{"id":"2626fa96-ab83-4ab6-8e33-86e0bbe0a246","arxiv_id":"2505.15481","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditioned on a fixed diploid pedigree, the ancestral process converges to an inhomogeneous (Ψ,c)-coalescent, mixing fixed-time multiple mergers from large families with constant-rate binary mergers.","lead":"This paper derives the large-population limit of gene genealogies conditional on a fixed population pedigree, and shows the result is a new type of coalescent process that differs from the classical averaged coalescent when reproduction is skewed. It provides the first general quenched coalescent theorem for arbitrary sample sizes in diploid populations, which changes how multi-locus genetic data may be interpreted and simulated.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.8's proof leans on the explicitly unproved Section 4.3 'leap of faith' that ancestral lineages mix to uniformity before every GLIP; this underpins Lemma 4.2 and the generator limits in Lemmas 6.10–6.11, so without it the main convergence result is not established.","rationale":"The paper's central claim is that the pedigree-conditional (quenched) coalescent converges to a specific inhomogeneous (Ψ, cpair)-coalescent. The proof strategy is to show that the true process is close, in mean square conditional on the pedigree, to an ε-naive coalescent driven by GLIPs (Lemma 5.9). The computation of the relevant generator rates in Lemma 6.10 (mixed coupling) and Lemma 6.11 (naive-naive coupling) uses Lemma 4.2, which replaces the one-generation transition probability conditional on offspring counts by the paintbox transition p(eV;ξ,η). Lemma 4.2 in turn relies on (35), which the authors derive from a 'leap of faith' in Section 4.3 — the assumption that ancestral positions are approximately uniform immediately before a GLIP. The paper explicitly says this is not currently rigorous. This is exactly the load-bearing step: without it, the mixed-coupling rate q_mix,ε may differ from q_pure, and the final comparison in Lemma 5.9 has no basis. We agree with the reader's identification of this as the weakest assumption. We do not see an independent error in the rest of the argument; the recovery of known results (Kingman case, n=2 examples) and the simulation figures are consistent with the claimed limit, making the theorem plausible but not proven. A concrete numerical test of the mixing claim, or an analytic proof of a quenched mixing lemma for ancestral lines, would settle whether the concern lands. We therefore recommend keeping the conditional accept: the paper is significant and likely correct, but the central proof lacks a key ingredient as written.","tokens_in":58235,"tokens_out":14372,"duration_ms":133690,"concrete_test":"Simulate the model of Section 7.5 for increasing N (e.g., 10^3, 10^4, 10^5). For many pedigree realizations, evolve n = 2 ancestral lineages through the pedigree using only Mendelian randomness until the first GLIP (a generation with ||eV|| ≥ ε). Just before the GLIP, record the ancestral individuals and genes. Compare the empirical joint distribution of these positions to the uniform distribution on pairs of distinct (gene, individual) states, using total variation distance or a χ² statistic. If the discrepancy does not decay like O(1/N) (or at least does not vanish as N → ∞), the Section 4.3 leap of faith is violated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result, Theorem 3.8, is proven by comparing the true pedigree-conditional coalescent Π^N to an auxiliary ε-naive coalescent Π^(N,ε) driven by the GLIP point process Ψ^(N)_ε (Section 4.4, Lemma 5.9). The comparison passes through the generator rates q_mix,ε and q_pure,ε computed in Lemmas 6.10 and 6.11. In particular, Lemma 6.10 evaluates the contribution of GLIP generations (||eV|| ≥ ε) as lim_{N→∞} (1/c_N) E[1_{||eV||≥ε} eπ_N(V;ξ,η) pε(eV;·,·)] = ∫_{||x||≥ε} p(x;ξ,η)p(x;·,·) Ξ(dx)/⟨x,x⟩. This identity invokes Lemma 4.2, which asserts eπ_N(V;ξ,η) = p(eV;ξ,η) + O(1/N). Lemma 4.2 is derived from equation (35), which rests entirely on the Section 4.3 'leap of faith': before a GLIP, ancestral gene positions are approximately uniformly distributed over diploid individuals and genes, independent of the