Cyclic Vertex Groups and Integral Toric Restriction

Let K be the dual boundary complex of a compact simple polytope, and let an integral matrix define a coordinate-torus quotient, with nonsingular restriction at each vertex. We show that restriction from coordinate-torus equivariant cohomology to the kernel subgroup is surjective over the integers exactly when all vertex cokernels are cyclic. Over F_p, the criterion is instead that every vertex determinant be prime to p. For a connected kernel, these also characterize the Borel Kirwan map. The proof calculates the regular-sequence condition in the established additive Koszul model by localizing at arithmetic fibers. We include fixed-sector criteria and an effective compact complex-dimension-three example with primitive normals and unit facet labels whose disconnected reducing group has a surjective integral Kirwan map despite noncyclic vertex groups. These are complete results in the stated Borel-cohomology scope, related to AIM-GEOMETRY-0315 and question 10.3 of the AIM moment-map list. They do not resolve the general disconnected Kirwan problem or classify coarse-space cohomology or integral Chen-Ruan products. Known weighted-projective, cyclic-necessity and Koszul-model antecedents are explicitly credited; no absolute priority claim is certified. The six-page English preprint is by Alper Ferudun, Mercury Software GmbH. It is AI-assisted, self-audited and unrefereed, with no independent review or proof-assistant certification claimed. The source package includes portable exact regression checks; finite computations do not replace the general proof. Version 1.0, 10 October 2026. CC BY 4.0.