Complete Opposite-Pair Evaluation Towers and the Hodge Conjecture for All Mixed Powers

We study the complete family of diagonal evaluation intersections of quadrics associated with pairs of opposite points, together with their hyperelliptic source curves, quotient curves, and Jacobians. For every number of opposite pairs, we construct exact algebraic correspondences with rational inverses and determine the connected joint monodromy of the complete reduced system of source curves. After accounting for the identifications and quotient decompositions forced by the opposite-pair geometry, the joint monodromy is an explicit product of symplectic groups. Using this calculation and the algebraicity of symplectic invariant tensors, we prove that the rational Hodge conjecture holds for every finite mixed product of varieties in the complete evaluation tower, their associated curves, and their Jacobians, with arbitrary repetitions of factors. The result holds simultaneously in every codimension at very-general parameters, outside a single countable union of proper closed algebraic subsets. For five opposite pairs, the theorem applies to a nonisotrivial family of Calabi–Yau fourfolds and all 45 associated complete intersection surfaces, including every mixed power of these varieties. We also determine the very-general Picard ranks of these surfaces: 105 for the five symmetric surfaces and 101 for the remaining forty. On every smooth fiber, the paper additionally establishes exact rational Chow-motive decompositions and proves that the evaluation tower motives are of abelian type and Kimura finite-dimensional. The all-mixed-powers Hodge conjecture conclusion is asserted for very-general parameters, rather than for every smooth fiber.