A Finite-Stage Proof of the Rational Hodge Conjecture via Calibration–Quantization

Finite-Stage Calibration–Quantization (Main Monograph: calibration-quantization.pdf) Claim (proved in the main paper) Theorem 14.6 (Chapter 14, p. 282) proves the Rational Hodge Conjecture in the paper’s frozen Hdg1 ledger form: for every smooth projective complex variety $X$ and every $p$, every rational Hodge $(p,p)$ class is a $\mathbb{Q}$-linear combination of codimension-$p$ algebraic cycle classes. Scope is explicit. This is the rational statement (no claim about the integral Hodge conjecture). The target and output spaces are fixed once-and-for-all (“Hdg1” containers), and the conclusion is the global container equality closing the ledger gap. The Method (finite-stage, exact, rigidity-driven) This is a finite-stage construction: once the correct stage is guaranteed, every downstream step is exact on frozen discrete data (no asymptotic “limit-and-hope” step). Proof spine (Route 3A — with check points) Frozen-stage exact feasibility — Theorem 10.163 (Chapter 10, p. 196): moment identities force the exact integer system $$Ax = 0,\quad Cx = b(h).$$ Stage existence — Theorem 11.66 (Chapter 11, p. 235): guarantees a finite stage $N_\star$ where the required separation/convex-hull condition holds, turning the pipeline from conditional to unconditional. Calibration → Quantization — Theorem 12.40 (p. 257): converts feasibility into a scaled integral calibrated cycle with exact class control. Analytic → algebraic upgrade — Theorems 12.54 and 12.56 (p. 263): holomorphic-chain recovery produces analytic cycles and Chow upgrades them to algebraic cycles in the projective setting. Bridge + final closure — the proof spine is summarized in reader-guide.pdf, which lists the complete theorem chain ending in Thms. 13.15, 13.18, 14.6. What the reader gets Input: a rational Hodge $(p,p)$ cohomology class. Output: an algebraic cycle class over $\mathbb{Q}$ representing it—produced by a finite chain of exact steps with explicit theorem checkpoints. (For a one-page map of the entire spine with theorem numbers, see reader-guide.pdf, page 1.)