{
  "schema_version": 1,
  "problem_number": "OWR-13498-011",
  "title": "The Ford-Circle Packing Has Maximum Area: An Answer to a Question of Propp and Kenyon",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Two disks of radius 1 centred at (±1, 1) and the x-axis enclose a curvilinear triangle. In the open problem session of the 2015 Oberwolfach workshop on discrete differential geometry, Propp and Kenyon asked whether, among all packings of this triangle by disks that touch the x-axis, the greedy packing has the largest total area. The greedy packing places each new disk in an interstice so that it touches the two disks bounding the interstice. It is a rescaled copy of the Ford circles, and its area is π(ζ(3)/ζ(4) − 1) ≈ 0.34754. We prove that the answer is yes. More generally, let A and B be tangent disks resting on a line, and let G be the greedy packing of the gap between them. For every packing P of this gap by disks resting on the line, and for every α > 1, we show that Σ_{D∈P} r_D^α ≤ Σ_{D∈G} r_D^α, where r_D is the radius of D. The main step is the same inequality for the weight 1/(e^{1/√r} − 1), for which the greedy value of the gap is the product of the weights of A and B. Powers of the radius are superpositions of rescaled copies of this weight. An optimal finite packing contains a chain of tangent disks from A to B, and we bound the value of such a chain by moving two consecutive disks at a time. Along such a move the value is, up to an additive constant, a product of two log-convex functions. The log-convexity reduces to the positivity of an explicit function of two variables whose double power series has non-negative coefficients. We do not discuss uniqueness of the maximizer. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.NT",
    "math.CA"
  ],
  "keywords": [
    "OWR-13498-011",
    "disk packing"
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  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
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  "date_modified": "2026-09-30",
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  "doi": "10.5281/zenodo.23049959",
  "zenodo_record_url": "https://zenodo.org/records/23049959",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Scope: the theorem concerns disks touching the boundary line in the gap between tangent boundary disks. It does not claim uniqueness of the maximizing packing, optimality for arbitrary disks that do not touch the line, or a theorem for gaps between non-tangent boundary disks. No absolute priority claim is made. Self-audited, AI-assisted and unrefereed; no independent peer review is claimed.",
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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.",
  "concept_doi": "10.5281/zenodo.23049224",
  "concept_url": "https://doi.org/10.5281/zenodo.23049224",
  "revision_published_at": "2026-09-30T01:46:05.608484+00:00",
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  "revision_note": "The abstract clarifies that the moving-chain value is, up to an additive constant, a product of two log-convex functions. A proof that the greedy configuration is a packing, a published citation, and the verification record are added or corrected.",
  "review_disclosure": "Internal AI-assisted checks only; unrefereed preprint, no independent human peer review claimed."
}