HSD/SQM AND A UNIVERSAL RESOLUTION OF THE HODGE CONJECTURE

We present a complete analytic–algebraic framework, the Harmonic Scale Decomposition (HSD) and ScaleQuantum Mechanics (SQM), that yields a universal resolution of the Hodge Conjecture. The frameworkcomprises: (i) a Gauss–Manin Bochner Laplacian with essential self-adjointness and compact resolvent onthe orthogonal complement of parallel sections, proved with weighted Sobolev estimates on quasi-projective bases with cusp metrics; (ii) algebraic identification of its finite-rank spectral projectors as Chow correspon-dences via polarizable Hodge modules; and (iii) an envelope–nef equivalence, Demailly regularization, and slicing scheme yielding effective cycles with quantitative bounds and exact equality. We prove the SpectralAlgebraicity Criterion: every rational Hodge class lies in a finite-rank spectral projector and has positiveenvelope, hence is algebraic. This closes the Hodge Conjecture unconditionally.