Elliptic Memory Energies and Forward-Invariant Admissibility in Arithmetic

This paper develops an invariant-based framework for the study of prime fluctuations in which the Riemann Hypothesis (RH) emerges as a rigidity consequence of forward invariance. A canonical second-order arithmetic invariant—the elliptic memory energy—is introduced as a positive quadratic form on regularized signals derived from the Riemann ξ-function. Using this invariant, an admissibility region is defined via a survivability functional combining elliptic energy, memory depth, and defect observables. A central result is a no-silent-escape principle: an approach to the boundary of admissibility necessarily produces detectable geometric events. Combined with an explicit boundary viability analysis, this yields forward invariance of the admissibility region under canonical windowed probing dynamics. Off-critical behavior would force exit from the invariant region, which is excluded by forward invariance; RH follows as a corollary. The argument is structural in nature and does not assume the Riemann Hypothesis or any zero-distribution results beyond the classical explicit formula. No spectral identification or zero-counting methods are used. This work provides the geometric enforcement layer within a broader admissibility-based program that includes diagnostic equivalence results, partial admissibility bounds, Lyapunov/survivability reductions, 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 Custom CC BY 4.0 (Scientific Theory Only).All code, simulations, implementations, derivative commercial use, or applied systems are explicitly reserved by the Trust.For implementation or licensing inquiries, contact: recursive.broomhead@proton.me.