{"id":"23a1c7fb-f2b8-4dba-b5b5-7628756e7261","arxiv_id":"2412.17482","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Large-lifetime persistence cycles in Poisson point clouds converge to Poisson point processes, jointly for centers, lifetimes, and deathtimes, in a sparse regime for dimensions d≥2.","lead":"The paper proves that the centers, lifetimes, and deathtimes of long-lived topological loops in random point clouds converge to a Poisson process, enabling statistical tests in topological data analysis. It covers the 2D torus without deathtime bounds and arbitrary dimensions in a sparse regime, under conditions that are fully verified only in low dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed rates in Theorems 2.3–2.5 omit an O(r_n) intensity error from Lemma 4.7; Assumption M does not imply r_n ≤ ρ_{n,m+1}, so the rates are not established as stated.","rationale":"The paper's central contribution is a quantitative Poisson approximation theorem, so the rate is part of the main claim. The proof of Theorem 2.3 contains a concrete unsupported absorption step: Lemma 4.7 gives an O(r_n) intensity error, and since dKR is bounded below by the total variation distance of the intensity measures, one cannot drop or absorb this term into a smaller O(ρ_{n,m+1}) unless r_n ≤ Cρ_{n,m+1}. Assumption M does not guarantee this, and the counterexample above satisfies all stated assumptions of Theorem 2.3: d=10, m=k=3 lies in the verified Čech case of Assumption U, Assumption P holds for a smooth bounded integrable κ on R^10, and Assumption T is available by Proposition 5.2. Thus this is not merely an open verification case; it is an internal gap in a stated theorem. The reader's CONDITIONAL verdict is therefore appropriate, but I do not think Assumption U is the weakest load-bearing point. Separately, I noticed an internal q inconsistency in the Vietoris–Rips example: Proposition 5.6 states q=4 for the diamond configuration, while its proof establishes lim h(u,1)/u^5∈(0,∞) and Remark 2.6(4) says q=5; Example 6.4 then uses a Weibull shape of 4. This reinforces the need for correction before the applications are taken at face value, but the rate gap is the more central concern.","tokens_in":46652,"tokens_out":32214,"duration_ms":313562,"concrete_test":"Run the proof's own machinery for d=10, k=3, m=3, Čech filtration, nonconstant smooth κ (e.g., a Gaussian density), and r_n=n^{-0.14}. Compute via Lemma 4.3 the intensity measure E[ξ^2_n] and compare it with ακ^3; in particular evaluate dTV(E[ξ^2_n], ακ^3) using the decomposition in Lemma 4.7. If this distance is Θ(n^{-0.14}) rather than O(n^{-0.2}), the rate in Theorem 2.3 is false as stated. As a quick algebraic check, verify whether n^m r_n^{dm-1} ≥ 1 follows from Assumption M; for this example n^3 r_n^{29}=n^{-1.06}<1, so the proof's absorption step is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 2.3, the authors absorb the O(r_n) term from Lemma 4.7 by asserting that 'r_n ⩽ ρ_{n,m+1} for all sufficiently large n'. This inequality is not a consequence of Assumption M. Counterexample: take d=10, m=3, and r_n = n^{-0.14}. Then n r_n^d = n^{1-1.4}=n^{-0.4}, so ρ_{n,m}=n(nr_n^d)^{m-1}=n^{-0.2?} no—ρ_{n,m}=n·n^{-0.8}=n^{0.2}→∞ and ρ_{n,m+1}=n·n^{-1.2}=n^{-0.2}→0, so Assumption M holds. But r_n=n^{-0.14}≫n^{-0.2}=ρ_{n,m+1}, so the asserted inequality fails. Lemma 4.7 only bounds the intensity error by O(r_n), and dKR(P,Q) is bounded below by dTV(E[P],E[Q]); for a nonconstant smooth κ the intensity error is generally of exact order r_n. Thus the displayed rates cannot be concluded from the proof as written. The same O(r_n) term is silently dropped in Theorems 2.4 and 2.5 after invoking Lemma 4.7. The convergence results may remain true with the rate replaced by O(r_n+ρ_{n,m+1}+ρ_{n,m}^{-1/q}), but the quantitative rates claimed in the theorems are not justified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Poisson approximation for large-lifetime cycles in persistent homology of Poisson point clouds. In the unbounded regime on the two-dimensional flat torus (Theorem 2.1), it proves convergence of the spatial centers of cycles with multiplicatively large lifetime to a stationary Poisson process, with an explicit rate. In the m-sparse regime on R^d (Theorems 2.3–2.5), it proves joint Poisson convergence of the centers, scaled lifetime deviations, and scaled deathtime deviations of large-lifetime cycles, under structural assumptions (F, M, P, T, U, G, H) on the filtration, sparsity, intensity, threshold, uniqueness, and regularity. The proofs combine stabilization, Malliavin–Stein methods, a U-statistic approximation, the continuous BK inequality, and detailed geometric analysis of maximal lifetime configurations. Section 5 verifies the thresholds and several model assumptions, Section 6 applies the results to largest lifetimes and persistence diagrams, and Section 7 contains simulations.","tokens_in":46961,"tokens_out":8186,"duration_ms":72542,"significance":"If the results hold as stated, this is a substantial contribution to extreme-value theory for persistent homology: it provides the first Poisson approximation for large-lifetime cycles in arbitrary dimensions, including marks for lifetimes and deathtimes, and it gives quantitative rates. The proofs are detailed and conditional on explicit assumptions, and the paper is honest about which assumptions remain open (notably Assumption U in general and Assumption H in d≥3). The applications to largest lifetimes and to a deathtime-based model check are attractive. However, the quantitative rates in Theorems 2.3–2.5 contain a load-bearing gap: an O(r_n) intensity error from Lemma 4.7 is absorbed without a valid justification under Assumption M. This does not necessarily invalidate the convergence statements, but it requires a correction of the stated rates or a strengthening of the assumptions.","major_comments":[{"comment":"The proof states 'As r_n ⩽ ρ_{n,m+1} for all sufficiently large n' and uses this to absorb the O(r_n) intensity error from Lemma 4.7. This inequality is not a consequence of Assumption M. For instance, with d=10, m=3 and r_n=n^{-0.14}, one has n(nr_n^d)^2=n^{0.2}→∞ and n(nr_n^d)^3=n^{-0.2}→0, so Assumption M holds, but r_n=n^{-0.14} is much larger than n^{-0.2}=ρ_{n,m+1}. Lemma 4.7 only gives the intensity error as O(r_n), so the displayed rate O(ρ_{n,m+1}) in Theorem 2.3 is not established as stated. The same O(r_n) term is silently dropped in the proofs of Theorems 2.4 and 2.5 after invoking Lemma 4.7. The convergence assertions may remain true, but the quantitative rates need to be replaced by at least O(r_n+ρ_{n,m+1}+ρ_{n,m}^{-1/q}), or Assumption M must be strengthened so that r_n is dominated by ρ_{n,m+1}.","section":"§4.3, Proof of Theorem 2.3"},{"comment":"The example takes k=d−1 and m=2d with the Vietoris–Rips filtration, but Proposition 5.4 verifies Assumption U only for m=k or for k=d+1 and m=2d. The combination k=d−1 is outside the verified range, and for d=2 it would give k=1, contradicting the standing assumption k≥3. If the intended choice is k=d+1, as the surrounding formulas and the later d=2 specialization suggest, then the text should be corrected accordingly.","section":"§6, Example 6.3"},{"comment":"The proposition statement says that, for the Vietoris–Rips case d=2, k=3, m=4, assumption H holds with q=4. However, the proof immediately after the statement aims to prove (i) lim_{u→0} h(u,1)/u^5∈(0,∞) and (ii) lim_{u→0} \\tilde h^{(0,1)}(u,v)/u^5∈(0,∞), concluding that q=5 works; this matches Remark 2.6(4), which also gives q=5 for the Vietoris–Rips case. The statement of Proposition 5.6 should be changed to q=5, and the associated asymptotic expression for ℓ_{n,α} should be corrected consistently.","section":"§5.2, Proposition 5.6"}],"minor_comments":[{"comment":"The limiting point process in the extension statement is described as having unit intensity on R^d × [0,∞), but in the unbounded torus setting the spatial coordinate should be on T^2; please check whether this should be T^2 × [0,∞).","section":"Remark 2.2(4)"},{"comment":"In the final displayed chain, the first term should presumably be dKR(ξ^1_n, \\hatξ^1_n) rather than dKR(\\hatξ^1_n, \\hatξ^1_n); as written the triangle-inequality step is not meaningful.","section":"§3.3, Proof of Theorem 2.1"},{"comment":"The displayed condition for j<q is garbled: it should state that \\tilde h^{(j,1)}(0,v)=0 for all j<q and all v, and that \\tilde h^{(q,1)}(0,v)>0; the current line 'when j < qfor all u, v' is not grammatical and could confuse readers.","section":"Theorem 2.5, Assumption H"},{"comment":"The intensity convention in the displayed intensity function appears to be (2π)^{-d^2} exp(-d∥y∥^2), consistent with κ(y)^{2d}; if so, it would be clearer to write it explicitly as κ(y)^m with m=2d.","section":"§6, Example 6.