{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0250",
  "title": "Minimal Geodesics under Finite Covers",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A minimal closed geodesic need not have a simple projection under a finite Riemannian cover. We give explicit flat double covers and primitive minimal geodesics whose primitive projections have exactly m double points, for every positive integer m. Both manifolds can be chosen orientable in every dimension at least three. These examples disprove the simple-projection clause of Theorem 2.5 in Contreras and Mazzucchelli's Closed geodesics and the first Betti number (DOI 10.1112/plms.70085). The obstruction is that a deck isometry need not preserve the minimizing cohomology class. We also prove a sufficient repair: if that class is pulled back from the base, the projected geodesic is minimal and its primitive traversal is simple. The authors' non-cover perturbation theorem then applies on the base. The general entropy-density question AIM 8.4.1 is not resolved, and the truth of the cited general Corollary 2.6 is not decided. This English preprint is AI-assisted, self-audited and unrefereed; novelty remains undetermined. No independent human review, formal verification or absolute-priority claim is made.",
  "result_type": "COMPLETE_AUXILIARY_COUNTEREXAMPLE_AND_DESCENT",
  "categories": [
    "math.DG",
    "math.DS"
  ],
  "keywords": [
    "closed geodesics",
    "finite Riemannian covers",
    "Aubry-Mather minimality",
    "flat manifolds",
    "cohomology",
    "counterexample",
    "math.DG",
    "math.DS"
  ],
  "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-geometry-0250/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0250/paper.pdf?v=dd4a585320b4",
  "doi": "10.5281/zenodo.23112889",
  "zenodo_record_url": "https://zenodo.org/records/23112889",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Explicit regular flat double covers carry primitive minimal geodesics whose primitive projections have exactly m double points for every m>=1. Both manifolds can be orientable in every dimension n>=3. This refutes the fixed simple-projection clause of Contreras and Mazzucchelli, Theorem 2.5, DOI 10.1112/plms.70085. For an arbitrary finite cover, a minimizing cohomology class pulled back from the base makes the projection minimal and its primitive traversal simple. The credited non-cover perturbation theorem then applies on the base. The original entropy-density question AIM 8.4.1 is not resolved, and the truth of the cited general Corollary 2.6 is not decided. AI-assisted, self-audited and unrefereed. Novelty remains undetermined.",
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    "source.zip": {
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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."
}