Spectral Admissibility and the Riemann Hypothesis

This paper establishes a coercive implication:admissibility of a canonically normalized prime residual implies that all nontrivial zeros of the Riemann zeta function lie on the critical line. The argument treats the Riemann Hypothesis as a boundary-stability consequence of a spectral admissibility condition, rather than as an equivalence, reformulation, or zero-counting statement. A σ-stability framework is introduced in which off-critical zeros generate exponential instability under canonical normalization, while critical-line modes remain neutrally stable. The paper is self-contained with respect to the implication proved. No arithmetic equivalence, Hilbert–Pólya construction, or necessity claim is made. The admissibility hypothesis is stated independently and precisely. Companion notes (I–III) analyze sufficient conditions under which admissibility holds, by controlling defect sets in measure and ruling out isolated instability via local rigidity. These notes are cited only to demonstrate that the admissibility hypothesis is non-vacuous within standard analytic frameworks; they are not logically required to evaluate the main implication.