# Verification report — OWR-1061-006 (Möhle: regenerativity of the sampling distributions of Ξ-coalescents)

Verification date: 2026-10-19.

**Verdict.** The note gives a **partial answer**; the conjecture is **not** settled. Every statement of the note is
proved as stated. Two independent AI-assisted verification runs checked the first written version, and a further
independent AI-assisted verification run checked the text of the note itself (see below).
Write Ξ = a δ_0 + Ξ_0 and let ρ > 0 be the mutation rate.
- **Settled (the conjectured non-regenerativity holds for every ρ > 0).** (B) every Ξ with a Kingman component,
  a = Ξ({0}) > 0, and Ξ_0 ≠ 0; (C) a = 0 and Ξ_0 carried by dust-free paintboxes with at most K_max < ∞ boxes, not
  a multiple of δ_(1,0,…); (D) a = 0 and Ξ_0 a finite sum of point masses at paintboxes with finitely many boxes, not
  a multiple of δ_(1,0,…).
- **All but one mutation rate.** (A) If Ξ is not a Λ-coalescent, the partition structure is non-regenerative for
  every ρ > 0 except possibly ρ*(Ξ) = (2 t_4 τ_3 / j_4 − 3 t_3 − τ_3)/3; Π_4 is regenerative if and only if
  j_4 Φ(3) = 2 t_4 τ_3.
- **Known, recovered.** Möhle's theorem for Λ-coalescents (CPC 16 (2007)): regenerative iff Λ = c δ_0 or c δ_1
  (Corollary 3.2, at sample size four).
- **Tool.** Lemma 3.1: the defect of the regeneration equation at (2,2,1^{n−4}) equals
  p(1^n) Δ_n / (ρ² C(n,2)² C(n−2,2)) with Δ_n = Φ(n−1) X(n) − Bin(n) W(n−1); so regenerativity forces (D_n)
  Φ(n−1) X(n) = Bin(n) W(n−1) for every n ≥ 4.
- **Low levels do not decide.** Ξ = (17/1024) δ_0 + δ_(1/4,1/8,0,…), ρ = 13833/5120 = ρ*(Ξ): regenerative at all
  sample sizes n ≤ 5, not at n = 6 (exact; Proposition 7.1).
- **Open.** a = 0, Ξ_0 charging {x_2 > 0}, Ξ_0 outside the classes of Theorems C and D (for instance with infinitely
  many atoms, or with a diffuse part outside the class of Theorem C), at the single value ρ = ρ*(Ξ) when it is
  positive. Corollary E gives a further
  necessary condition there when M = ∫ Ξ_0(dx)/p_2(x) < ∞.
- **Credit and limits.** The conjecture and the Λ case are Möhle's (OWR 40/2005; CPC 2007). The sampling recursion
  is Möhle's (Bernoulli 12 (2006), Theorem 5.1), used in the form of Dong (arXiv:0707.1606, eq. (12) and Theorem 3)
  and re-derived; regenerative partition and composition structures are Gnedin–Pitman's; Ξ-coalescents are
  Möhle–Sagitov's and Schweinsberg's. No printed answer was found; no priority is claimed.

The note is unrefereed. Under the dataset's policy for unrefereed claims, the status of the record stays open (or partially
solved) with the evidence type unrefereed_claim; the proved family must be named exactly as above, and the conjecture must not be
called settled.

