Convia Mathematical Program I: Ricci–Schr¨odinger Correspondence (RSC) and the Hodge Problem

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.