Architectural Isomorphism: Geometric Verification of Connes' Spectral Triple via Null-Metric Conditions and Resonance Theory

The Hilbert-P´olya conjecture posits the existence of a self-adjoint operator whose spectrum corresponds to the non-trivial zeros of the Riemann zeta function ζ(s). Historically, the search for such an operator has been conducted within the framework of L2-spectral theory on hyperbolic manifolds; however, the presence of cusps leads to wave leakage and the formation of a continuous spectrum, rendering the naive search for discrete eigenvalues untenable. In this paper, we propose a paradigm shift: a transition from the search for an L2-spectrum to the analysis of scattering resonances (poles of the scattering matrix) in the spirit of Helmholtz acoustics and Lax-Phillips theory. We formulate a geometric framework (the Artemov Conjecture) in which the critical line Re(s) = 1/2 is realized as a null-metric (isotropic) surface of a pseudo-Riemannian manifold, and prime numbers are encoded by a network of hyperbolic resonators. The central result of this work is the formulation of a rigorous architectural isomorphism between this geometric construction and Alain Connes’ abstract noncommutative spectral triple on the adelic space of classes. We demonstrate that the geometric null-metric conditions constitute the exact classical (commutative) shadow of the degeneration of Connes’ spectral distance. In conclusion, we formulate the “Artemov Factorization Conjecture,” which reduces the proof of the Riemann Hypothesis to verifying an identity between the Fredholm determinant of the perturbed operator and the Selberg zeta function.