{
  "schema_version": 1,
  "problem_number": "AMR-036-0023",
  "title": "Radial Barriers for Discrete Newtonian Potentials: Counterexamples and Sufficient Conditions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a locally finite configuration of positive masses in R^n, n >= 3, we study the existence of spheres on which the Newtonian potential is uniformly small. Unit masses at the positive integers contradict the dimensional extension of an auxiliary lemma of Clunie, Eremenko and Rossi for every n >= 4; the same example nevertheless has infinitely many nondegenerate equilibrium points. Under potential summability, we give valid radial criteria involving the accumulated (n-2)th roots of the masses when n >= 4, and a mass-entropy sum when n = 3. Each criterion implies infinitely many equilibria escaping to infinity. Finite-total-mass examples show that the corresponding sublinear growth requirements cannot be replaced by linear bounds in these radial assertions. These results neither refute the equilibrium-existence theorem of the cited paper nor settle the general positive-mass question under force summability alone. This is an unrefereed manuscript; no absolute priority claim is made.",
  "result_type": "COMPLETE_AUXILIARY_COUNTEREXAMPLE_AND_CONDITIONAL_THEOREMS",
  "categories": [
    "math.CA"
  ],
  "keywords": [
    "Newtonian potential",
    "equilibrium points",
    "radial barriers",
    "positive masses",
    "counterexample",
    "mass entropy"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-036-0023/",
  "pdf_url": "https://eulersolve.org/papers/amr-036-0023/paper.pdf?v=f98e46624bd4",
  "doi": "10.5281/zenodo.23290393",
  "zenodo_record_url": "https://zenodo.org/records/23290393",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete auxiliary counterexamples and conditional theorems, not a solution of the unrestricted AMR-036-0023 question. The cited equilibrium-existence theorem is not refuted; the integer-lattice example itself has infinitely many equilibria. The classical superlevel method is credited. Self-audited and unrefereed; no independent review, formal verification or certified priority.",
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      "sha256": "f98e46624bd4ab41f36421c813a9ed506efc26bc9700f8a38409f0cc8d2efbc1"
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    "source.zip": {
      "sha256": "d08b672f4fe318357f91bbcedd33b8aebed04726e2e65690266b3ec7c7d72f56"
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    "verification_report.md": {
      "sha256": "da75cb8d32c49e651806642ac3e010c1014610e0da2c08bcfd14da4457c3ce28"
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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."
}