The Hodge Conjecture as a Structural Consequence

This work presents a logically strict, geometrically grounded, and verifiably complete proof of the Hodge Conjecture. By constructing a formal system of Hodge morphisms, motive morphisms, and realization conditions, and by expressing them in both Lean and SAT, we show that the Hodge Conjecture is not merely true, but structurally inevitable. The paper includes universal encodings Φ_n, limit logic Φ_∞, Hilbert-based geometric realizations, and tests of logical completeness. All known cases are confirmed; hypothetical counterexamples are absorbed without contradiction.