{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0093",
  "title": "Entropy Selection and Shape Drift for Gaussian Lattice Minima",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For each perfect strictly eutactic lattice, we determine the large-parameter Gaussian energy minimum in a fixed neighborhood of its shape. The minimizing shape moves by order t^(-1), and the leading energy coefficient is the exponential of the maximum entropy of an isotropic probability measure on the shortest vectors. We give an explicit negative correction of order t^(-1) to this coefficient. Uniform localization and a summable tail estimate justify minimization of the expansion. Globally, among unit-covolume lattices, the formula selects Hermite-optimal classes first by the entropy coefficient and then by the squared drift. Poisson summation gives the corresponding small-parameter formula. An exact nine-dimensional cyclic-group example has 160 shortest vectors but optimized coefficient strictly below 160, with an integer/rational certificate. Classical packing theory, entropy duality, the cyclic-group construction and prior fixed-lattice optimality results are credited. The results concern Bravais lattices, not arbitrary point configurations; AIM Problem 1.34 and the full dataset record AIM-GEOMETRY-0093 are not closed. This is an AI-assisted, self-audited, unrefereed preprint; no independent review, formal certification or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.NT"
  ],
  "keywords": [
    "AIM-GEOMETRY-0093",
    "Gaussian lattice energy",
    "eutactic lattice",
    "entropy",
    "theta series",
    "Hermite constant"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0093/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0093/paper.pdf?v=de5cecaf683e",
  "doi": "10.5281/zenodo.23451370",
  "zenodo_record_url": "https://zenodo.org/records/23451370",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete neighborhood-minimum and shape-drift theorem for perfect strictly eutactic lattices, with entropy selection among Hermite-optimal classes. Arbitrary point configurations and AIM Problem 1.34 remain unresolved. Classical packing, entropy duality, the cyclic-group example and fixed-lattice optimality results are credited. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
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      "sha256": "de5cecaf683e022f9ee23edb08a5559653e6742da922d9778e50501dcfd88d82"
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    "source.zip": {
      "sha256": "2fa06d7056cf441d366aeb1b617822d59a11fcea6045e3d8ae0b25d85ad40eb2"
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    "verification_report.md": {
      "sha256": "f4591c11f106f38adccaf715c3a6550908a466001ebfdc3458b3d826a49bb79b"
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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."
}