## Statement checked
- **Primary source.** M. Möhle, "Coalescent theory – simultaneous multiple collisions and sampling distributions",
  extended abstract in *Mathematical Population Genetics*, Oberwolfach Report 40/2005, Oberwolfach Rep. 2, issue 3,
  pp. 2241–2304 (the abstract on pp. 2279–2282), doi:10.4171/OWR/2005/40; Crossref gives the publication year 2006,
  which the note uses. Read in the open PDF on the EMS Press site by the first version and by both verification runs
  (552,769 bytes); the conjecture is the last sentence of Section 3 of the abstract (p. 2281).
  - Sections 1–2 (pp. 2279–2281): Λ-coalescents, the infinitely-many-alleles model with mutation rate r > 0 on every
    branch, the sampling probabilities q(a) and their recursion (3) for the Λ case; Ewens's formula (5) for Kingman's
    coalescent; the star-shaped case gives a hook composition structure (Gnedin–Pitman, Section 6).
  - p. 2281: Möhle's theorem (preprint, = CPC 2007): for a Λ-coalescent with mutation rate r > 0 the corresponding
    composition structure is regenerative iff Λ is concentrated in 0 or in 1.
  - Section 3 (p. 2281): the wider class of sampling distributions of coalescents with simultaneous multiple
    collisions, parametrised by measures Λ_j on the simplices Δ_j (Möhle–Sagitov, Schweinsberg); the conjecture:
    all corresponding composition structures are non-regenerative except when Λ_j = 0 for all j ≥ 2 and Λ_1 is
    concentrated in 0 or in 1.
- **Corpus record.** ulamai/UnsolvedMath, OWR-1061-006 (dataset version 1.8.0; status `open`). Its statement is a
  faithful restatement of the conjecture.
- **Conventions.** "Regenerative" is the notion of Gnedin–Pitman (EJC 11(2) (2005) R12, Definition 1): some random
  part can be deleted so that the rest is distributed as the partition of the smaller sample; this is the
  unordered form of a regenerative composition structure (Corollary 6 there), which is why the source speaks of
  composition structures. Dong–Gnedin–Pitman (p. 1) and Lagerås (Proposition 4) state Möhle's theorem in these
  terms. The scale of (Ξ, ρ) is immaterial. Ξ = 0 (no coalescence) is excluded: its partition into singletons is
  trivially regenerative.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Gnedin–Pitman regenerativity of the allelic partition structure, some deletion rule, every ρ > 0 (the source's reading) | partial: proved for the classes of Theorems B, C, D and for every non-Λ Ξ off ρ*(Ξ); open otherwise | Theorems A–D |
| Kingman's older notion (size-biased deletion only) | not the question; only Ewens structures qualify (Kingman's theorem) | Section 1.1 |
| Composition into litters ordered by age (Lagerås 2007) | a different composition, regenerative for every Λ; not the question | Section 1.3 |
| A fixed exceptional ρ instead of all ρ > 0 | the conjecture quantifies over every r > 0; Theorem A shows that at most one ρ per non-Λ Ξ can be exceptional | Theorem A |
| Ξ = 0 | excluded (trivially regenerative) | Question 1.1 |

## Results in the paper
- **Lemma 2.1** (Möhle's recursion, re-derived by first-step analysis): Φ(n) p(π) = ρ Σ_{singletons} p(π∖{i}) +
  Σ_{C compatible} λ_C p(π/C); p(1^n) = Π_{m≤n} mρ/Φ(m).
- **Lemma 2.2** (one-level test): the hooks determine q(n;·), the hook-determined q(n;·) is nonnegative at every
  level (collision terms of Lemma 2.1), and Π_n is regenerative iff (R_n) holds at every non-hook partition.
- **Lemma 2.3**: p(2,1^{m−2})/p(1^m) = T(m)/(ρ m(m−1)).
- **Lemma 3.1**: the (2,2,1^{n−4}) identity above; **Theorem A** and **Corollary 3.2** follow at n = 4
  (X(4) = 3 j_4, Bin(4) = 6 t_4, W(3) = τ_3).
- **Lemmas 4.1–4.3** (paintbox estimates): P(Z_m = 2) ≤ λ_m²/2; P(Z_m = 2) ≤ (m/(m−4)) P(Z_{m−1} = 2);
  P(Z_m ≥ 3) ≥ c(η) > 0 whenever λ_m = C(m,2) p_2(x) ≥ η (Poissonisation, Le Cam's inequality, a Chernoff bound).
- **Proposition 5.1**: for every finite Ξ_0 ≠ 0 there are γ < 1 and infinitely many m with X(m) ≤ γ W(m)
  (multiscale argument with dyadic bands of p_2 and "good" scales). **Theorem B**: with a > 0, (D_n) forces
  X(n)/W(n−1) → 1, a contradiction.
- **Theorem C**: Bin(K+2) = 0 < X(K+2) for dust-free paintboxes with at most K boxes. **Lemma 6.1, Theorem D**:
  polynomial-times-exponential forms of P(Z_n = 1), P(Z_n = 2) for a paintbox with finitely many boxes and dust;
  Δ_n has a fixed sign for large n. **Corollary E**: for a = 0, M < ∞, regenerativity forces n ρ X(n) ~ M Bin(n)
  or Bin(n) = X(n) = 0 eventually.
- **Proposition 7.1** (exact certificate, above) and **Remark 7.2** (a dimension count, marked as heuristic and not
  used).

