{
  "schema_version": 1,
  "problem_number": "OWR-15427-019",
  "title": "Log-Free Supersaturation of Collinear Triples in Three-Dimensional Grids",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let F(n,m) be the minimum number of unordered collinear triples in an m-point subset of [n]^3. We prove that F(n,m) has order m^4/n^6 whenever 12n^2<=m<=n^3, with absolute constants. The lower bound removes the logarithmic loss in the Balogh-Solymosi supersaturation estimate. Its proof counts all short primitive positive directions and uses the exact partition of the grid into maximal lattice chains. For the upper bound, we adapt Roche-Newton's periodic finite-field cap construction to arbitrary side lengths and exact cardinalities, using an anisotropic quadratic form and translation averaging. In particular, the minimum has order n^(6-4s) for every fixed 0<=s<1 and m=ceil(n^(3-s)).\n\nThis is a complete proof of a dense-regime order theorem connected to OWR-15427-019 in UnsolvedMath v1.6.0. It does not determine exact minimum values, optimal leading constants or the threshold near m=n^2. The upper method is credited to prior work, and the separate rigidity question in the source is already solved in the literature.\n\nAI-assisted, self-audited and unrefereed preprint. No independent review, formal proof-assistant verification or absolute priority is claimed. Exact finite regression programs accompany the source; they do not replace the general written proof.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "keywords": [
    "collinear triples",
    "supersaturation",
    "integer grid",
    "primitive direction",
    "finite-field cap",
    "discrete geometry",
    "OWR-15427-019"
  ],
  "manuscript_version_date": "2026-10-09",
  "publication_date": "2026-10-09",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-09",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15427-019/",
  "pdf_url": "https://eulersolve.org/papers/owr-15427-019/paper.pdf?v=451899374b92",
  "doi": "10.5281/zenodo.23250910",
  "zenodo_record_url": "https://zenodo.org/records/23250910",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For 12n^2<=m<=n^3, the minimum number F(n,m) of unordered collinear triples in an m-point subset of [n]^3 is of order m^4/n^6. The lower bound removes the logarithmic loss. The cap upper method is credited to Roche-Newton. Exact minima, optimal constants and the threshold near m=n^2 remain unresolved; the separate rigidity question is already known. The broader source record remains partial. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
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      "sha256": "451899374b92bafc49f58eafe48be5221c297881d9c6b56e0607611d66f9e8d2"
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    "source.zip": {
      "sha256": "83a2da436f056a6c19cf9fd81ad53625ed515ac8e96ed6913cdc2624aa48e5fb"
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    "verification_report.md": {
      "sha256": "a7d8ab75ba0a4da71568db86251303cb7972048d6b5f94c906175fb0ea99319e"
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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."
}