{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0195",
  "title": "Angle-Data Reconstruction for Triangulated Polyhedral Surfaces and a Genus Deficit",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An AIM problem asks for a characterization of the face-angle and dihedral-angle data of a triangulated polyhedral surface in ℝ³ and conjectures that the realizable data have dimension E−1 in every genus. We use the realization convention later made explicit by Hempel: a labeled, simplexwise-linear, simplexwise-injective map of a consistently oriented triangulated closed surface, modulo similarities. The combined face-angle and oriented-dihedral map is injective and has a local real-analytic left inverse at every nondegenerate realization. Its actual image therefore has dimension 3V−7 = E−1−6g. Thus the AIM formula is correct for the sphere and fails in every positive genus. The embedded Császár torus gives the concrete count 14 rather than 20. We also give a finite necessary-and-sufficient realizability test: intrinsic sine-law compatibility followed by vertex and non-tree-hinge closure in a dual-spanning-tree development. At generic realizations, Fogelsanger rigidity shows that face angles alone already have rank E−1−6g, identifying the missing 6g as global extrinsic closure codimension. This note is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "keywords": [
    "polyhedral surfaces",
    "angle data",
    "dihedral angles",
    "rigidity",
    "triangulations",
    "AIM-GEOMETRY-0195",
    "open mathematics",
    "mathematical proof",
    "math.MG",
    "math.CO"
  ],
  "manuscript_version_date": "2026-08-29",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0195/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0195/paper.pdf?v=2c4aecc22f5d",
  "doi": "10.5281/zenodo.22245788",
  "zenodo_record_url": "https://zenodo.org/records/22245788",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
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    "source.zip": {
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  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}