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 $(p,p)$-classesAX,p := H²ᵖ(X,Q) ∩ Hp,p(X,C)equals the algebraic subspace generated by codimension-p cycle classes:AX,p = AᵃˡᵍX,p. 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 AX,p and AᵃˡᵍX,p, 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 H²ᵖ(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.