{
  "schema_version": 1,
  "problem_number": "OWR-12485-006",
  "title": "A Closed Curve with Finite Critical Menger Curvature and No Tangent at One Point",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give an explicit embedded planar loop of length one with finite integral Menger curvature M_3 whose arclength parametrization is not differentiable at one point. The curve is smooth everywhere else. Its local graph has slope a sin(log log(1/|x|)) plus a smooth term. A direct three-point estimate bounds the critical Menger energy of a Lipschitz graph by the W^(1/3,3) seminorm of its slope. The oscillating slope has finite seminorm, while its secants have two distinct limiting directions. A smooth circle-arc replacement closes the example. This gives a complete negative answer to the everywhere-differentiability subquestion in OWR 22/2013, p. 1353, and Problem 5.3 of Strzelecki and von der Mosel. It does not close the bundled circle-minimization or stationary-regularity questions. Earlier Mobius-energy counterexamples and fractional-Sobolev methods are credited. This is an unrefereed, AI-assisted preprint; priority is not certified.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CA",
    "math.DG",
    "math.GT"
  ],
  "keywords": [
    "OWR-12485-006",
    "integral Menger curvature",
    "critical energy",
    "Lipschitz curve",
    "differentiability",
    "fractional Sobolev space"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12485-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-12485-006/paper.pdf?v=35c71728d99d",
  "doi": "10.5281/zenodo.23633041",
  "zenodo_record_url": "https://zenodo.org/records/23633041",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "An explicit embedded planar loop of length one has finite critical integral Menger curvature M_3, is smooth off one point, and has no geometric tangent or arclength derivative there. This completely answers the everywhere-differentiability subquestion (2) on printed page 1353 of OWR 22/2013, also Problem 5.3 in Strzelecki-von der Mosel. A direct endpoint graph bound with constant 12(1+B^2)^(3/2), logarithmically oscillating slope, closed-curve gluing, all mixed triples and unit arclength normalization are proved. The bundled minimization and stationary-regularity questions are not closed by this manuscript. A complete negative answer to the everywhere-differentiability subquestion (2) in OWR 22/2013, p. 1353, also Strzelecki–von der Mosel Problem 5.3. The bundled circle-minimization and stationary-regularity questions are not closed by this manuscript. Earlier Möbius-energy counterexamples and fractional-Sobolev methods are credited; absolute priority is not certified. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "35c71728d99d507a93498d99a4fbd4c8bbc84c20833b359013be65082694ffe1"
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    "source.zip": {
      "sha256": "a2de2c2838d14963052d1b251cc8d4a31e78e7e89e8c8bd482bb766aabb43054"
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    "verification_report.md": {
      "sha256": "c621cc604b5173a428900ab880c57b7204a8791676c055c5db780d4fe99a3195"
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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."
}