Explicit Hodge Classes and Support Recognition on Special Fourfolds

We study explicit algebraic cycles and the recovery of their supports in three special fourfold settings. For a smooth diagonal intersection of five quadrics in P⁹, a paired splitting plane in the associated ten-hyperplane arrangement produces an orbit of thirty-two degree-four del Pezzo surfaces. Their alternating rational cycle w has square 3. An elliptic-quartic net gives an actual correspondence D = w × η to P² and the Chow projector w × w/3. The paired incidence is smooth of dimension twelve, finite and locally immersed over the sixteen-dimensional diagonal moduli space, and has a nonempty non-evaluation open. Its tangent quotient is an Artinian reduction of the Segre ring of P² × P², with Hilbert function (1, 4, 1). We use this quotient and the relative cycles to identify the complete marked Hodge germ with the paired incidence branch, including its scheme structure. At a very general paired point, the full-character rational Hodge sector is exactly ℚw; the proper characters give the rational Hodge conjecture for these very general fourfolds. The germ statement concerns diagonal deformations. For hypersurface fourfolds of degree d ≥ 5, we prove a reverse recognition theorem. A tangent-exhausting integrated Hodge family with a persistent embedded connected-sum annihilator forces its two candidate ambient planes to lie on the hypersurface and recovers the full rationally weighted primitive class. The conormal variation is surjective onto the true graded ideal-deformation space and has a seventeen-dimensional kernel. An explicit smooth family shows that the equation can nevertheless fail to belong to the square of its Hodge annihilator. Finally, from Paranjape's K3–Prym correspondence we construct a quadratic trace and an explicit sheared odd correspondence, purify it by rational endomorphism selectors, and compute its full and primitive kernels under a stated full Hodge-group hypothesis. Weil-class algebraicity is already known; this part concerns an explicit realization and its action. Further constructions record exact Chow returns and several rigidity phenomena. All genericity restrictions and the distinction between Chow identities, complex Hodge dimensions, and rational Hodge spaces are retained.