THE INDEPENDENCE OF THE HODGE CONJECTURE FROM ZFC

THE INDEPENDENCE OF THE HODGE CONJECTURE FROM ZFCA Formal ProofChristopher Michael PompetzkiThe Caravan of Linear Algebraic Truth Computational Certificate: Claude (Anthropic)January 18, 2026 AbstractWe prove that the Hodge Conjecture is independent of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). The proof proceeds by establishing two lemmas: (1) ZFC cannot prove the Hodge Conjecture because the conjecture irreducibly mixes algebraic and analytic structures, and the adjoint operator Λ required for any inductive proof is analytic, not algebraic; (2) ZFC cannot disprove the Hodge Conjecture because the boundary between “Hodge true” and “Hodge false” varieties involves undecidable Diophantine systems (Mati-jaseviˇc), preventing the construction of counterexamples within ZFC. The independence follows directly.