{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0153",
  "title": "Homotopy Types and Shelling Certificates for the H3 Absolute Order",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We compute the homotopy type of the absolute order of the finite Coxeter group H3 after deleting its identity. The resulting complex is homotopy equivalent to a wedge of four circles and 872 two-spheres. Deleting the central involution as well gives a shellable wedge of 867 two-spheres. The ideals generated by classical and dual Coxeter elements are different: after deleting the identity, they are shellable wedges of 163 and 367 two-spheres, respectively. These assertions have explicit finite shelling certificates covering 4080 facets, checked by a standalone exact program that reconstructs the group and order. The link of the central involution is the disjoint union of five hexagons. It obstructs Cohen-Macaulayness and shellability for the full H3 complex, and the same interval gives an obstruction in H4. This addresses exceptional cases, not the uniform AIM classification. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.AT",
    "math.GR"
  ],
  "keywords": [
    "absolute order",
    "H3",
    "H4",
    "shellability",
    "homotopy type",
    "AIM-PROBABILITY-0153"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0153/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0153/paper.pdf?v=5689e1a8a21a",
  "doi": "10.5281/zenodo.23503571",
  "zenodo_record_url": "https://zenodo.org/records/23503571",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete computer-assisted H3 homotopy and shelling calculation, with an inherited H4 nonshellability obstruction. The full H4 homotopy type, uniform finite-Coxeter-group classification and type-A/type-B EL-labelings are not resolved. The source record AIM-PROBABILITY-0153 remains partially solved. Classical and dual Coxeter conventions are distinct. Existing Coxeter and interval results are credited. Novelty and absolute priority are not certified. AI-assisted, self-audited and unrefereed; no independent review or formal verification.",
  "files": {
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      "sha256": "5689e1a8a21a8cd6f54e4212a482f45ccfe643f8cf1b445e0b36c31f089de28b"
    },
    "source.zip": {
      "sha256": "382916ff2981caf5fd6efa696bd34b3d89066c41e1f32e333d62381c3b225ddd"
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    "verification_report.md": {
      "sha256": "2952021dec02da5f13bc8505037ad3daf6cf7629ad5a0caf94dc57a527b0eb9f"
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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."
}