Global Conifold Hodge Attachments and Tropical Curve Relations

We study the global cohomology of corrected forward mixed-Hodge-module extensions at ordinary double points. For a projective nodal threefold admitting a projective small resolution, derived adjunction, purity, and forgetting supports identify the global attachment map, up to marked local normalization, with the Gysin map of exceptional curves. Its pure-Tate kernel is also realized as a quotient of the cohomology of the extension object. For Wuebben’s four-node X9 geometry, a strict tropical belt and integral bounding chain realize the primitive Thom-calibrated relation (1,−1,−1, 1); the Hodge evaluation has rank three. A four-nodal S4-equivariant cubic contrasts full-rank formal incidence with vanishing global attachment, while periodic tropical grids exhibit primitive relations and integral-equivariant obstructions. We identify the classical vanishing-cycle rank consequence for a compatible smoothing of the original X9 fiber, and distinguish it from the mirror geometry. Compatibility with the corrected variation cone and limiting mixed Hodge structures remains a morphism-level problem.