The Hodge Conjecture as a Problem of Prismatic Descent for Algebraic Cycles

We propose a structural reduction of the Hodge conjecture for smooth projective varieties over number fields to the conjunction of two independent statements concerning arithmetic and geometric descent properties. The first (Arithmetic Criterion) requires that crystalline reductions of Hodge classes are algebraic for almost all primes—a weak form of the Tate conjecture. The second (Geometric Criterion) is a descent problem for higher Chow groups: if a Hodge class has algebraic reductions on a Zariski-dense set of special fibers, it should be globally algebraic. Using the prismatic cohomology of Bhatt–Scholze and specialization theory for higher cycles, we formulate a framework in which these two criteria together imply the existence of a global prismatic cohomology class lifting the Hodge class (Theorem 4.1). This framework separates complex-geometric and arithmetic obstructions, linking the former to variational Hodge theory and the latter to crystalline variants studied by Morrow. All technical assumptions are explicitly stated, and dependencies on unpublished results are marked as Gap Statements. The paper concludes with a three-phase research program aimed at resolving each criterion independently. Author's Note: This work represents an undergraduate research project by a first-year student, aimed at exploring foundational structures in logic and arithmetic geometry. These papers are intended as working drafts to establish priority of ideas and invite feedback from the community. They are not claimed as final, peer-reviewed proofs of major conjectures. Acknowledgment of AI Assistance: During the preparation of this work, the author used large language models (specifically Qwen and DeepSeek) as research assistants for brainstorming, technical drafting, LaTeX formatting, and literature navigation. The author takes full responsibility for all mathematical claims, logical structure, and final content. AI tools were used to accelerate exploration, not to replace critical thinking or verification. All errors, omissions, and conceptual gaps are the author's own. Correspondence, constructive criticism, and collaboration proposals are welcome at poslpytka@gmail.com.