{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0315",
  "title": "Cyclic Vertex Groups and Integral Toric Restriction",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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.\n\nThese 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.\n\nThe 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.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AT",
    "math.SG"
  ],
  "keywords": [
    "toric orbifold",
    "Kirwan map",
    "integral cohomology",
    "moment-angle complex",
    "cyclic isotropy",
    "Koszul complex",
    "Smith normal form"
  ],
  "manuscript_version_date": "2026-10-10",
  "publication_date": "2026-10-10",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-10",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0315/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0315/paper.pdf?v=373aeac1a24c",
  "doi": "10.5281/zenodo.23281434",
  "zenodo_record_url": "https://zenodo.org/records/23281434",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The stated Borel restriction criteria and counterexample are proved. The general disconnected Kirwan problem, coarse cohomology and integral Chen-Ruan products are not classified; the whole AIM question is not closed. Self-audited and unrefereed; established antecedents are credited and absolute priority is not certified.",
  "files": {
    "paper.pdf": {
      "sha256": "373aeac1a24c86e5918db87454e6da4d11cd66117a932b9b4b8ade751fe147bc"
    },
    "source.zip": {
      "sha256": "dc30301db0238a1dc9824c5517ef27b3ec9b1c5c72be48b5434c105c3e06b031"
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    "verification_report.md": {
      "sha256": "adc027fcc368dead376b3445f57afa5d74c4e7325b6d60c719922e0baeb0f74b"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}