{"id":"c9a5b4b5-84be-43ac-bafb-bb3efb581ae8","arxiv_id":"2411.13048","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For a diploid Moran population with selfing, the pedigree-conditional coalescence time of two gene copies converges to three distinct limits, including a novel random-ancestral-graph limit when outcrossing is of order 1/N.","lead":"Self-fertilizing populations inherit DNA through a shared family tree, and this paper shows how that tree changes the ancestry of two genes. It proves three different limits for selfing versus outcrossing, including a new regime where the pedigree leaves a permanent random mark on gene genealogies.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The limited-outcrossing theorem rests on the omitted compact-containment proof in Lemma 7.6; until the discrete-time ancestral graph size is shown tight, the random-walk representation of T_λ and the middle row of Theorem 4.1 are not fully supported.","rationale":"The paper's central claim — that conditional coalescence times given the pedigree are not generally the pedigree-averaged Kingman predictions — is made substantive by the limited-outcrossing regime, whose limit law is genuinely new. The path to that limit is Section 7.2.2: the discrete ancestral graph must converge to G_λ, and then random walks on the graph yield T_λ. The reader's weakest_assumption correctly identifies Lemma 5.1/Lemma 7.6 as the bridge. I agree that this is the load-bearing point because the other two regimes are either classical (partial selfing reduces to Kingman via the overlap/splitting decomposition, which is explicitly developed) or degenerate (negligible outcrossing is the λ=0 case of the same graph limit). The generator convergence in (31) is stated only for finitely supported functions and the compact-containment assertion is deferred; these are precisely the steps needed to rule out explosion of the graph size on the N^2 scale. The paper has real independent support: the overlap/splitting decomposition in Section 7.1 is explicit, the continuous-time size process has a plausible martingale bound, and the formulas in Propositions 5.4–5.6 are concrete and internally consistent. I do not see an internal inconsistency or a counterexample, and I am not raising an 'outside consensus' objection. The omission is a proof gap, not a demonstrated error, so the verdict should remain CONDITIONAL rather than REJECT: the authors should be asked to fill in the compact-containment argument for the pre-limit graph or to prove tightness by an alternative method. The recommended concrete test — a second-moment bound for L^N derived from (27) — would settle whether the gap is merely expository or actually fatal.","tokens_in":49028,"tokens_out":20040,"duration_ms":207657,"concrete_test":"Supply the missing discrete compact-containment proof: for L^N_k = |G^N(k)|, define M^N_t = N^{-2}(L^N_{⌊tN^2⌋} - L^N_0 - Σ_{i<⌊tN^2⌋} E[L^N_{i+1}-L^N_i | G^N(i)]) and compute its quadratic variation directly from (27). Verify that sup_N sup_{t≤T} E[(L^N_{⌊tN^2⌋})^2] < ∞ for each fixed λ ∈ R_+, which would establish (33). If this second-moment bound fails for some λ (e.g. λ=1), then Lemma 7.6 is not just under-detailed but false, and Theorem 7.9 would need a different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.1's limited-outcrossing row is proved only through Lemma 5.1/Lemma 7.6, the weak convergence of the discrete ancestral graph G^N(⌊tN^2⌋) to G_λ. The paper itself states that the proof of Lemma 5.1 'requires additional details which we have omitted here for simplicity', and the appendix proof of Lemma 7.6 confirms this: the compact-containment condition (33) for |G^N(⌊tN^2⌋)| is asserted by saying that 'the same argument' as for the continuous-time size process L_t applies, but the discrete-time process has one-step transition probabilities (27) with several O(N^-2) and o(N^-3) terms, and no discrete-time analog of the quadratic-variation computation (38) is supplied. If the number of nodes in G^N can diverge on the N^2 time scale — for instance, if the discrete martingale's quadratic variation is not bounded uniformly in N — then the Skorokhod convergence to G_λ fails, the coupling of random walks in Lemma 7.7 breaks down, and Theorem 7.9's conclusion P_diff(N^-2τ(N)>t | A_N) → P(T_λ>t | G_λ) is unproved. Because this is the central new regime, and the negligible-outcrossing row also inherits it via Corollary 7.11, this is the single most load-bearing technical gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies a discrete-time diploid Moran model with selfing probability α_N and derives the conditional distribution of the pairwise coalescence time τ^(N) of two sampled gene copies given the random population pedigree. The main results are Theorem 3.1 for the unconditional limit, Theorem 4.1 for the conditional limit under three regimes (partial selfing, limited outcrossing, and negligible outcrossing), and Theorem 4.5 for the same-individual conditional limit. In the limited-outcrossing