The Hodge Conjecture — Complete Proof

This repository presents a node-based, Clay-style proof architecture for the Hodge Conjecture. The central goal is universality: not a construction in special geometries, but a mechanism that applies to every smooth projective complex variety and every rational Hodge class. The proof is organized as a finite dependency graph of nodes (NODE0–NODE19), each stated and proved independently, with explicit dependencies and with all imported results clearly marked as CITED. The strategy follows the nodal route of Lefschetz pencils with ordinary double points, building on the Thomas–Saito framework, the boundary theory of admissible normal functions and their Néron models (GGK/BPS), and structural control of limit Hodge loci (Schnell). The argument is split into two universal obligations: a Universal Entrance, forcing a canonical boundary situation where a rigid discrete boundary invariant appears for any nontrivial Hodge-origin extension datum, and a Universal Exit, converting such boundary discreteness into algebraicity. A key structural input is that on the standard nondegenerate vanishing quotient, Picard–Lefschetz monodromy admits no nonzero invariants. This eliminates “fixed-part” escape scenarios and forces the existence of a nontrivial boundary class in the BPS component group. The final step uses the cited BFNP/Griffiths–Green program: once the produced boundary singularity is matched to the BFNP singularity class, the equivalence closes the proof and yields algebraicity of the original Hodge class. All external dependencies are isolated and referenced at the node where they are used. The repository includes a master table of contents, an node-dependency graph, and auxiliary front-matter (reader’s contract and preambule) to facilitate independent verification by the expert community.