissibility and the Riemann Hypothesis

This version corresponds to the manuscript submitted to Communications in Mathematical Physics. The paper establishes a coercive implication: under a precisely defined spectral admissibility condition on a canonically normalized prime residual, all nontrivial zeros of the Riemann zeta function must lie on the critical line. The argument treats the Riemann Hypothesis as a boundary-stability consequence of a functional-analytic condition, rather than as an equivalence, reformulation, or zero-counting statement. No form of the Riemann Hypothesis is assumed. The manuscript is self-contained with respect to the implication proved. All definitions, assumptions, and estimates required to verify the argument appear explicitly in the paper. In particular, the admissibility hypothesis is stated independently and is not derived from auxiliary results. Companion notes (Papers I–III), cited in the manuscript, analyze sufficient analytic conditions under which the admissibility hypothesis holds via mean control, local rigidity, and fixed-threshold closure. These companions are not logically required to evaluate the implication proved here; they are included to demonstrate that the admissibility condition is non-vacuous within standard analytic frameworks.