{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0211",
  "title": "A Pivot-Independent Toric Model for Framed Carlsson-Weinshall Duality",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An ordered projective anchor simplex and n further marks in its complement determine a configuration torus modulo diagonal translation. Choosing a pivot mark identifies this quotient with a product of tori, which has a product-permutohedral compactification. We construct a pivot-independent normal projective model dominating all these compactifications. Its fan is the arrangement of rectangle characters, or equivalently the normal fan of an explicit lattice zonotope. Inversion extends regularly; in dimension three it gives a biregular Carlsson-Weinshall duality on the closures of the reduced-measurement graphs. For ambient dimension d >= 1 and n >= 2 marks, the common model is smooth exactly when d = 1 or n = 2. Otherwise it contains a four-dimensional cone with five extreme rays and is not Q-factorial. Classical CW duality and standard toric machinery are credited. This is a complete theorem about the specified anchor-framed model, not a stable compactification of unframed camera moduli or a full closure of AIM Algebraic Vision Problem 5.3. AI-assisted, self-audited and unrefereed; no absolute-priority claim is made.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "keywords": [
    "toric geometry",
    "Carlsson-Weinshall duality",
    "configuration spaces",
    "hyperplane arrangements",
    "permutohedral variety",
    "algebraic vision"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0211/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0211/paper.pdf?v=8dfbfd59c8eb",
  "doi": "10.5281/zenodo.23308121",
  "zenodo_record_url": "https://zenodo.org/records/23308121",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for the specified anchor-framed configuration torus: its minimal common normal projective toric model over every pivot, the sharp smoothness and non-Q-factoriality criterion, and regular CW symmetry on the reduced-measurement graph closures. This is not a stable boundary model for unframed cameras or a full closure of AIM Algebraic Vision Problem 5.3. Classical duality and toric results are credited. No independent review, formal verification or absolute-priority certificate is claimed.",
  "files": {
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      "sha256": "8dfbfd59c8eb0dcd36404ba191f364eb51f21bb5045e51a65ae8a63886fbd370"
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    "source.zip": {
      "sha256": "4a64953a69cc8d1119450a5aeb75445153649e669e6d87cba3be2db180977f91"
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    "verification_report.md": {
      "sha256": "d8e1f3f05258ce157f30c457e91a30f07ab0880f45cab160df18134fc1b22290"
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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."
}