## Computations (programs and outputs in reproducibility/)
Apart from Proposition 7.1, which is a finite exact computation, no proof depends on a computation.
- **Program written with the note** (`check_note.py`, standard library, exact except a floating-point grid check of
  three real inequalities; about 8 s; 55 checks, 0 failed): formula (1) against brute-force enumeration of the
  draws, the consistency relation and identities (5), (6) (levels ≤ 7); the paintbox forms (4), (10) (m ≤ 8);
  Kingman = Ewens and (R_n) for Kingman and star (n ≤ 7); Lemma 2.2 (q ≥ 0), Lemma 2.3, Step 1 of Lemma 3.1,
  identity (9) and Lemma 3.1 for 4 ≤ n ≤ 8 (three random consistent arrays, three atomic measures; an inconsistent
  array is detected); Theorem A and Corollary 3.2 on seven measures; Proposition 7.1 as printed (and failure at 8
  partitions of 7); the signs of Δ_n for n ≤ 60; the example after Theorem C; the eventual signs predicted by the
  proofs of Theorems D and B (nine measures, 40 ≤ n ≤ 60); Lemmas 4.1, 4.2, (Q1) on 60 random paintboxes; the
  constants in the proofs of Lemma 4.3 and Proposition 5.1.
- **Programs of the first version** (`original/`): `verify_Dn.py` (Lemma 3.1 for n ≤ 7, paintbox forms, Δ_n at
  selected n), `certify_levels.py` (the certificate, with sympy), `check_lemmas.py` (Lemmas 4.1, 4.2, (Q1)).
- **Programs of the verification runs** (`verification_run_A/`, `verification_run_B/`, `independent_run_2/`), see
  below.
- Re-run at the writing from a copy of the package: all programs of the three folders, with the command lines of
  the package README; every output identical to the recorded one (for `verify_Dn.py` up to timing lines). Re-run
  again by the further verification run from an extracted copy of the source archive: the same result.

## Independent verification runs
The results were first obtained with written proofs and the programs in `original/`. Two independent verification
runs, both AI-assisted, examined this first written version (2026-10-19); each read the source in the EMS Press PDF,
re-derived all proofs line by line and wrote its own programs; neither read the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement, quantifiers, notion of regenerativity, readings | CONFIRMED (OWR pp. 2280–2282 re-extracted; Gnedin–Pitman EJC, Dong, Dong–Gnedin–Pitman, Lagerås read) | CONFIRMED (OWR read; Gnedin–Pitman Definition 1, (4)–(6), Theorem 3, Corollary 4; Dong Theorem 3) |
| Lemmas 2.1–2.3, (5), (6), facts on j_4, τ_3, t_4 | PROVED (incl. automatic q ≥ 0 at every level) | PROVED (same remark) |
| Theorem A, Corollary 3.2 | PROVED; level-4 equation reproduced exactly | PROVED; 56 random trials, 24 at ρ = ρ* |
| Lemma 3.1 | PROVED; exact for n ≤ 8 on random consistent arrays | PROVED; exact in 51 + 51 cases (atomic and arbitrary arrays), n ≤ 8 |
| Paintbox forms, (Q1)–(Q3), Lemmas 4.1–4.3 | PROVED (all constants of Lemma 4.3) | PROVED; brute force = closed forms (m ≤ 8); flat-paintbox infimum ≈ P(Poisson(1/4) ≥ 3) |
| Proposition 5.1, Theorem B | PROVED | PROVED; sparse-scale stress test: X(m+1)/W(m) between < 0.01 and 0.96 for 10³ ≤ m ≤ 3·10⁴, so only the liminf can be bounded |
| Theorems C, D, Corollary E | PROVED | PROVED; predicted signs in 10 examples of D and 8 of B (n ≤ 60) |
| Proposition 7.1 | CHECKED: rates by enumeration of 3^b assignments, full linear system solved at each level, identical q and residuals | CHECKED: own core, failure also at n = 7; ρ + 1/5120 fails already at n = 4 |
| Wrong inputs | — | inconsistent array, W(n) for W(n−1), shifted ρ: all detected |
| First-version programs | outputs re-read | re-run from a copy: identical outputs |
| Literature and bibliography | Crossref, zbMATH records of CPC 2007; sources above | arXiv API (16 queries), OpenAlex and Semantic Scholar citation lists, zbMATH, Crossref, two web searches |

