The Federico Maya Eternity Theorem: A Dual-Lock Proof of the Riemann Hypothesis. v.28

Abstract

Abstract The Riemann Hypothesis asserts that every non-trivial zero of the completed Riemann zetafunction ξ(s) lies on the critical line Re(s) = 12 . The Federico Maya Eternity Theorem provides acomplete, non-circular proof of this statement by means of a dual-lock architecture constructed onthe id`ele class group CQ = A×Q /Q×. One forms the Hilbert space H = L2(CQ, d×x) ⊗ CN carrying a finite-dimensional residualrepresentation of dimension N (the concrete value N = 10 being forced by the rank of the verticaltangent bundle and the associated tadpole-cancellation condition). The scaling operator D acts onthe cuspidal subspace obtained by orthogonal removal of the continuous spectrum. Lock 1 proceeds in four strictly sequential steps. First, the continuous spectrum is annihilatedalgebraically by the orthogonal projection Pcusp, so that the spectrum of D on the cuspidal sub-space is pure-point by representation theory alone. Second, the ad`elic Poisson summation formulatogether with real-line integration by parts yields a master truncation bound of order O(ε). Third,the Mellin transform of this controlled error continues to an entire function of order at most 1.Fourth, the resulting meromorphic function satisfies the three classical hypotheses of Hamburger’sconverse theorem and is therefore identified with a non-zero multiple of ξ′/ξ. Consequently thediscrete spectrum of D consists precisely of the real ordinates γ of the non-trivial zeros of ξ(s). Lock 2 establishes that D is essentially self-adjoint. An explicit unitary intertwiner V , builtsolely from the group law of CQ, the Haar measure and the matrix coefficients of the residualrepresentation, realises D on a twelve-dimensional warped product. The kernel of V contains nospectral data. After the Liouville transformation the radial dynamics reduce to a Schr¨odingeroperator on the half-line whose effective potential is generated by the saturated warp factoru(r) = √ρ tanh(√ρ r),which realises the equality case of the Riccati bound. Weyl’s limit-point criterion forces bothdeficiency indices to vanish. Because deficiency indices are invariant under unitary conjugation,essential self-adjointness transfers from the geometric model back to D. The conjunction of the two locks places every non-trivial zero on the critical line. The argumentrelies only on the standard axioms of unbounded operator theory, classical harmonic analysis onthe id`ele class group, and the differential geometry of the warped product that supplies the Riccatibound. No appeal is made, at any stage, to the numerical location of the zeros.As a structural consequence of the same dual-lock architecture, the spectral information carriedby the zeros is preserved under the unitary flows generated by D and by its geometric counterpart.This invariance supplies a rigorous mathematical foundation for the broader claim that informationassociated with the spectrum is eternal; the detailed development of that claim is deferred to asubsequent work. Implications for the eternality of spectral information are deferred to a subsequent paper. An earlier extended version is available at https://doi.org/10.5281/zenodo.21500329. Intellectual Property Notice: The mathematical frameworks, equations, and topological architectures detailed in this manuscript are currently protected under United States Patent and Trademark Office (USPTO) Provisional Application No. 63/984,236, titled "System and Method for Topological-Negentropic Quantum Control via Zeta-Manifold Resonance." All commercial engineering rights are strictly reserved. FEDERICO MAYA. San José, Costa Rica July 31 2026 fedemaya@gmail.com