{
  "schema_version": 1,
  "problem_number": "OWR-2041-002",
  "title": "Learning Capacity of Totally Geodesic Maps into the Circle",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We classify the squared geodesic loss class of smooth totally geodesic maps from a connected Riemannian manifold without boundary into the unit circle. Its pseudodimension is two exactly when there is no nonzero integral-period parallel one-form; otherwise its fat-shattering dimension at margin 1/24 is infinite. On any closed domain admitting a nonconstant map, a Dirichlet-energy cutoff gives matching logarithmic orders of capacity. The proof combines lattice packing and phase thresholds with an explicit base-four shattering construction. We also determine the closure of affine-circle interpolation values and give a sharp two-point obstruction to an unrestricted interpolation assertion in the 2008 report. The known infinite capacity of circle-to-circle maps is credited. Arbitrary target manifolds in OWR-2041-002 remain unclassified here. AI-assisted, self-audited preprint; no independent review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DG",
    "cs.LG",
    "math.ST"
  ],
  "keywords": [
    "OWR-2041-002",
    "totally geodesic maps",
    "learning capacity",
    "pseudodimension",
    "fat-shattering",
    "Dirichlet energy",
    "circle regression"
  ],
  "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-2041-002/",
  "pdf_url": "https://eulersolve.org/papers/owr-2041-002/paper.pdf?v=ef88761ee947",
  "doi": "10.5281/zenodo.23644457",
  "zenodo_record_url": "https://zenodo.org/records/23644457",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Exact finite-versus-infinite capacity classification for squared geodesic losses of smooth totally geodesic maps from nonempty connected Riemannian manifolds without boundary into the unit circle. The class has pseudodimension two exactly when there is no nonzero integral-period parallel one-form; otherwise its fat-shattering dimension at margin 1/24 is infinite. On closed domains with a nonconstant map, energy cutoffs yield matching logarithmic orders. Also an exact evaluation-closure description and a sharp two-distinct-input obstruction to the unrestricted interpolation assertion in the 2008 report. Arbitrary target manifolds are not classified. The complete classification is for circle targets and squared geodesic losses. Closed-domain energy cutoffs have logarithmic capacity. A two-point counterexample corrects an auxiliary interpolation quantifier; the arbitrary-target manifold question remains unresolved here. The known unrestricted circle infinite-capacity example is credited. 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": "ef88761ee9471ee85aa4fd099a7f107572a180fcd8d5175b9a9cb6542088eee3"
    },
    "source.zip": {
      "sha256": "aa7561b91ce950758e51d682f7175d19e1acd6928adc15367c06882920f9b750"
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    "verification_report.md": {
      "sha256": "fc7db9bef09d970af8acfcb0fa5c2b3e5f2692a660470adf5ccd9f8c842aca74"
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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."
}