{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0190",
  "title": "Exact Positive-Focal Certificates for Fundamental-Matrix Pencils",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For an independent rational 3-by-3 matrix pencil, we give an exact procedure that counts its real projective rank-two members compatible with two positive finite focal lengths, or reports that there are infinitely many. Principal points are known, skew is zero and aspect ratio is one. A positive-kernel formulation of the two-view Kruppa relation yields a division-free polynomial sign predicate. Classical Hermite trace forms evaluate this predicate without constructing or isolating determinant roots. Repeated roots, rank-one members, the projective point at infinity and critical calibration families are included. For identically singular pencils, derivative-root sampling also detects bounded feasible intervals. The complementary count identifies incompatible rank-two members. On a guarded generic chart, the count reduces to four Tarski queries. The construction has polynomial bit complexity for this fixed-size input; no empirical speed advantage over numerical reconstruction is asserted. The exact pencil theorem is complete, but the open-ended AIM Algebraic vision Problem 1.5 is not marked fully resolved. Classical methods and the inherited generic indicator are credited. This AI-assisted, self-audited preprint is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "cs.CV",
    "cs.SC"
  ],
  "keywords": [
    "self-calibration",
    "fundamental matrix",
    "positive focal length",
    "Kruppa equations",
    "matrix pencil",
    "Hermite matrix",
    "Tarski query",
    "positive circuit",
    "exact arithmetic",
    "AIM-ALGEBRAIC_GEOMETRY-0190"
  ],
  "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-0190/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0190/paper.pdf?v=cc5475a9075a",
  "doi": "10.5281/zenodo.23381788",
  "zenodo_record_url": "https://zenodo.org/records/23381788",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of the explicitly stated rational-pencil counting theorem, not full resolution of the open-ended AIM Algebraic vision efficiency question. Known principal points, zero skew, unit aspect ratio and two independently varying finite positive focals. No empirical RANSAC speedup, scene cheirality or arbitrary-intrinsics claim. Classical methods and the inherited generic indicator are credited. Self-audited, unrefereed and AI-assisted; no absolute novelty, independent review or formal verification claimed.",
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      "sha256": "cc5475a9075a7e9a2f7900b5317db83be42a05e57c7899bc8e4d4a40795ec16e"
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    "source.zip": {
      "sha256": "3310f390669dfff67be342131bcad79c12af9959ffba497f513b8b51d617c6b1"
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    "verification_report.md": {
      "sha256": "2b3c61d85450abad036bae9d37a94b836f8c01f94b2ad29a63b690c18feec146"
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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."
}