This paper proposes a structural reformulation of the Hodge Conjecture based on a principle of globalrigidity for cohomological realizations. Instead of constructing algebraic cycles directly, it is introduced acategory of compatible cohomological structures (ECC), an adelic cohomology formalism encoding all classicalrealizations simultaneously, and a group of global symmetries acting across realizations (the meta-Galois group).It is formulate a Uniqueness Theorem asserting that admissible adelic cohomology classes invariant under thissymmetry are uniquely algebraic. This framework absorbs known rigidity phenomena (Lefschetz, Mumford–Tate, absolute Hodge classes) and explains the failure of all known near-counterexamples to the HodgeConjecture. This work identify precisely which structural ingredients remain unproven for this principle tobecome a theorem.
Rodolfo Moroz (Sat,) studied this question.