pedigree. The authors write: 'we presently do not know how to make it rigorous.' If the true position distribution before a GLIP is not uniform (e.g., because a previous GLIP left many lineages concentrated in a few individuals and the intervening generations are too few for mixing), then the quenched transition probability through the GLIP can differ from p(eV;ξ,η) by an amount that survives the rescaling, and q_mix,ε and q_pure,ε would not converge to q_pure. Consequently the final step of Lemma 5.9, which concludes that the squared L2-distance between the conditional finite-dimensional distributions tends to zero, would fail. The theorem may still be true, but as written the proof has a genuine missing ingredient at its core.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the coalescent for a fixed diploid pedigree generated by the exchangeable bi-parental Cannings model of Birkner et al. (2018), conditioned on the pedigree. Under assumptions (6), (7), (12), and (13), it states that the rescaled pedigree-conditional n-coalescent converges to an inhomogeneous (Ψ, c_pair)-coalescent, where Ψ is a Poisson point process encoding the timing and scale of generations with large individual progeny and c_pair is the residual binary merger rate. The paper constructs the limiting process via a coagulator-based stochastic flow, proves the convergence by coupling two conditionally independent coalescents on the same pedigree, computes the generator limits in Section 6 and Appendix B, works through several examples including diploid Beta-coalescents, and illustrates pedigree effects on the site-frequency spectrum and total tree length.","tokens_in":58675,"tokens_out":7529,"duration_ms":74949,"significance":"If Theorem 3.8 is correct, this is a substantial contribution: it provides the first quenched coalescent limit for arbitrary sample size in a general diploid exchangeable population model, demonstrates a genuine quenched-annealed distinction when multiple mergers are present, and introduces a technically useful construction of inhomogeneous coalescents with dense potential jump times. The coagulator construction, the coupling strategy adapted from random walks in random environments, and the explicit examples are valuable and go well beyond previous sample-size-two or Kingman-only results. The paper is also commendably explicit about its main gap. However, because the proof of the central theorem depends on an unproved heuristic, the significance is currently conditional on closing that gap.","major_comments":[{"comment":"The main theorem is not established as written because the proof of Lemma 4.2 and the subsequent generator computations in Lemmas 6.10 and 6.11 rely on the 'leap of faith' stated in Section 4.3: before a generation with large individual progeny, ancestral gene positions are asserted to be approximately uniformly dispersed and independent of the pedigree. The authors explicitly write that they do not currently know how to make this rigorous. In the quenched setting, the positions X_j(g) are functions of the pedigree being conditioned on, so the passage from the left-hand side of (34) to the conditional law given V is exactly the point that needs proof; without it, the O(1/N) error in Lemma 4.2 is not justified. Since Lemma 4.2 feeds into the limits q_mix,epsilon and q_pure,epsilon that are used in Lemma 5.9 to control the squared L2 distance in (48), Theorem 3.8 currently rests on an unproved approximation rather than on the stated assumptions (6), (7), (12), and (13).","section":"Section 4.3, Eq. (35), Lemma 4.2"},{"comment":"The asymptotic computation of q_pure in Lemma 6.9 is load-bearing, but its proof in Appendix B delegates several nontrivial facts to phrases such as 'a moment's thought' (after Eq. (72)) and 'it is not hard to see' (in the treatment of the case s > 0). These steps involve the limit of (N)_{d downarrow}/(c_N (N)_{b downarrow}) multiplied by products of falling factorials of bV_i, with K_l containing up to four contributions, and the combinatorial reorganization via the injections varrho. The recursion for the aggregate transition probabilities is stated without a fully written verification. Please provide complete details or a precise