{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0111",
  "title": "Soft Density Profiles Do Not Determine Vacant Percolation",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For any packing of unit balls in three-dimensional Euclidean space, we construct two fixed modifications preserving its upper and lower soft-density profiles at every fixed outer radius. One modification has an unbounded vacant component at every finite radius. The other has infinitely many nonempty bounded vacant components, and no unbounded one, at every radius at least 2/sqrt(3). Below this radius the vacancy of every unit-core packing is path connected and unbounded. The constructions use a widening empty corridor and lacunary triangulated octahedral barriers, whose fixed-width neighborhoods have volume O(R^2) in an observation ball of radius R. An elementary attainment argument then gives percolating and nonpercolating unrestricted upper-density optimizers at every radius at or above the threshold.\n\nThis is a complete scoped three-dimensional theorem related to AIM Soft Packings Problem 1.4 and AIM-GEOMETRY-0111 in UnsolvedMath v1.6.0. It does not settle periodic, saturated, stationary or higher-dimensional versions, determine FCC/BCC optimality, or prove the existence of a fully path-connected optimizer above the threshold. Classical density-insensitivity and triangular-throat geometry are credited to prior literature. No identical whole-profile modification theorem was located in a bounded review; absolute priority is not certified.\n\nAI-assisted, self-audited and unrefereed preprint. No independent peer review or proof-assistant verification is claimed. Exact finite regression programs accompany the source; they do not replace the general written proof.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.AT"
  ],
  "keywords": [
    "soft ball packing",
    "vacant percolation",
    "packing density",
    "density profile",
    "octahedral barriers",
    "saturation",
    "AIM-GEOMETRY-0111"
  ],
  "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-0111/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0111/paper.pdf?v=4226ca4ac531",
  "doi": "10.5281/zenodo.23264461",
  "zenodo_record_url": "https://zenodo.org/records/23264461",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for arbitrary unit-core packings in three dimensions: fixed modifications preserve every fixed-radius upper and lower soft density while producing opposite vacant-percolation behavior at and above the classical radius 2/sqrt(3). Below that radius, all such vacancies are path connected and unbounded. Both kinds of unrestricted upper-density optimizer therefore exist above the threshold. This does not settle periodic, saturated, stationary or higher-dimensional versions, FCC/BCC optimality, or existence of a fully path-connected optimizer above the threshold. The broader AIM record remains partial. Classical density-insensitivity and throat geometry are credited; no absolute priority, independent review or formal verification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "4226ca4ac5316e4f508372767d34017d94e8125d1b88591ceda9d2668539b8cc"
    },
    "source.zip": {
      "sha256": "f87bc7ff76dc44e325a6d26d666550be3dac14262a2c68f6442af86987636b62"
    },
    "verification_report.md": {
      "sha256": "3ddcf94a52875c1a152be0d1e4e7c904bf8f3071d98d8fbf5da75dc29b546fc0"
    }
  },
  "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."
}