{
  "schema_version": 1,
  "problem_number": "OWR-15988-004",
  "title": "The Scaling-Rotation Chain Metric for Positive-Definite Two by Two Matrices",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The scaling-rotation distance between positive-definite matrices minimizes the distance between their eigendecomposition fibers. Its associated chain metric also allows changes of eigenframe at repeated eigenvalues. For real two by two matrices we compute that chain metric explicitly. Writing m for the normalized logarithmic determinant, r and s for the normalized logarithmic spectral gaps, and alpha for the acute angle between the top eigenlines, its square is (m_Y-m_X)^2 + min{(r-s)^2+k alpha^2, (r+s)^2}. We characterize all minimizing paths in the finite-piece class and determine exactly when a minimal smooth scaling-rotation curve remains globally shortest, including the orthogonal-eigenline exception. We also prove determinant splitting and a sufficient spectral-gap condition in arbitrary dimension. These calculations use the established Groisser-Jung-Schwartzman framework. They answer the two-dimensional case of Jung's Oberwolfach question, not its general-dimensional classification. This is an unrefereed preprint with AI assistance disclosed; no absolute priority, independent review or formal verification is claimed. Possible overlap with the original authors' separately listed existence manuscript is not excluded.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.DG"
  ],
  "keywords": [
    "OWR-15988-004",
    "positive-definite matrices",
    "scaling-rotation distance",
    "chain metric",
    "eigendecomposition",
    "shortest paths",
    "spectral collision",
    "stratified geometry"
  ],
  "manuscript_version_date": "2026-10-16",
  "publication_date": "2026-10-16",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-16",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15988-004/",
  "pdf_url": "https://eulersolve.org/papers/owr-15988-004/paper.pdf?v=ca7f0795b082",
  "doi": "10.5281/zenodo.23415195",
  "zenodo_record_url": "https://zenodo.org/records/23415195",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for the two-dimensional chain metric, all finite-piece minimizing paths and the exact smooth-curve global-minimality criterion, plus determinant splitting and a sufficient spectral-gap condition in arbitrary dimension. The general OWR-15988-004 classification remains unresolved. The Groisser–Jung–Schwartzman framework is credited. Possible overlap with their separately listed existence manuscript is not excluded. AI-assisted, self-audited and unrefereed; no certified historical priority, independent review or formal verification.",
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      "sha256": "ca7f0795b08295286dac6ca952daf1a8853fa5c3bf884ba942db85c9b84610de"
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    "source.zip": {
      "sha256": "4063894f421380078a705a77cb39d0c762a510b534eae9b425b65af51818fd44"
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    "verification_report.md": {
      "sha256": "67b36286a66115ce209fd50e5d8b6b4f7b26034a62c460e544b9ced32effbc1d"
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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."
}