Geometric Construction of Riemann-Type Spectral Structure Based on the Hayashi Model: Operator Candidates via Hyperbolic Critical Values, Connection Laplacians, and Torus Cycles, with Trace Formulas

We report on a research program that geometrically reconstructs the completed Riemann zeta function as the spectrum of a self-adjoint operator on a hyperbolic surface. Under Hypothesis 13.1 the spectral zeta function of the Hayashi operator coincides with the completed zeta function and the underlying geometric structure is uniquely determined. Principal contributions include the rigorous computation of the difference kernel between Selberg and Weil explicit formulas, identification of cusp scattering matrices as the realization mechanism, and a conditional reduction of the Riemann Hypothesis via Lax-Phillips scattering theory. This paper does not prove the Riemann Hypothesis but constructs a geometric framework and clearly identifies the remaining gap.