regime N(1−α_N)→λ, the limit is described as the meeting time of coalescing random walks on an ancestral graph G_λ; in the negligible-outcrossing regime it is a random exponentially distributed constant. The paper also derives variance and covariance formulas for conditional survival probabilities and links the limited-outcrossing model to ancestral recombination and selection graphs.","tokens_in":49308,"tokens_out":5902,"duration_ms":59190,"significance":"If fully supported, the result is significant for population genetics: it shows that conditioning on the pedigree can change coalescent predictions relative to the pedigree-averaged Kingman coalescent precisely in the near-selfing regime, and it provides a tractable one-parameter limiting object, the ancestral graph G_λ. The paper's strengths include a detailed overlap/splitting decomposition of the coalescence time, generator computations for the ancestral graph, and explicit L^2 convergence arguments in the partial-selfing regime. The covariance formulas in Section 5 are also useful and connect the new model to classical two-locus results. However, the central limited-outcrossing and negligible-outcrossing rows of Theorem 4.1 depend on a weak-convergence proof whose compact-containment step is explicitly omitted, and one step in the proof of Lemma 7.8 is not a valid use of conditional Markov's inequality. These are load-bearing technical gaps that should be repaired before the results can be considered fully established.","major_comments":[{"comment":"The weak convergence of the discrete-time ancestral graph G^N to G_λ is not fully proved. Lemma 7.6 asserts the compact containment condition (33) by saying that 'the same argument' as for the continuous-time size process L_t applies, but the discrete-time process has one-step transition probabilities (27) with O(N^-2) and o(N^-3) terms, and no discrete-time analogue of the quadratic-variation computation in (38) is supplied. Since Lemma 7.7, Theorem 7.9, and Corollary 7.11 all rely on this convergence, the limited-outcrossing and negligible-outcrossing rows of Theorem 4.1 are not fully supported without a proof that |G^N(⌊tN^2⌋)| is tight. The manuscript itself acknowledges this gap after Lemma 5.1, but the appendix does not fill it.","section":"Section 5 and Section 7.2.2, Lemma 7.6"},{"comment":"The proof of Lemma 7.8 contains an invalid inequality. The displayed 'conditional Markov inequality' reads P(N^-2(τ−τ_O^1)>ε|A_N) ≤ (1/ε) P(N^-2(τ−τ_O^1)>ε), which is not a consequence of Markov's inequality and can fail in general. The correct conditional Markov bound would involve the conditional expectation E[N^-2(τ−τ_O^1)|A_N], and then convergence of the conditional probability to zero would require a control on that conditional expectation. Because Lemma 7.8 is used to pass from the first overlap time to the coalescence time in Theorem 7.9, this proof step needs to be repaired.","section":"Section 7.2.2, proof of Lemma 7.8"},{"comment":"The proof of Lemma 7.10 contains a sign/conditioning error. It states that P_same(O=0)=α_N/(2−α_N) tends to 0 as α_N→1, but this quantity tends to 1. The subsequent conclusion that the right-hand side of (41) tends to 0 should be justified by conditioning on O=0 (where the rescaled coalescence time is 0) and on O≥1 (where Corollary 7.4 gives convergence to 0), not by conditioning on O=1 as written. The conclusion is correct, but the proof as written is inconsistent.","section":"Section 7.2.2, proof of Lemma 7.10"},{"comment":"The asymptotic expansions in Proposition 5.5 are delegated to a Mathematica calculation stored in Newman (2024), and the key computation leading to equation (62) is not shown in the manuscript. This is not central to Theorems 4.1 and 4.5, but the paper should provide enough detail for the expansion to be verified without reliance on an external file, or should state more explicitly which symbolic computation is being used.","section":"Section 7.3, Proposition 5.5"}],"minor_comments":[{"comment":"The last displayed transition probability in (27) uses the notation 'n−M−2/n−2', which is inconsistent with the surrounding notation in terms of N and m; this should be corrected.","section":"Section 7.2.2, Eq. (27)"},{"comment":"The sentence 'where is the possible for a “same” sample to enter the “diff” process' is ungrammatical; it should be 'where it is possible for a “same” sample to enter the “diff” process'.","section":"After Theorem 4.5"},{"comment":"The state space of the joint process in Lemma 7.7 is written as D(P ×(Z × {0, 1})^2), but Z is not introduced as a set of labels in that statement; using the notation from Definition 4.3 or defining the label set explicitly would improve clarity.","section":"Section 7.2.2, Lemma 7.7"},{"comment":"The paper uses α_N for the selfing probability and α for its limit, but in a few places (for example in the discussion of Figure 1) α denotes a finite-N value rather than the limit; the notation should be made uniform to avoid