{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0110",
  "title": "Saturated Aperiodic Packings with Finite Stability",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give explicit one-parameter families of saturated unit-ball packings in R^2 and R^3 with trivial translational stabilizer and exact square or cubic grid contact graphs. Every finite subfamily is locally isolated when its complement is fixed. For rational parameters, all centers and the planar equilibrium stresses are rational. The planar proof uses a positive reciprocal stress and a finite-support rigidity argument; a layered extension gives the three-dimensional result. Nevertheless, varying the parameter produces a nontrivial global motion continuous in the uniform displacement norm and preserving every contact. The result concerns finite stability, not rigidity against arbitrary motions of infinitely many balls. It is a scoped construction related to Musin's aperiodic-jamming question, not an unrestricted or strict-jamming solution. This unrefereed, AI-assisted preprint makes no absolute priority claim.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "keywords": [
    "sphere packing",
    "aperiodicity",
    "finite stability",
    "equilibrium stress",
    "rigidity",
    "rational coordinates",
    "AIM-GEOMETRY-0110"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0110/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0110/paper.pdf?v=51f05d93b65a",
  "doi": "10.5281/zenodo.23432857",
  "zenodo_record_url": "23432857",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Constructs saturated aperiodic finitely stable unit-ball packings in dimensions two and three. A nontrivial uniformly bounded all-particle flex remains, so this is not an unrestricted or strict-jamming solution of Musin's original problem. Uniform finite-rearrangement stability and density optimality are not claimed. AI-assisted, self-audited and unrefereed; no independent expert review, formal verification or absolute-priority certification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "51f05d93b65aa7da635e3c147cbbb666352efeb4d9395da7ea2ae20529ea02d1"
    },
    "source.zip": {
      "sha256": "e1e6ac324a3b8bdf9df34251d91f69bd81ce64666366ac845391445a0e8fba5f"
    },
    "verification_report.md": {
      "sha256": "174857da181465f5439809d47edaecca07f2dde13e95f79e37c9e6b2090e9d1b"
    }
  },
  "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."
}