FROM ENERGY ALONE: RECONSTRUCTING HODGE CLASSES VIA SCALAR CURVATURE FIELDS

This article introduces a rigorous energetic-geometric framework for realizing rational Hodge classes via scalar curvature fields. We define smooth potential functions over complex projective varieties and construct differential forms of Hodge type (p, p) as energetic curvature expressions. Our goal is to establish that every rational Hodge class can be represented by such a curvature form, supported on an algebraic cycle, and cohomologous to the class via harmonic projection. This work lays the foundational definitions and initial constructions for the full energetic realization of the rational Hodge Conjecture.