Mathematical proof establishes the location of non-trivial zeros of the Riemann hypothesis, suggesting robust arithmetic structures.
Complete dual-lock proof of the Riemann Hypothesis on the idèle class group CQ=AQ×/Q× CQ=AQ^×/Q^× CQ=AQ×/Q×. Lock 1 realises the arithmetic identification of the discrete spectrum of the scaling operator D D D with the non-trivial zeros of ξ(s) ξ(s) ξ(s) via the adèlic Poisson summation formula and Hamburger’s converse theorem, with explicit Plancherel remainder estimates (Cρ≤2.15 C_ρ≤ 2.15 Cρ≤2.15, master bound 8.4 8.4 8.4). Lock 2 realises essential self-adjointness of D D D by means of a spectrally blind unitary intertwiner V V V and a global saturated Riccati bound on a twelve-dimensional warped product; Weyl’s limit-point criterion forces the deficiency indices to vanish. The conjunction of the two locks places every non-trivial zero on the critical line Re(s)=12 Re(s)=12 Re(s)=21. The argument is non-circular: Lock 1 never uses self-adjointness and Lock 2 never uses the location of the zeros. 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 27 2026 fedemaya@gmail.com
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