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

Finite-Stage Calibration–Quantization Main monograph: calibration-quantization. pdf A finite-stage, exact certificate turning a rational Hodge (p, p) class into a Q-algebraic cycle class—through a fixed chain of theorem checkpoints, with no asymptotic “limit-and-hope” step once the stage is chosen. Main theorem Corollary 14. 6 (Main Theorem; Chapter 14, p. 287). For every smooth projective complex variety X and every codimension p, the analytic container of rational (p, p) -classesAₗ, ₏: = H^2p (X, Q) H^p, p (X, C) equals the algebraic subspace generated by codimension-p cycle classes: Aₗ, ₏ = A^algₗ, ₏. Equivalently, every rational (p, p) Hodge class is a Q-linear combination of algebraic cycle classes; Lemma 14. 7 explicitly identifies this statement with the Clay formulation. Scope Rational statement only. No claim is made about the integral Hodge conjecture; integrality appears only after denominator clearing / scaling in the route. Fixed setup throughout. The proof works inside the once-and-for-all cohomological containers Aₗ, ₏ and A^algₗ, ₏, then closes the gap by producing geometric representatives in the fixed class. What makes the method sharp The argument is built as a finite-stage pipeline. A single capture stage N_ is guaranteed by a compactness–separation theorem; after N_ is fixed, all downstream steps are exact on frozen, finite coefficient data, with no post-selection refinement loop. Proof spine Checkpoint map (the visible certificate chain): 11. 70 ;; 12. 42 ;; 12. 55 ;; 12. 57;; 13. 2 ;; ;; 14. 6. This chain is stated explicitly as the stable-norm calibration route certificate in §12. 1. 1. 1. Frozen-stage exact feasibility — Theorem 10. 186 (Ch. 10, p. 196) Moment identities force an exact integer linear system on atom coefficients: Ax = 0, Cx = b (h). 2. Stage existence — Theorem 11. 70 (Ch. 11, p. 240) A compactness–separation argument yields a finite stage N_ satisfying the required convex-hull capture condition. 3. Calibration–quantization — Theorem 12. 42 (p. 269) Feasibility is converted into a scaled integral calibrated cycle with exact class control. 4. Holomorphic-chain recovery — Theorem 12. 55 (p. 273) Calibrated rigidity recovers holomorphic-chain representatives (and records projective algebraic via Chow at the upgrade step). 5. Analytic algebraic upgrade — Theorem 12. 57 (p. 274) In the projective setting, the recovered representatives yield effective algebraic cycles whose classes realize the target rational Hodge class. 6. Bridge + final closure Theorems 13. 16–13. 17 (p. 283) Theorem 14. 5 Corollary 14. 6 (p. 287). The comparison bridge locks the identification, the Hodge-defect collapses, and the main equality—and hence the Rational Hodge Conjecture—follows. Reader-facing input / output Input: a rational Hodge (p, p) cohomology class in H^2p (X, Q). Output: an algebraic cycle class over Q representing it—constructed by a finite chain of exact steps, each pinned to an explicit theorem checkpoint. Quick start: §12. 1. 1 (p. 255) gives the one-page “certificate view” of the entire route.