Universal Kernel Operators and Seed Correspondences in the Direction of the Hodge Conjecture

This record contains a Coherence Geometry manuscript studying universal kernel operators and seed correspondences in the direction of the Hodge Conjecture. The manuscript begins with a natural operator associated with the wedge product on exterior powers and shows that its kernel has a universal representation-theoretic structure. In the symplectic setting, the invariant sector reduces to a canonical two-dimensional plane, allowing an explicit description of the kernel and yielding a universal hyperplane relation. The operator structure is then realized geometrically through algebraic correspondences on polarized abelian varieties. This produces a distinguished seed class whose behavior is governed by the operator kernel. On products \ (XN\), these seed classes generate a canonical subspace \ (WN H^2, 2 (XN) \), identified with the irreducible \ (SN\) -representation of type \ ( (N-2, 2) \). The manuscript shows that the seed generates a correspondence algebra whose action on primitive cohomology produces a graded orbit isomorphic to \ (Sym^ (WN) \). This orbit captures the balanced representations \ ( (N-k, k) \) and gives an explicit representation-theoretic model for the primitive interaction sector. This manuscript is part of the Coherence Geometry Clay-problem research series and is categorized under the Hodge Conjecture. It is presented as a structural coherence-geometric investigation of kernel operators, algebraic correspondences, and primitive cohomology, not as an accepted resolution of the Hodge Conjecture.