Admissibility of the Smoothed Normalized Prime Residual

This paper develops the admissibility framework introduced in Spectral Admissibility and the Riemann Hypothesis: A Conditional Equivalence by analyzing the smoothed, canonically normalized prime residual as a long-horizon arithmetic field. Using the smoothed explicit formula and spectral decomposition, it shows that admissibility excludes exponentially growing spectral modes and motivates admissibility as a natural second-order stability condition. No argument is given establishing admissibility unconditionally; the purpose is to place admissibility on structural footing within the framework defined in Paper 0. 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