Primitive Exterior-Square Frobenius Moments for a Paired-Root Genus-Two Pencil

For a genus-two pencil whose six branch points form three paired roots, we study the rank-five primitive exterior-square local system and a proposed exact finite-field Frobenius moment. We formulate a conditional first-moment identity and an associated degree-fifteen Frobenius polynomial. The formal cohomological derivation uses a tame-sheaf boundary-defect criterion and geometric inputs specifying the based Picard--Lefschetz action, compatible prime-to-characteristic specialization of Jacobian two-power torsion, and Frobenius-equivariant identification of five inertia-fixed stalks with nearby-cycle modules. We provide explicit braid-path, nodal and quartic infinity calculations, together with reproducible finite-field tests. The family-specific moment formula is conditional on the stated comparison hypotheses; no unconditional closure is claimed.