{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0184",
  "title": "Four- and Five-View Thresholds for Calibration with Known Focal Scales",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Suppose projective camera matrices are available and the two diagonal focal scales are known in every view, while skew and principal point may vary independently between views. We give explicit sharp generic thresholds for admissible rank-three metric upgrades over the complex numbers: four views give a finite fiber of degree at least two, whereas five compatible views give a unique upgrade. The four-view ambiguity persists on a real open set, with two positive-semidefinite upgrades, affinely independent camera centers, and scene points in front of all cameras in both reconstructions. The proof uses classical dual-quadric calibration, an irreducible incidence variety, two exact regular rational solutions, and a separator for every false upgrade. In the fifth view the product of the two focal scales alone is sufficient. We do not compute the four-view degree or settle other interpretations of the open-ended AIM question on partial calibration. This is an AI-assisted, self-audited, unrefereed preprint; no independent review, proof-assistant formalization or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "cs.CV"
  ],
  "keywords": [
    "AIM-ALGEBRAIC_GEOMETRY-0184",
    "partial calibration",
    "dual absolute quadric",
    "known focal scales",
    "generic identifiability",
    "cheirality"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0184/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0184/paper.pdf?v=80f42291352a",
  "doi": "10.5281/zenodo.23390147",
  "zenodo_record_url": "https://zenodo.org/records/23390147",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for the explicitly stated model: projective cameras are available, both diagonal focal scales are known separately in each view, and skew and principal point may vary. Five-view uniqueness is for generic compatible data. The exact four-view degree and the known-center branch are not resolved; the entire open-ended AIM record is not claimed solved. Classical calibration methods are credited. Self-audited, unrefereed and AI-assisted; no absolute novelty, independent review or formal verification claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "80f42291352aa9d32bf07d1fd88481f6007c4488b7f73c6594037650bd90e379"
    },
    "source.zip": {
      "sha256": "a3ef01b760686dcbe4fe6405b39bc6fefc64c1587d108061a93d9bc21f99d086"
    },
    "verification_report.md": {
      "sha256": "04ba57146f785f30d0c0b673df677978ff8de865859952f65474a5becf5cf30a"
    }
  },
  "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."
}