Hodge Algebraicity by Projective K-Obstruction No-Ghost

We present a projective K-obstruction framework to address the rational Hodge algebraicity conjecture. By employing a rational Chow ledger as a quotient base, we propagate nonzero residual Hodge classes through a finite faithful tower of data. This tower encompasses base Hodge–Chow data, same-parent variation detectors, branchial Chow–Hilbert incidence, derived/formal and limiting mixed Hodge structures, and concludes with a topological K-theoretic Tate envelope. At each height, the residual class is shown to either close via a trace-effective Chow row, descend through a certified lower-complexity port, or lift faithfully to the subsequent height. The terminal obstruction is rigorously formulated as a Projective K-Obstruction Null-Form estimate. We demonstrate that the preserved top memory class determines a rational topological K-ghost via the rational Chern character. The terminal repair problem is identified as the derived fiber of a bounded same-parent repair stack over the corresponding primitive observer point. We show that its cotangent-obstruction complex yields a universal primitive K-obstruction parent, where reduced branch projectors lie in a finite rational central-support selector algebra with bounded denominators. Consequently, all local obstruction packets are proven to be orthogonal projections of a single total obstruction vector. Finally, we invoke a branching-or-closing argument, which generates arbitrarily large reduced active antichains. Uniform projected charge and Bessel packing forces the inequality N εK(T) ≤ \|OK(T)\|K² for every N, leading to a contradiction. This minimal-counterexample argument excludes the existence of residual Hodge ghosts, thereby establishing the rational algebraicity of Hodge classes.