{
  "schema_version": 1,
  "problem_number": "OWR-13346-003",
  "title": "Border Observations and the Number of Rank-Two Matrix Completions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let H be a connected bipartite graph specifying interior observations of a generic rank-two matrix. Suppose every entry in a border row and a border column is also observed, except their intersection. We determine the number of completions from the cycle structure of H. A tree gives a one-dimensional generic fiber; a unicyclic graph whose cycle has length 2k gives exactly k-1 complex completions; and a graph with at least two independent cycles gives a unique completion. For a cycle with the border intersection restored, we give an irreducible circuit polynomial of degree 2k-1, of degree k-1 in that intersection and degree one in each other variable. The proof uses a Schur complement and a greatest common divisor calculation for two cycle polynomials. The rank-one ingredients and small circuit examples are known and credited. This restricted theorem does not classify all rank-two circuits or settle random-regular matrix completion. This is an unrefereed preprint prepared with AI assistance; no independent review, formal verification, or absolute priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "keywords": [
    "OWR-13346-003",
    "matrix completion",
    "determinantal matroid",
    "circuit polynomial",
    "Schur complement"
  ],
  "manuscript_version_date": "2026-10-24",
  "publication_date": "2026-10-24",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-24",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-13346-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-13346-003/paper.pdf?v=ebcabb36d1ad",
  "doi": "10.5281/zenodo.23623353",
  "zenodo_record_url": "https://zenodo.org/records/23623353",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Generic characteristic-zero rank-two completion with a connected interior observation graph, an observed border row and column, and their intersection missing: tree fiber dimension one; a unique 2k-cycle gives k-1 distinct complex completions; at least two independent cycles give a unique completion. The restored-anchor cycle is an explicit irreducible 4k+1-entry circuit with the stated polynomial and coordinate degrees. The general rank-two circuit and random-regular completion questions remain open; the full source stays partially solved. Standard ingredients and known small circuits are credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
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    "source.zip": {
      "sha256": "dc3f16ac57e9c9ec3445f1bfe5f3193a7d633529a5d432b54e9301dccab211b6"
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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."
}