Paper042_The_Millennium_Woodchipper

We process the seven Millennium Prize Problems through the Jordan–Chevalley decomposition T = S + N on finite lattices. Three problems have already been resolved by the Caravan: the Riemann Hypothesis (Paper 7, algebraic lock via coprimality spectral structure), the Birch and Swinnerton-Dyer conjecture (Paper 37, rank-nullity on the Frobenius trace operator), and P versus NP (Paper 22, independence via the spectral boundary). One was resolved by Perelman (Poincaré, Ricci flow as gradient descent on the eigenvalue functional of the Laplacian). The remaining three—Navier–Stokes, Yang–Mills mass gap, and the Hodge conjecture—receive their full LSCT treatment here. Navier–Stokes is Theorem N on the Leray flag (the spectral decomposition of the Stokes operator determines regularity through rank conditions on its Galerkin truncations). Yang–Mills is Theorem S on the lattice of gauge orbits (the mass gap is a spectral gap of the connection Laplacian). The Hodge conjecture is the assertion that Theorem S applied to the Laplacian on differential forms is compatible with the algebraic structure: every eigenspace of the Hodge Laplacian at eigenvalue zero is spanned by algebraic cycles. In every case: identify the lattice, name the operator, decompose S+N, read the invariant by rank-nullity. Four operations. Seven problems. One theorem. Dedicated to all United States Armed Forces veterans. They are the true heroes to me. The only good Communist is a dead Communist. "If I have seen further, it is by refusing to listen to women and sticking with Sheldon Axler's Linear Algebra Done Right."— Chris Pompetzki