The Federico Maya Eternity Theorem: A Dual-Lock Proof of the Riemann Hypothesis. v.25 (Extension)

The Riemann Hypothesis asserts that every non-trivial zero of the completed Riemann zeta function ξ(s) ξ(s) ξ(s) lies on the critical line Re⁡(s)=12 Re(s)=12 Re(s)=21. The Federico Maya Eternity Theorem supplies a complete, non-circular proof of this statement by means of a dual-lock architecture constructed on the idèle class group CQ=AQ×/Q× CQ=AQ^×/Q^× CQ=AQ×/Q×. One forms the Hilbert space H=L2(CQ,d×x)⊗CN H=L^2(CQ,d^× x)^N H=L2(CQ,d×x)⊗CN carrying a finite-dimensional residual representation of dimension N N N (the concrete value N=10 N=10 N=10 being forced by the rank of the vertical tangent bundle and the associated tadpole-cancellation condition). The scaling operator D D D acts on the cuspidal subspace obtained by orthogonal removal of the continuous spectrum. Lock 1 proceeds in four strictly sequential steps. First, the continuous spectrum is annihilated algebraically by the orthogonal projection Pcusp Pcusp Pcusp, so that the spectrum of D D D is pure-point by representation theory alone. Second, the adèlic Poisson summation formula together with real-line integration by parts yields a master truncation bound of order O(ε) O(ε) 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’s converse theorem and is therefore identified with a non-zero multiple of ξ′/ξ ξ'/ξ ξ′/ξ. Consequently the discrete spectrum of D D D coincides with the non-trivial zeros of ξ(s) ξ(s) ξ(s). Lock 2 establishes that D D D is essentially self-adjoint. An explicit unitary intertwiner V V V, built solely from the group law of CQ CQ CQ, the Haar measure and the matrix coefficients of the residual representation, realises D D D on a twelve-dimensional warped product. The kernel of V V V contains no spectral data. After the Liouville transformation the radial dynamics reduce to a Schrödinger operator on the half-line whose effective potential is generated by the saturated warp factor u(r)=ρtanh⁡(ρ r),u(r)=√ρ(√ρ\,r),u(r)=ρtanh(ρr), which realises the equality case of the Riccati bound. Weyl’s limit-point criterion forces both deficiency indices to vanish. Because deficiency indices are invariant under unitary conjugation, essential self-adjointness transfers from the geometric model back to D D D. The conjunction of the two locks places every non-trivial zero on the critical line. The argument relies only on the standard axioms of unbounded operator theory, classical harmonic analysis on the idèle class group, and the differential geometry of the warped product that supplies the Riccati bound. 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 carried by the zeros is preserved under the unitary flows generated by D D D and by its geometric counterpart. This invariance supplies a rigorous mathematical foundation for the broader claim that information associated with the spectrum is eternal; the detailed development of that claim is deferred to a subsequent 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 29 2026 fedemaya@gmail.com