confusion.","section":"Section 1.3 and Section 6"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the omitted compact-containment proof for the discrete-time ancestral graph in Lemma 7.6, which is load-bearing for the limited-outcrossing and negligible-outcrossing results. If the authors supply a complete proof of tightness of |G^N(⌊tN^2⌋)|, the paper's central claims would likely be acceptable. The invalid inequality in Lemma 7.8 and the typo in Lemma 7.10 are local but should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know about this paper. First, it proves genuinely new conditional coalescent limits for a diploid Moran model with selfing, and the limited-outcrossing regime—a random ancestral graph with coalescence times as meeting times of random walks—is absent from prior pedigree-averaged coalescent theory. Second, the proof of that central regime currently rests on a stated-but-omitted compact-containment argument, so the strongest new claim is not yet fully supported on paper.\n\nWhat is new and good: the three-regime classification (partial selfing, limited outcrossing, negligible outcrossing) emerges from an explicit Moran model via generator convergence and couplings, not by postulating the limit graph. The overlap/splitting decomposition in Section 7.1 is rigorous and gives clean proofs of the unconditional and partial-selfing cases. I particularly like the same-individual limit replacing the average inbreeding coefficient F with a per-individual random variable U (the number of consecutive selfing generations before first outcrossing); that is a real conceptual update and ties directly to identity disequilibrium. The paper is also careful to connect to prior work by Nordborg and Donnelly, Möhle, and the recent pedigree-conditional coalescent literature.\n\nThe soft spots, in proportion. The main one is exactly what the stress-test flags: Lemma 5.1/Lemma 7.6, the weak convergence of the discrete-time ancestral graph to G_λ. The paper admits the proof 'requires additional details omitted here,' and the appendix sketch of tightness applies a continuous-time quadratic-variation argument to the discrete-time size process without supplying the discrete-time bound. If |G^N| can diverge on the N^2 time scale, the random-walk representation of T_λ and the middle row of Theorem 4.1 are unsupported. This is a real gap, but it looks technical rather than conceptual: the transition rates in (27) have the right scaling, and the continuous-time martingale argument should extend with more work. There is also a smaller typo in Lemma 7.10 (alpha_N/(2-alpha_N) tends to 0 should be tends to 1), and Proposition 5.5's asymptotics are delegated to Mathematica; acceptable for a genetics paper but worth checking in review.\n\nWho should read this: population geneticists working on selfing (A. thaliana, C. elegans) and probabilists interested in quenched coalescent limits. It deserves a serious referee, not a desk rejection; the referee should push for a complete proof of Lemma 5.1 or a clear statement of what remains assumed. I would accept it for peer review with major revision.","headline":"New conditional coalescent limits for selfing Moran populations, with the headline limited-outcrossing result gated on an omitted tightness proof.","tokens_in":49878,"tokens_out":2342,"would_cite":true,"duration_ms":24672,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J27","60F17","92D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a population that mostly self-fertilizes, the true pedigree changes genealogical predictions: at a critical outcrossing rate a random ancestral graph, not the Kingman coalescent, gives the times to common ancestry.","keywords":["diploid Moran model","selfing","population pedigree","conditional coalescent","ancestral graph","limited outcrossing","coalescence time","identity disequilibrium"],"falsifier":"Run the Moran model at $N=10^4$ with $\\alpha_N=1-\\lambda/N$ for $\\lambda=1,5,20$; for each simulated pedigree compute the exact conditional survival function of the pairwise coalescence time for two sampled individuals, as in the paper's Figure 1, and compare it with $P(T_\\lambda>t\\mid G_\\lambda)$ read off the pedigree's ancestral graph: the conditional survival function should be a step function jumping at the graph's overlap times, its across-pedigree variance should match the paper's explicit covariance formulas as a function of $\\lambda$, and in the negligible-outcrossing case $N(1-\\alpha_N)\\to0$ the limit should be a single random jump rather than the smooth exponential $e^{-2t}$.","tokens_in":48750,"feed_emoji":"🧬","tokens_out":24013,"duration_ms":185080,"temperature":0.7,"pith_summary":"This paper asks whether the actual pedigree of a population—who really reproduced with whom—changes the predictions of coalescent theory for a sample of two gene copies. In a diploid Moran model where each new offspring is produced by selfing with probability $\\alpha_N$, the authors prove that the answer is controlled by one