This note inaugurates the Convia Mathematical Program series. It presents a programmatic, conditional research framework named the Ricci–Schrödinger Correspondence (RSC), which couples the Ricci flow with a Hodge heat evolution on differential forms. The system is designed to share a heuristic monotonic energy structure that performs both curvature smoothing and Hodge harmonic regularization, placing the regularization of the Hodge structure on complex 3-dimensional projective manifolds within a dynamic analytic setting. Scope statement: All statements in this note are programmatic and conditional; no claim of a complete proof of the Hodge Conjecture is made. We establish the conditional implication: if the coupled flow forces the cohomology class of the residual current R in the Siu decomposition to vanish, i.e. R = 0, then the Hodge Conjecture holds in complex dimension three. The core unsolved question remains why the coupled flow should guarantee R = 0.
Kim Hak-Jun (Mon,) studied this question.