reference, because this lemma is one of the three generator limits on which Lemma 5.9 depends.","section":"Appendix B, proof of Lemma 6.9"},{"comment":"The paragraph after Eq. (42) asserts that N2_loc is a closed subset of N under the stated weak convergence, arguing by mollifications of indicators of (0,u) x (Delta \\ {0}). This is not immediate, because condition (40) involves an infinite sum of <x,x> and weak convergence of point processes does not in general preserve second-moment summability when atoms can accumulate near x = 0. Since the proof of Lemma 5.4 relies on the limiting configuration lying in N2_loc, please supply a proof of the closure claim or replace it with a topology that makes the domain closed.","section":"Section 5, N2_loc closure claim"}],"minor_comments":[{"comment":"The statement that cd(Π^{N,n}) converges in the Skorokhod topology is made without proof or reference; if this claim is retained, it should be proved or explicitly labeled as a conjecture.","section":"Remark 3.10"},{"comment":"The simulation section would be more reproducible if the number of loci per pedigree, the number of pedigrees, and Monte Carlo standard errors were stated in the text rather than only in figure captions.","section":"Section 8"},{"comment":"There are numerous typographical and formatting artifacts, including 'c` adl` ag', 'L´ evy-Prokhorov', and several missing or scrambled diacritics in the references; a careful copyedit is needed.","section":"Throughout"},{"comment":"The phrase 'leap of faith' is appropriate, but the paper should make clear at the statement of Theorem 3.8 that the proof currently depends on this unproved heuristic; the current organization places this caveat only in the proof outline, which is easy to miss.","section":"Section 4.3"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper is within scope and the ideas are strong, but the main theorem is not fully proved because of the Section 4.3 heuristic and the deferred computations in Appendix B. I recommend major revision rather than rejection: the authors have identified the main gap themselves, and the remaining work, while substantial, may be achievable. I would ask for either a rigorous proof of the mixing approximation or a revised theorem statement that makes it an explicit assumption, and for complete details in Appendix B before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know about arXiv:2505.15481. First, it is a real step forward: for the first time in a general diploid exchangeable Cannings model, it states a quenched (pedigree-conditional) coalescent limit for arbitrary sample size n, covering multiple-merger regimes. The limit is a new object, the inhomogeneous (Ψ,c)-coalescent, and the construction via coagulator flows is the right way to handle dense potential jump times. Second, the proof has an acknowledged hole at its core. Section 4.3 makes a 'leap of faith': before a generation with large individual progeny (GLIP), the ancestral gene positions are approximately uniform on the pedigree, independent of the pedigree. The authors write they do not know how to make it rigorous. That assumption enters equation (35), and Lemma 4.2 (eπN = p(eV) + O(1/N)) follows from it. Lemma 4.2 is then used to evaluate the GLIP part of the generator limits in Lemmas 6.10 and 6.11, which feed the final L2 argument in Lemma 5.9. So the main theorem is not actually established by the printed proof.\n\nWhat's good: the paper is honest about the gap. The special cases check out: the Kingman case recovers Tyukin (2015), and the n=2 case recovers Diamantidis et al. (2024). The examples — diploid Beta coalescents, occasional large families, the two-sex model — are concrete and useful. Adapting RWRE coupling ideas to coalescents is a substantial technical contribution, and the coagulator flow is likely to be reused.\n\nThe soft spots beyond the main gap: Appendix B, which proves Lemma 6.9, contains a couple of 'a moment's thought' and 'it is not hard to see' where the reader has to fill in non-trivial combinatorial bookkeeping. That's more annoying than fatal, but it does mean the most technical part of the paper is not fully checkable. The simulations in Section 8 are illustrative; no code or parameter files are provided. That's a minor issue for a math.PR paper, though.