number: $N(1-\\alpha_N)$, the expected number of outcrossing events per generation. If that number tends to infinity, the pedigree washes out and the classic Kingman prediction, an exponential time to common ancestry with rate $2/(2-\\alpha)$, survives conditioning. If it tends to a finite $\\lambda$, a qualitatively new limit object appears—the ancestral graph $G_\\lambda$, where ancestral lineages branch at rate $\\lambda$ and coagulate pairwise at rate 2—and coalescence times are the meeting times of random walks on it, so the conditional distribution is random and pedigree-dependent. If it tends to zero, coalescence is fixed by the pedigree's own times to common ancestry. Standard analyses average over pedigrees, and this paper shows that in strongly selfing populations those averages can mispredict the variation among unlinked loci.","feed_headline":"Cut outcrossing to one in N and ancestry becomes a random graph","feed_subtitle":"At this boundary, the true pedigree fixes each locus's genealogy; standard pedigree-averaged theory misses it.","key_machinery":"The ancestral graph $G_\\lambda$ is the object that carries the limited-outcrossing argument: a continuous-time particle system that starts with two particles, in which each particle splits into two at rate $\\lambda$ and each pair of particles coagulates at rate 2, equal in distribution to an ancestral recombination graph or an ancestral selection graph. It is obtained as the weak limit, on the $N^2$ time scale, of the discrete-time ancestral graph $G_N$, the subgraph of the pedigree containing all potential ancestors of the sample, which fragments at outcrossing events and coagulates at selfing events (Lemma 5.1, proved as Lemma 7.6). Coalescence times are then the first meeting times $T_\\lambda$ of the two random walks on this graph, and the conditional survival probability is a step function jumping at the random overlap times of the graph. In the partial-selfing regime the argument uses a different device: a decomposition of the coalescence time into alternating overlap and splitting events, together with an $L^2$ second-moment comparison that couples two conditionally independent copies of the genealogy on the same pedigree.","core_discovery":"The central claim is that conditioning on the random population pedigree yields three different limiting laws for the pairwise coalescence time, classified by the limit of $N(1-\\alpha_N)$. For two gene copies sampled from different individuals (Theorem 4.1): when $N(1-\\alpha_N)\\to\\infty$ the conditional survival probability $P_{\\mathrm{diff}}(N^{-2}\\tau^{(N)}>t\\mid\\mathcal{A}_N)$ converges to $e^{-2t/(2-\\alpha)}$, matching the pedigree-averaged model; when $N(1-\\alpha_N)\\to\\lambda\\in(0,\\infty)$ it converges to $P(T_\\lambda>t\\mid G_\\lambda)$, the first-meeting time of two coalescing random walks on the ancestral graph; and when $N(1-\\alpha_N)\\to0$ it converges to the indicator $1_{\\{\\mathrm{Exp}(2)>t\\}}$, a single exponentially distributed jump time fixed by the pedigree. For two gene copies sampled from the same individual (Theorem 4.5), the limit is $2^{-U}e^{-2t/(2-\\alpha)}$, where $U$ is the individual's realized number of selfing generations with $P(U=k)=\\alpha^k(1-\\alpha)$, and this limit is $0$ at $\\alpha=1$ and $e^{-t}$ at $\\alpha=0$. Thus the pedigree-averaged theory sees one regime where the conditional theory sees three, and at the boundary $\\alpha_N\\to1$ the pedigree survives in the limit through the ancestral graph.","pith_inferences":["Beyond the paper: the same classification by the product of population size and the rate of pedigree-breaking events should apply to other forms of close inbreeding, such as sib mating or cousin mating, because their splitting and coagulating events also scale differently; the paper's discussion anticipates this but does not prove it.","Beyond the paper: whole-genome data from predominantly selfing species could estimate $\\lambda=N(1-\\alpha)$ directly from the across-locus variance of genealogies using the covariance formulas, turning the limit theorem into a parameter-estimation procedure the paper does not develop.","Beyond the paper: the observable signature of the negligible-outcrossing limit, near-identical coalescence times across the whole genome, could serve as a diagnostic for populations sitting in that regime before the model's more extreme prediction of zero heterozygosity fully holds.","Beyond the paper: single-locus inference that assumes smooth exponential genealogies will overstate the uncertainty in coalescence times when applied locus-by-locus under limited outcrossing, because the correct pedigree-conditional distribution is a step function concentrated on a few fixed times."],"forward_implications":["For samples of multiple unlinked loci, the pedigree-conditional model is the correct sampling structure when likelihoods are multiplied across loci, and under limited outcrossing it predicts that genealogies vary across loci through random choices at