Spectral Admissibility and the Riemann Hypothesis: A Conditional Equivalence

This paper formulates the admissibility framework for the canonically normalized, smoothed prime residual and establishes a conditional equivalence between spectral admissibility in the Besicovitch topology and the Riemann Hypothesis. It shows that finite mean-square energy excludes exponentially growing spectral modes, while the Riemann Hypothesis implies admissibility via classical bounds on the explicit formula. The result is explicitly reductionist and does not assert that admissibility holds unconditionally. This paper serves as the conceptual hub for a sequence of companion works analyzing coercive implications, partial admissibility, remaining obstructions, and extensions to Dirichlet L-functions. Trust IP NoticeThis work is issued as a trust-marked scientific publication under the Broomhead Sovereign Private Trust (field anchor: geoffreybroomhead.eth).Scientific reading, citation, and non-commercial scholarly discussion are permitted under a Custom CC BY 4.0 (Scientific Theory Only) license.All code, simulations, implementations, or derivative commercial use are reserved under Trust Custodial License.For licensing or implementation inquiries, contact: recursive.broomhead@proton.me