{
  "schema_version": 1,
  "problem_number": "AIM-LINEAR_ALGEBRA-0007",
  "title": "Regular Totally Nonnegative Loops Without Entire Matrix Logarithms",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct an explicit family of regular, entire, two-periodic totally nonnegative loops with determinant one whose inverses are also entire. Each loop is the product of two positive exponential phases. We prove that this product admits an entire complex matrix logarithm if and only if the phase generators commute. Noncommuting phases give a counterexample to Lemma 8.4 of the inspected arXiv version 0812.0840v3. The obstruction compares the trace parity forced on a logarithm at nontrivial positive and negative unipotent values. Coefficient asymptotics prove regularity, and an explicit rational-parameter example has an exact Jordan certificate. The known general two-exponential factorization theorem is sharp within this regular totally nonnegative class. This does not settle the broader AIM block-Toeplitz factorization request.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CV",
    "math.CA",
    "math.CO"
  ],
  "keywords": [
    "total nonnegativity",
    "periodic matrices",
    "block Toeplitz matrices",
    "matrix logarithms",
    "entire functions",
    "regular loops",
    "exponential factorization",
    "AIM-LINEAR_ALGEBRA-0007",
    "math.CV",
    "math.CA",
    "math.CO"
  ],
  "manuscript_version_date": "2026-10-03",
  "publication_date": "2026-10-03",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-03",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-linear-algebra-0007/",
  "pdf_url": "https://eulersolve.org/papers/aim-linear-algebra-0007/paper.pdf?v=44462f6624af",
  "doi": "10.5281/zenodo.23122702",
  "zenodo_record_url": "https://zenodo.org/records/23122702",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of the stated two-phase regular-loop classification and a counterexample to Lemma 8.4 of arXiv:0812.0840v3; no claim about an uninspected journal passage. The original AIM Problem 1.2 asks for a broad AESW-type factorization and is not counted as resolved. The Lam-Pylyavskyy curl-core-whirl factorization is not refuted. General matrix-logarithm failure and the two-exponential upper bound are known; the Kutzschebauch-Studer result is credited. Novelty of the regular positive two-phase classification remains undetermined; no absolute-priority claim. AI-assisted, self-audited and unrefereed; no independent human review or proof-assistant verification.",
  "files": {
    "paper.pdf": {
      "sha256": "44462f6624af395d80e1311f69e04a2ad5b4c78abd686ba62ce68c6b638c6dc4"
    },
    "source.zip": {
      "sha256": "898ff39b10e56bd55d525b55d973635065db16488e2d3523df95893f01b967f7"
    },
    "verification_report.md": {
      "sha256": "b3954611de7d138b880ac1495bfc3ce02593b12661fbd3d5bdb161dcb5742473"
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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."
}