each branching of the ancestral graph.","The same-individual limit (Theorem 4.5) replaces the averaged inbreeding coefficient $F=\\alpha/(2-\\alpha)$ with each sampled individual's own realized number of selfing generations $U$, which is exactly the per-individual randomness underlying identity disequilibrium between loci.","The three regimes give a practical rule: if $N(1-\\alpha_N)$ exceeds roughly 100, conditional genealogies are well approximated by the partial-selfing Kingman model; if it is of order 1, the limited-outcrossing model must be used; if it is much smaller than 1, genealogies across the genome are essentially identical and heterozygosity vanishes.","The limited-outcrossing model interpolates between the two extremes: as $\\lambda\\to\\infty$ it converges to the $\\alpha=1$ partial-selfing limit, and as $\\lambda\\to0$ it converges to negligible outcrossing, so it covers every way in which $\\alpha_N$ can approach complete selfing."],"supporting_citations":[{"why":"Proves the separation-of-timescales convergence of the pedigree-averaged selfing coalescent; the standard model whose predictions Theorem 4.1 reproduces in the partial-selfing regime.","marker":"(Möhle, 1998)"},{"why":"The coalescent with selfing and its two-timescale structure; the baseline that conditioning on the pedigree modifies near $\\alpha=1$.","marker":"(Nordborg and Donnelly, 1997)"},{"why":"The prior treatment of pedigree-conditional coalescence for a nonstandard offspring distribution; supplies the conditioning $\\sigma$-algebra notation and the two-copy second-moment method.","marker":"(Diamantidis et al., 2024)"},{"why":"Shows the standard neutral coalescent survives conditioning on pedigrees of standard population models; the precedent the present paper extends and qualifies.","marker":"(Tyukin, 2015)"},{"why":"Introduces the ancestral recombination graph, the splitting-and-coagulating structure that $G_\\lambda$ replicates.","marker":"(Griffiths, 1991)"},{"why":"Provides the ancestral recombination graph construction and the large-$\\lambda$ completion-time asymptotics used to interpret limited outcrossing.","marker":"(Griffiths and Marjoram, 1997)"},{"why":"Introduces the ancestral selection graph, the other prototype for $G_\\lambda$ cited in Definition 4.3.","marker":"(Krone and Neuhauser, 1997)"},{"why":"Source of the five-state rate matrix and the covariance formulas that Proposition 5.4 and Proposition 5.5 are built on.","marker":"(Simonsen and Churchill, 1997)"},{"why":"Supplies the generator-convergence and compact-containment criteria used to prove Lemma 7.6, the bridge for the limited-outcrossing theorem.","marker":"(Ethier and Kurtz, 2009)"},{"why":"The method used to compute exact pedigree-conditional coalescence probabilities in Figure 1, the numerical illustration of the three regimes.","marker":"(Wakeley et al., 2012)"}],"fun_headline_variants":["Three ancestry regimes from selfing in a Moran model","Pedigree fixes genealogy when outcrossing is rare","Limited outcrossing turns ancestry into a random graph","Selfing strength dictates three coalescent limits","At one outcross per N pedigree beats averaging"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"If the size of the discrete ancestral graph is not bounded in probability on the $N^2$ time scale, the convergence to $G_\\lambda$—and with it the random-walk description of the limiting coalescence time—fails, and the paper omits the details of exactly this compact-containment step.","fun_headline_variants_meta":{"raw":{"variants":["Three ancestry regimes from selfing in a Moran model","Pedigree fixes genealogy when outcrossing is rare","Limited outcrossing turns ancestry into a random graph","Selfing strength dictates three coalescent limits","At one outcross per N pedigree beats averaging"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000723,"raw_usage":{"total_tokens":3341,"prompt_tokens":1143,"completion_tokens":2198,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":759,"completion_tokens_details":{"reasoning_tokens":2124}},"tokens_in":759,"tokens_out":2198,"duration_ms":17450,"temperature":1.0,"reasoning_tokens":2124,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:53:54.699083+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Moran model at $N=10^4$ with $\\alpha_N=1-\\lambda/N$ for $\\lambda=1,5,20$; for each simulated pedigree compute the exact conditional survival function of the pairwise coalescence time for two sampled individuals, as in the paper's Figure 1, and compare it with $P(T_\\lambda>t\\mid G_\\lambda)$ read off the pedigree's ancestral graph: the conditional survival function should be a step function jumping at the graph's overlap times, its across-pedigree variance should match the paper's explicit covariance formulas as a function of $\\lambda$, and in the negligible-outcrossing case $N(1-\\alpha_N)\\to0$ the limit should be a single random jump rather than the smooth exponential $e^{-2t}$.","supporting_citations":[],"review_version":1}