\n\nMy take: the theorem is probably true, and the paper deserves to be in the literature. But as it stands, it should not be accepted without either (a) a rigorous proof of the uniform dispersion statement, at least under extra assumptions, or (b) a version of the theorem that does not need it, or (c) a clearly labeled open problem splitting the theorem into the proven part and the heuristic part. I would send it to a serious referee, with a specific request to scrutinize Section 4.3 and Appendix B. I would not rely on the proof as a black box yet.","headline":"Genuinely new quenched coalescent limit with a new limiting process, but the proof's central mixing assumption is explicitly heuristic, so the main theorem is not yet proven as written.","tokens_in":59224,"tokens_out":3800,"would_cite":true,"duration_ms":33500,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J90","92D10","60K37","60J95"],"pacs":[],"model":"deepseek-v4-flash","headline":"Conditioning on a fixed diploid pedigree changes the limiting coalescent: multiple mergers occur only at pedigree-fixed times of large families, whereas the classical coalescent averages over them.","keywords":["coalescent theory","population pedigree","quenched limit","inhomogeneous coalescent","multiple mergers","site-frequency spectrum","diploid exchangeable model","generations with large individual progeny"],"falsifier":"Simulate the occasional-large-family model (Section 7.4, $\\gamma=1$) at increasing population sizes $N$, fix one realization of the pedigree, and estimate the probability that three sampled lineages undergo a simultaneous triple merger at rescaled times away from generations with large offspring numbers; Theorem 3.8 predicts all such multiple mergers occur exactly at GLIP times, so a nonzero triple-merger rate away from those times that does not vanish as $N\\to\\infty$ would refute the convergence.","tokens_in":58011,"feed_emoji":"🧬","tokens_out":16366,"duration_ms":132268,"temperature":0.7,"pith_summary":"Classical coalescent theory predicts genetic variation by averaging over all possible reproductive histories, which erases the population pedigree. This paper asks what happens instead when the pedigree—the complete record of which individuals were parents of whom—is fixed, and answers it for the standard diploid exchangeable bi-parental (Cannings) model in the large-population limit. The answer is a new object, the inhomogeneous $(\\Psi,c_{\\mathrm{pair}})$-coalescent: multiple mergers happen only at times fixed by generations with unusually large individual offspring numbers (GLIPs), recorded in a Poisson point process $\\Psi$, while ordinary binary mergers happen at constant rate $c_{\\mathrm{pair}}$. This pedigree-conditional (quenched) limit differs from the pedigree-averaged (annealed) $\\Xi$-coalescent whenever the offspring distribution has mass at nonzero frequencies, so the paper concludes that a fixed pedigree leaves observable signatures in multi-locus statistics such as the site-frequency spectrum. The proof works for samples of arbitrary size and covers several concrete models, including two kinds of diploid Beta-coalescents.","feed_headline":"Big families freeze multiple mergers at fixed pedigree times","feed_subtitle":"Genealogies on a fixed pedigree place multiple mergers at fixed times, changing the site-frequency spectrum.","key_machinery":"The central object is the inhomogeneous $(\\Psi,c_{\\mathrm{pair}})$-coalescent, a partition-valued process built from two ingredients: a Poisson point process $\\Psi$ whose atoms $(t,x)$ trigger 'paintbox mergers' with probabilities $p(x;\\xi,\\eta)$ given in (15), and independent pair mergers at rate $c_{\\mathrm{pair}}$. Because $\\Psi$ may be dense in time, the construction uses a coagulator: each partition $\\alpha$ defines an operator $\\mathrm{Coag}_\\alpha$ that merges blocks of the current partition according to the blocks of $\\alpha$, and the coalescent is the chronologically ordered product of these operators over the atoms of the driving point process, giving a stochastic flow. The bridge from the finite population to this limit is the $\\varepsilon$-naive coalescent