From Numerical Evidence to Conditional Structure: A Unified Program for the Riemann Hypothesis

This article synthesizes and unifies a multifaceted investigation program of the Riemann Hypothesis (RH), transforming scattered numerical and geometric evidence into a rigorous conditional logical structure. We demonstrate that RH is equivalent to the existence of certain self-adjoint operators whose spectra, under specific transformations, coincide with the non-trivial zeros of the zeta function. We present three concrete candidates for such operators--an integral operator constructed from the prime distribution, the Laplacian on the Enneper minimal surface, and a quantum operator emerging from a conformal transformation of the hydrogen atom--and show how all satisfy, numerically with extreme accuracy (10^(-7) to 10^(-12)), the necessary conditions of the conditional theorem. The underlying geometric structure, encapsulated in the symmetry of a real-analytic function F(s) derived from the Gamma function, provides the unifying bridge between the approaches. We conclude by explicitly stating the open mathematical theorems whose proof would finalize a proof of RH.