A Finite-Stage Proof of the Rational Hodge Conjecture via Calibration–Quantization
Authors/Creators
Description
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)
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Frozen-stage exact feasibility — Theorem 10.163 (Chapter 10, p. 196): moment identities force the exact integer system
$$Ax = 0,\quad Cx = b(h).$$
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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.
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Calibration → Quantization — Theorem 12.40 (p. 257): converts feasibility into a scaled integral calibrated cycle with exact class control.
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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.
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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.)
Other
Applications to Hodge Classes on Abelian Varieties (Validation Paper)
This companion preprint applies the feasibility–calibration framework to abelian varieties as a controlled testbed. It develops an explicit reconstruction-and-verification workflow for Hodge classes in this setting: the method produces a concrete algebraic cycle representative and then checks, by an independent cohomological route, that the resulting cycle class matches the target Hodge class. The goal of this paper is validation-by-application: it isolates a high-symmetry class of varieties where the framework can be stress-tested, and it documents the full input→output map from rational Hodge data to an algebraic representative in a form designed to be reproducible and checkable.
Other
Reconstruction of Lefschetz (1,1) (Codimension-One Validation Paper)
This paper presents a codimension-one validation of the Hdg1 framework by recovering Lefschetz (1,1) behavior as a rigidity “unit test.” In the divisor case, the target statement is classical and structurally constrained; the paper shows that the same finite-stage pipeline—capture, exact feasibility, and calibration—specializes cleanly to codimension one and reproduces the expected algebraicity of (1,1) classes. The emphasis is not novelty of the end theorem, but the diagnostic value: codimension one provides a precise environment to verify that the framework’s normalization, invariants, and exactness mechanisms behave correctly before deploying them in higher codimension.
Other
Reader Guide: Hodge-to-Chow Reconstruction (Navigation + Proof Map)
This reader guide is a structured roadmap to the full project. It summarizes the world assumptions and frozen normalizations, states the main theorem in “ledger” form (analytic Hodge classes vs algebraic cycle classes), and then maps the proof into a small number of named modules with explicit dependency order. It is designed for fast navigation: where the key definitions live, where each bridge step is proved, what is imported vs proved internally, and how the auxiliary validation papers relate to the main proof spine. The guide is intended for referees and collaborators who want a high-level entry point without losing the precise internal structure of the argument.
Files
reader-guide.pdf
Additional details
Additional titles
- Alternative title
- Applications of the Feasibility–Calibration Framework to Hodge Classes on Abelian Varieties: Validation via Exact Cycle-Class Reconstruction
- Alternative title
- A Feasibility–Calibration Reconstruction of Lefschetz (1,1): Codimension–One Validation of the Hdg1 Framework
- Alternative title
- Reader Guide: Hodge-to-Chow Reconstruction — From Hodge Classes to Algebraic Cycles
- Alternative title
- Proof of the Rational Hodge Conjecture for Smooth Projective Complex Varieties
- Alternative title
- Finite-Stage Rigidity in Hodge Theory: Proof of the Rational Hodge Conjecture