driven by the cutoff point process $\\Psi^{(N)}_\\varepsilon$, built from offspring frequencies through $\\widetilde V=(V_{(1)}/4N,V_{(1)}/4N,V_{(2)}/4N,\\dots)$; Lemma 4.2 shows that the coarse-grained pedigree transition probabilities equal the paintbox probabilities $p(\\widetilde V;\\xi,\\eta)$ up to $O(1/N)$. Coupling two conditionally independent copies of the coalescent on the same pedigree, interpreted as two unlinked loci, and applying a separation-of-timescales argument isolates the limiting jump rates $q_{\\xi,\\eta}$ that define the quenched process.","core_discovery":"The paper's central claim is Theorem 3.8: under the diploid Cannings model with no selfing, for every sample size $n$, every finite set of times $t_1,\\dots,t_k$, and every sequence of partition states $\\xi_1,\\dots,\\xi_k$, the pedigree-conditional probabilities $P^{(N)}(\\Pi^{N,n}_{\\lfloor t_1/c_N\\rfloor}=\\xi_1,\\dots,\\Pi^{N,n}_{\\lfloor t_k/c_N\\rfloor}=\\xi_k \\mid \\mathcal{A}^{(N)})$ converge to $P(\\Pi^n_{t_1}=\\xi_1,\\dots,\\Pi^n_{t_k}=\\xi_k\\mid\\Psi)$, where $\\Pi^n$ is the inhomogeneous $(\\Psi,c_{\\mathrm{pair}})$-coalescent of Definition 5.6. Here $\\Psi$ is a Poisson point process on $[0,\\infty)\\times(\\Delta\\setminus\\{0\\})$ with intensity $dt\\,\\langle x,x\\rangle^{-1}\\Xi(dx)$, encoding the times and sizes of GLIPs, and $c_{\\mathrm{pair}}=1-\\Xi(\\Delta\\setminus\\{0\\})$ is the limiting rate of binary mergers. Because a realized pedigree fixes which generations had large offspring numbers, those multiple-merger times are frozen in the quenched limit, whereas the annealed limit averages over them; hence the two limits differ exactly when $\\Xi$ has atoms at nonzero offspring frequencies. The paper also shows that finite-dimensional distributions are the right notion of convergence: the prelimiting process spends time in states with two ancestral genes in one individual on the fast dispersal timescale, so it does not converge in the usual path-space topology, but its completely dispersed version does.","pith_inferences":["The paper leaves open whether its 'leap of faith' in Section 4.3 can be upgraded to a theorem; if it can, the same limit should hold under weaker mixing assumptions, while a failure would require replacing formula (35) and Lemma 4.2.","A testable extension the paper only sketches is a quenched ancestral recombination graph for linked loci, with recombination rate $\\rho$; the two-copy coupling developed here is the natural tool for that proof.","A statistical consequence the paper does not spell out is that shared GLIP times across unlinked loci make multi-locus site-frequency spectra informative about the times and sizes of past large families.","For the infinite-intensity Beta cases, the paper notes the simulation difficulty; a concrete check is whether truncating small paintbox events at a sample-size-dependent threshold preserves the quenched site-frequency spectrum."],"forward_implications":["If the theorem is right, the quenched and annealed limits agree only in the Kingman case $\\Xi=\\delta_0$; whenever the marginal coalescent has genuine multiple mergers, conditioning on the pedigree changes genealogies for every sample size.","The genealogy is fully described by a Poisson point process of large-family times plus a constant pair-merger rate; in the occasional-large-family example with $\\gamma=1$ this is explicit: a $(\\psi/4,\\psi/4,\\psi/4,\\psi/4,0,\\dots)$-merger at rate $4/(\\psi^2+2)$ and Kingman mergers at rate $2/(\\psi^2+2)$.","Multi-locus summaries inherit the pedigree effect: the paper's simulations show pedigree-specific bumps in the expected site-frequency spectrum and show via the law of total variance that the pedigree can explain a substantial fraction of variation in total tree length.","In the diploid Wright-Fisher model and finite-variance random-fitness models the quenched limit reduces to the Kingman coalescent, recovering the earlier Kingman result as a special case.","The coagulator-flow definition makes the limiting process meaningful even when potential jump times are dense, so the quenched limit does not require ordering infinitely many GLIPs."],"supporting_citations":[{"why":"Defines the diploid exchangeable Cannings