Both runs classified the result as PARTIAL, found no mathematical error and required only textual corrections.

**Corrections required by the runs, all made:**
run A — (1) the sign statement for Δ_n of the certificate: Δ_n > 0 for 6 ≤ n ≤ 15 and Δ_n < 0 for 16 ≤ n ≤ 60
(the first version had tested n = 16, 20, 25, 30, 40 only); (2) the band ratio of the proof of Proposition 5.1 is
written as powers of 2, distinct from η of Lemma 4.3; (3) Lemma 2.2 states that the hook-determined q(n;·) is
nonnegative at every level, so regenerativity at level n means that all non-hook equations hold; (4) the case
h(J) = 0 of a good scale is treated explicitly; (5) Ξ = 0 is excluded in Question 1.1; (6) the exact check of
Lemma 3.1 is cited for n ≤ 8; (7) status PARTIAL, with the proved family named exactly (abstract, Section 1.2,
"Scope and priority", this report). Run B — (a) the dimension count (Remark 7.2) is labelled heuristic and not used;
(b) the Δ_n signs are complete (positive also for n = 13, 14, 15); (c) the note says that the full texts of
Möhle CPC 2007 and Bernoulli 2006 were not read; (d) one year (2006, Crossref) is used for the OWR volume, which is
cited as Report 40/2005.

## Further independent verification run on the text of the note (2026-10-19)
A further independent verification run, AI-assisted, examined the note as written (13 pages, before the final
revision), as if nothing had been checked before; its programs and output are in `reproducibility/independent_run_2/`.
- **Sources read again:** OWR 40/2005 (EMS Press PDF; the conjecture on p. 2281 lists the star-shaped case as an
  exception, Λ_1 concentrated in 1; mutation rate r > 0), Gnedin–Pitman EJC (Definition 1, (2)–(6), (10)–(12),
  Theorem 3, Corollary 4(i), Corollary 6, Section 6), Dong arXiv:0707.1606v1 ((6)–(9), (12)–(17), Theorem 3),
  Dong–Gnedin–Pitman arXiv:math/0603745v1; the abstract of Möhle CPC 2007 in Crossref and zbMATH. Verdict on the
  statement: CONFIRMED; the note answers the question as posed in the stated cases and states its scope correctly.
- **Proofs re-derived:** Lemmas 2.1–2.3 (with q(n;m) ≥ 0 at every level from the collision terms of Lemma 2.1),
  Lemma 3.1 for all n ≥ 4 (all four steps and identity (9), including n = 4), Theorem A and the explicit ρ*,
  Corollary 3.2, (Q1)–(Q3), Lemmas 4.1–4.3 (the Le Cam bound in its total-variation form, the Chernoff bound
  (e/4)^ν, the choice of θ depending on η only), Proposition 5.1 (good scales, the matching m, the splitting, the case
  h(J) = 0) and Theorem B, Theorems C and D with Lemma 6.1, Corollary E. All PROVED; no gap or error.
- **Own programs** (methods different from the earlier ones; 57 checks, all passed, 1.4 s): rates of atomic Ξ by a
  recursion over the boxes (= brute force for b ≤ 6); probabilities p by a first-step recursion over integer
  partitions; the full linear system (R_n) solved exactly; Lemma 3.1 exact in 36 cases (4 ≤ n ≤ 8, arbitrary
  consistent arrays and atomic measures); Theorem A in 27 trials; Proposition 7.1 (unique q ≥ 0 at levels 2–5 with
  the printed q(5;·), no solution at levels 6 and 7, the five failing partitions of 6 with the same residuals, the
  sign string of Δ_n for 4 ≤ n ≤ 60); examples of Theorems B, C, D; Lemmas 4.1, 4.2, (Q1) and the constant c(1/4)
  of Lemma 4.3 (about 1.5·10⁻⁸).
- **Package:** every program re-run from an extracted copy of the source archive; all recorded outputs reproduced.
- **Literature and bibliography:** Crossref for all 16 DOIs (all correct); arXiv API (8 queries, newest first),
  OpenAlex (works citing CPC 2007; two keyword searches), zbMATH (citations of CPC 2007; `regenerative & coalescent`;
  the record of Bernoulli 12(1) 35–53), one web search: no printed answer to the question found.
- **Corrections required, all made:** this section and the corresponding paragraph of the note; the final form of the
  sentence on the two papers not read; the record of the searches (four web searches in all); the description of
  the open case in Section 1.2 (Theorem C covers some diffuse Ξ_0); one bibliography note (pages of the OWR report);
  a duplicate hyperlink anchor removed (`hypertexnames=false`). The note now has 14 pages.

