{
  "schema_version": 1,
  "problem_number": "AMR-021-0002",
  "title": "Finite Equilibria for Rational Inverse-Power Fields in Any Dimension",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We extend the compact-curve method of Enciso and Peralta-Salas for Coulomb equilibria in three dimensions. For distinct real sites, nonzero real charges, and any rational exponent s > 1, the field E_s(x) = sum_i q_i (x-a_i) |x-a_i|^{-s} has only finitely many equilibria where S_s(x) = sum_i q_i |x-a_i|^{-s} is nonzero, in every ambient dimension. In particular, all equilibria are finite in number for same-sign charges. A scalar projection replaces the three-dimensional cross-product step, and a local field-pole argument treats charge sites. An algebraic lift proves complex isolation and a nonsharp degree bound. The three-dimensional Coulomb theorem is prior work; this preprint presents an explicitly attributed extension. AI-assisted, self-audited and unrefereed, with no formal verification or absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CA",
    "math.AG",
    "math-ph"
  ],
  "keywords": [
    "Coulomb equilibria",
    "point charges",
    "inverse-power fields",
    "algebraic curves",
    "AMR-021-0002"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-021-0002/",
  "pdf_url": "https://eulersolve.org/papers/amr-021-0002/paper.pdf?v=6e1196899cb8",
  "doi": "10.5281/zenodo.23494420",
  "zenodo_record_url": "https://zenodo.org/records/23494420",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of the stated rational-exponent theorem and the same-sign Coulomb finiteness question in arbitrary dimension; it also covers Newtonian potentials in dimensions at least three. An explicitly attributed extension of Enciso and Peralta-Salas, arXiv:2609.03965v1, whose three-dimensional Coulomb theorem is prior work. The exponent must be rational and greater than one; mixed signs require nonzero total inverse-power weight. No irrational-exponent theorem, sharp Maxwell bound, certified novelty or absolute priority is claimed. AI-assisted, self-audited and unrefereed; no independent review or formal verification.",
  "files": {
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      "sha256": "6e1196899cb833e931fe23de7dfb7ba11ac0cb9a92f2fdee7ea0e574ca9e3f07"
    },
    "source.zip": {
      "sha256": "1fa984e155895e04d77f3fcb1f4dbab47492367cf973bed000c81c5fd8e9eed4"
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    "verification_report.md": {
      "sha256": "54b3b9d2c6dafdb7211a93ff3652694f09c0289180e579d3e07a6d0dc9f9cc51"
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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."
}