{
  "schema_version": 1,
  "problem_number": "AMR-099-0003",
  "title": "Equilibria of Inverse-Square Repulsion on the Line Are Arithmetic Progressions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Benjamini asked whether every configuration of points on the real line that is in equilibrium under the inverse-square repulsive force must be an arithmetic progression. Georgakopoulos and Kolountzakis proved this when some gap between consecutive points has maximal or minimal length, and described the general (aperiodic) case as open. We show that the answer is yes. More generally, let 1 < s ≤ 2, and let X ⊂ ℝ be a locally finite set with at least two points such that, for every x ∈ X, the total force Σ_{y∈X∖{x}} |y − x|^(−s) is finite and the net force Σ_{y∈X∖{x}} sgn(y − x) |y − x|^(−s) is zero. Then X is an arithmetic progression. No assumption on the gaps is needed. Subtracting the equilibrium equations of two consecutive points shows that the gaps g_n form a positive harmonic function for an explicit reversible random walk on ℤ with long-range jumps. Equilibrium also bounds the ratio of consecutive gaps, by 1.5386… when s = 2. With this bound, an energy estimate shows that the Doob transform of the walk by g is recurrent. Since 1/g is a positive harmonic function of the transformed walk, it is constant. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR",
    "math.CA",
    "math-ph"
  ],
  "keywords": [
    "equilibrium configurations",
    "inverse-square force",
    "Riesz interaction",
    "arithmetic progression",
    "crystallization",
    "reversible random walk",
    "electrical networks",
    "Doob transform",
    "recurrence",
    "harmonic functions",
    "Benjamini",
    "UnsolvedMath",
    "AMR-099-0003",
    "math.PR",
    "math.CA",
    "math-ph",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-03",
  "publication_date": "2026-10-03",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-03",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-099-0003/",
  "pdf_url": "https://eulersolve.org/papers/amr-099-0003/paper.pdf?v=c195175ff29b",
  "doi": "10.5281/zenodo.23113437",
  "zenodo_record_url": "https://zenodo.org/records/23113437",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Benjamini's question (Points in equilibrium, Question 0.1; Warwick problem list 2015) affirmatively: every locally finite configuration on the line in equilibrium under the inverse-square force, with finite one-sided forces and increasing labelling (the convention of Georgakopoulos and Kolountzakis, Proc. AMS 2017), is an arithmetic progression; more generally for forces |x|^-s with 1 < s <= 2. Georgakopoulos and Kolountzakis had proved the case of an attained extremal gap and called the general case open. Not covered: general decreasing force laws, s > 2, and a principal-value reading with infinite one-sided forces.",
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    "source.zip": {
      "sha256": "447f170a84c602f901b17ad0157896d531bf9a6e6796c8e33ee251871be9c2ad"
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    "verification_report.md": {
      "sha256": "ce3d6db6e323c5271d368b662288cedb4543f3c4bae2f5143d7587ce98f00281"
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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."
}