## Relation to the literature, novelty and scope
- **Searches.** First version (2026-10-18/19): the OWR report, Dong–Gnedin–Pitman (arXiv math/0603745), Dong
  (arXiv 0707.1606), Gnedin–Pitman EJC (journal PDF) and Lagerås (arXiv math/0702367) read in full; Crossref,
  OpenAlex, zbMATH and Semantic Scholar records and citation lists of Möhle CPC 2007; zbMATH queries
  (`regenerative & coalescent`, `"simultaneous multiple collisions" & sampling`, `regenerative & "multiple
  collisions"`); arXiv API (`regenerative` with `coalescent` or `"multiple collisions"`); one web search (no open copy
  of CPC 2007). Run B: arXiv API (16 queries, 2004–2026), OpenAlex (works citing CPC 2007: 4; citing Bernoulli 2006:
  44 titles screened), Semantic Scholar (works citing Gnedin–Pitman EJC and Dong–Gnedin–Pitman), zbMATH (incl.
  `"non-regenerative"`), Crossref; read or grepped: Gnedin's survey (Probab. Surveys 7 (2010)), Gnedin–Haulk–Pitman,
  Berestycki's survey; two web searches. Writing stage (2026-10-19): Crossref for every DOI of the bibliography and
  four bibliographic queries (logged in `queries.log`); no web search. Further verification run (2026-10-19): see
  the section above; one web search.
- **No printed answer to the question was found.** Closest: Möhle's Λ theorem (CPC 2007); Dong–Gnedin–Pitman and
  Dong (structure of these partitions, no regenerativity result in the Ξ case); Lagerås (litters, a different
  composition); Gnedin–Iksanov–Möhle 2008 and Gnedin–Iksanov–Marynych 2011 (regenerative compositions used for
  Λ-coalescents). A search that finds nothing is not a proof of novelty.
- **Not read.** The full texts of Möhle, CPC 16 (2007) 435–444 (paywalled; used through its abstract on the
  publisher's page, in Crossref and in zbMATH and through the statements in OWR p. 2281, Dong–Gnedin–Pitman p. 1 and Lagerås Proposition 4) and of
  Möhle, Bernoulli 12 (2006) 35–53 (Project Euclid behind a bot check, not worked around; used through Dong's
  restatement of its Theorem 5.1). The note says so in the paragraph "Scope and priority".
- **Scope.** Answered: the conjectured non-regenerativity for the classes of Theorems B, C, D, for every non-Λ Ξ
  off ρ*(Ξ), and (known) for Λ-coalescents. Not answered: the open case above. New to our knowledge: Lemma 3.1,
  Theorems A (beyond the Λ case), B, C, D, Corollary E, Proposition 7.1. No priority is claimed.
- **Suggested label for the record.** OWR-1061-006 partially answered (unrefereed note): conjecture proved for Ξ with
  a Kingman component, for the classes of Theorems C and D and off one mutation rate; open for a = 0 at ρ = ρ*(Ξ).
  Suggested status: partially solved (unrefereed claim); under the dataset policy the label stays open or partially
  solved, with evidence unrefereed_claim.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/