{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0112",
  "title": "Exact Small-Softness Stabilization of Euclidean Lattice Packings",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a Euclidean lattice whose nonzero vectors have length at least two, consider the density of the union of balls of radius 1+lambda centered at its points. We prove that, in every fixed dimension d >= 2 and for all sufficiently small lambda > 0, the maximizing lattices are exactly the densest hard-sphere lattices with the smallest kissing number among the hard-optimal lattices. The proof compares a linear local covolume margin with a vanishing Lipschitz constant for pair overlaps, and uses Mahler compactness to exclude all other competitors. In dimension three this proves the lattice alternative of Bezdek and Langi's small-softness FCC conjecture and gives the optimal density pi(1+3lambda-6lambda^2-5lambda^3)/(3sqrt(2)). The global softness threshold is existential. The result does not treat general nonlattice packings.\n\nThis is a complete theorem in the stated Euclidean Bravais-lattice setting, related to AIM-GEOMETRY-0112 (Stability) in ulamai/UnsolvedMath v1.6.0. It does not close the whole open-ended AIM record. Classical hard FCC optimality, Mahler compactness, the no-triple-intersection threshold and earlier local soft FCC optimality are credited. A bounded literature review found no identical global stabilization theorem; absolute priority is not certified. The manuscript is unrefereed and self-audited, with disclosed AI assistance, and is not formally verified. Exact rational regression checks are included with their finite domains and limitations.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.NT"
  ],
  "keywords": [
    "soft ball packing",
    "Euclidean lattice",
    "packing density",
    "FCC",
    "lattice stability",
    "kissing number",
    "AIM-GEOMETRY-0112"
  ],
  "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/aim-geometry-0112/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0112/paper.pdf?v=e7a076747ad2",
  "doi": "10.5281/zenodo.23266221",
  "zenodo_record_url": "https://zenodo.org/records/23266221",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for Euclidean Bravais lattices in each fixed dimension d >= 2 and sufficiently small positive softness: the union-density maximizers are the hard-optimal lattices with minimum kissing number among hard optima. In three dimensions, FCC is the unique maximizer up to orthogonal transformations. The global softness threshold is existential; explicit numerical estimates in the paper are local only. This proves the lattice alternative of Bezdek and Langi's small-softness conjecture, not the nonlattice alternative or the entire open-ended AIM Stability record. Classical hard FCC optimality, local soft FCC optimality, Mahler compactness and overlap geometry are credited. Unrefereed and self-audited; no independent review, formal verification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "e7a076747ad2ee87ecf1a99b38efaa7835f00c0064ec20c0de983032cc0c4574"
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    "source.zip": {
      "sha256": "2b272d53057a29f1a8ef7edaac1f4067abf482215382beb6332fe58121e4bc58"
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    "verification_report.md": {
      "sha256": "25801ccf9492c16c3d2d257eff3920f211dd87e0f7cd7cbce2f8efcbbc7d6f27"
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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."
}