A Rigidity Reduction for the Rational Hodge Conjecture

The Rational Hodge Conjecture predicts that every rational cohomology class of Hodge type (k, k) on a smooth projective complex variety is a rational linear combination of algebraic cycle classes. In this paper, we isolate a rigidity condition satisfied by all algebraic cycles—namely invariance under the action of algebraic correspondences—and show that this condition sharply restricts the space of admissible Hodge classes. We prove that correspondence-invariant rational Hodge classes are motivated in the sense of André, and that algebraicity follows from a single additional rigidity input: that the correspondence-fixed part of cohomology is Tate. This reduces the Rational Hodge Conjecture to an explicit motivic fixed-part statement, thereby localizing the remaining difficulty to a single, well-defined structural gap.