{
  "schema_version": 1,
  "problem_number": "OWR-1061-006",
  "title": "Non-Regenerative Sampling Distributions of Ξ-Coalescents: Möhle's Conjecture with a Kingman Component, for Finitely Many Atoms or Boundedly Many Dust-Free Boxes, and Off One Mutation Rate",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "M. Möhle (Oberwolfach Report 40/2005) conjectured that the sampling distributions of the infinitely-many-alleles model along a coalescent with simultaneous multiple collisions (a Ξ-coalescent), with mutation rate ρ > 0, are never regenerative in the sense of Gnedin and Pitman, except for Kingman's coalescent and the star-shaped coalescent. He had proved this for coalescents with multiple but not simultaneous collisions (Λ-coalescents; Combin. Probab. Comput. 16 (2007)). We prove the conjecture for three classes of Ξ and for all but at most one mutation rate in general; the conjecture is not settled. Write Ξ = aδ_0 + Ξ_0. The main tool is an identity: the regeneration equation at the partition (2,2,1^{n−4}) holds if and only if Φ(n−1)X(n) = Bin(n)W(n−1), where X(n), Bin(n) and W(n) are the total rates of the mergers of two pairs, of one pair, and of all other kinds, among n lineages, and Φ(n) is nρ plus the total merger rate. Consequences: (A) if Ξ is not a Λ-coalescent, the partition structure is non-regenerative for every ρ > 0 except possibly one explicit value ρ*(Ξ), and Möhle's theorem is recovered at sample size four; (B) every Ξ-coalescent with a Kingman component (a > 0), other than Kingman's coalescent, is non-regenerative for every ρ > 0; (C, D) the same holds for a = 0, the star-shaped coalescent excepted, when Ξ_0 is carried by paintboxes without dust with boundedly many boxes, or is a finite sum of point masses at paintboxes with finitely many boxes. The measure Ξ = (17/1024)δ_0 + δ_(1/4,1/8,0,…) with ρ = 13833/5120 is regenerative at all sample sizes n ≤ 5 but not at n = 6, so finitely many sample sizes do not decide the question. Open remains the case a = 0, Ξ_0 charging {x_2 > 0} and outside the classes of (C) and (D), at the single value ρ = ρ*(Ξ). The conjecture and the Λ case are Möhle's; his recursion for the sampling probabilities is re-derived. No printed answer to the question was found in the literature searched, which is not a proof of novelty. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.PR"
  ],
  "keywords": [
    "Xi-coalescent",
    "coalescent with simultaneous multiple collisions",
    "infinitely-many-alleles model",
    "sampling formula",
    "exchangeable partition probability function",
    "regenerative partition structure",
    "regenerative composition structure",
    "Möhle's conjecture",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-1061-006",
    "math.PR",
    "q-bio.PE",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1061-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-1061-006/paper.pdf?v=0e4026fa664a",
  "doi": "10.5281/zenodo.23483631",
  "zenodo_record_url": "https://zenodo.org/records/23483631",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to Moehle's conjecture: proved with a Kingman component, for two classes without one, and for every non-Lambda coalescent off one explicit mutation rate; the general case stays open.",
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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."
}