model, the annealed Xi-coalescent limit, and the assumptions (6), (7), (12), (13) with the pair-coalescence rate c_N that this paper conditions on.","marker":"Birkner et al. (2018)"},{"why":"Supplies the separation-of-timescales lemma used to turn fast dispersal plus slow coalescence into the continuous-time limit of the coupled Markov chains.","marker":"Möhle (1998)"},{"why":"Proved the quenched Kingman coalescent limit for broad diploid models; the paper's Theorem 3.8 reduces to this in the pure Kingman case.","marker":"Tyukin (2015)"},{"why":"Established the sample-size-two quenched limit with highly reproductive couples and contributed the two-copy coupling technique the paper adapts to arbitrary sample sizes.","marker":"Diamantidis et al. (2024)"},{"why":"Provides the Xi-coalescent framework and the paintbox merger probabilities p(x;xi,eta) that define the limiting transitions.","marker":"Schweinsberg (2000)"},{"why":"Supplies the state space S_n, the complete-dispersion map, and the expected-transition setup for diploid exchangeable coalescents.","marker":"Möhle and Sagitov (2003)"}],"fun_headline_variants":["Pedigree fixes when multiple mergers happen in coalescent","Quenched coalescent pins multiple mergers to fixed times","Given pedigree, multiple mergers occur at deterministic times","Inhomogeneous coalescent arises from fixed pedigree","Multiple-merger times frozen by pedigree in coalescent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that before a generation with unusually large offspring numbers, ancestral gene copies are approximately uniformly scattered across the population's individuals—a step the paper explicitly labels a 'leap of faith' whose rigorous justification is left open, and on which the auxiliary $\\varepsilon$-naive coalescent rests.","fun_headline_variants_meta":{"raw":{"variants":["Pedigree fixes when multiple mergers happen in coalescent","Quenched coalescent pins multiple mergers to fixed times","Given pedigree, multiple mergers occur at deterministic times","Inhomogeneous coalescent arises from fixed pedigree","Multiple-merger times frozen by pedigree in coalescent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000509,"raw_usage":{"total_tokens":2589,"prompt_tokens":1166,"completion_tokens":1423,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":782,"completion_tokens_details":{"reasoning_tokens":1348}},"tokens_in":782,"tokens_out":1423,"duration_ms":10276,"temperature":1.0,"reasoning_tokens":1348,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:16:23.362649+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the occasional-large-family model (Section 7.4, $\\gamma=1$) at increasing population sizes $N$, fix one realization of the pedigree, and estimate the probability that three sampled lineages undergo a simultaneous triple merger at rescaled times away from generations with large offspring numbers; Theorem 3.8 predicts all such multiple mergers occur exactly at GLIP times, so a nonzero triple-merger rate away from those times that does not vanish as $N\\to\\infty$ would refute the convergence.","supporting_citations":[{"cited_title":"Coalescent results for diploid exchangeable population models","cited_arxiv_id":null,"evidence_quote":"Defines the diploid exchangeable Cannings model, the annealed Xi-coalescent limit, and the assumptions (6), (7), (12), (13) with the pair-coalescence rate c_N that this paper conditions on."},{"cited_title":"Quenched limits of coalescents in fixed pedigrees","cited_arxiv_id":null,"evidence_quote":"Proved the quenched Kingman coalescent limit for broad diploid models; the paper's Theorem 3.8 reduces to this in the pure Kingman case."},{"cited_title":"Bursts of coalescence within population pedigrees whenever big families occur","cited_arxiv_id":null,"evidence_quote":"Established the sample-size-two quenched limit with highly reproductive couples and contributed the two-copy coupling technique the paper adapts to arbitrary sample sizes."},{"cited_title":"Coalescents with simultaneous multiple collisions","cited_arxiv_id":null,"evidence_quote":"Provides the Xi-coalescent framework and the paintbox merger probabilities p(x;xi,eta) that define the limiting transitions."}],"review_version":1}