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically rich and the conditional theorems are valuable, but the rate issue in Theorems 2.3–2.5 is load-bearing and must be resolved before publication. The inconsistencies in Example 6.3 and Proposition 5.6 are local but should be corrected in the same revision. I am not recommending rejection because the central Poisson-convergence framework appears sound and the gaps are fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing to know: this is the first paper to get Poisson limits for large-lifetime cycles beyond codimension 1, and it adds joint convergence with lifetime and deathtime marks. Theorems 2.3–2.5 are substantial and, under their stated assumptions, the arguments look right. The U-statistic approximation plus the continuous BK inequality is a real technical advance. Credit also for verifying the model assumptions in nontrivial cases (Cech with m=k, VR with k=d+1 and m=2d, and assumption H for d=2), and for shipping code and simulations.\n\nSoft spots are real but not fatal. The rates as printed are over-claimed. In the proof of Theorem 2.3 the authors assert r_n ≤ ρ_{n,m+1} without argument, and Lemma 4.7 only gives an O(r_n) intensity error. Assumption M does not imply that inequality; for example, d=10, m=3, r_n=n^{-0.14} satisfies M but r_n ≫ ρ_{n,m+1}=n^{-0.2}. The same O(r_n) is silently dropped in Theorems 2.4 and 2.5. The qualitative Poisson convergence should still hold, but the stated rates need an extra O(r_n) term or an additional assumption linking r_n to ρ_{n,m+1}. This is a fixable gap, but the theorems as written are not correct.\n\nAlso, Example 6.3 applies Theorem 2.3 to k=d−1 with m=2d, but Proposition 5.4 does not verify uniqueness U for that combination. That looks like a typo in the example, not a flaw in the main results. The simulations for m=5,6 explore unverified territory, and the paper says so.\n\nThe paper is honest about what remains open: assumption U is only verified in special cases and H only for d=2. For a first paper in arbitrary dimensions that is acceptable scope, but readers should not take the broader claims at face value until those assumptions are addressed or clearly flagged as hypotheses.\n\nWho this is for: anyone working on extreme-value theory for TDA, null distributions for persistent homology tests, or Poisson approximation for stabilizing functionals. It deserves serious refereeing. I would send it to review and ask for a revision that fixes the rate statements and the example, rather than desk-reject.","headline":"First Poisson limits for large-lifetime cycles in arbitrary dimensions, but the claimed rates in Theorems 2.3–2.5 are not justified as written.","tokens_in":47501,"tokens_out":3767,"would_cite":true,"duration_ms":33610,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55U10","60D05","60G55","60F05","60G70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that near-maximal-lifetime cycles in a Poisson point cloud converge to a Poisson point process, jointly for centers, lifetimes and deathtimes, under explicit conditions.","keywords":["persistent homology","Poisson point process","large-lifetime cycles","Cech complex","Vietoris-Rips complex","extreme value theory","Poisson approximation","persistence diagram"],"falsifier":"For a parameter triple (d,k,m) not covered by Proposition 5.4, search numerically for an m-point set containing two different negative simplices with deathtime at most 1 and lifetime within epsilon of ell_max(m). Such a configuration would violate Assumption U and remove the Poisson limit in Theorems 2.3-2.5, while a proof that no such configuration exists would extend the theorems to that parameter triple.","tokens_in":2006,"feed_emoji":"⭕","tokens_out":2014,"duration_ms":104815,"temperature":0.7,"pith_summary":"Persistent homology singles out cycles—loops, cavities—that survive over a long range of scales, and topological data analysis treats these as candidates for genuine structure. This paper proves that, in a growing Poisson point cloud, the rare cycles with near-maximal lifetime are asymptotically independent and Poisson-distributed: their centers converge to a Poisson point process, and in a sparse regime the centers, scaled lifetime deviations, and deathtime deviations converge jointly to a Poisson process with an explicit intensity. On the two-dimensional flat torus the authors prove Poisson convergence of the centers for Cech complexes with multiplicative lifetimes. In dimensions d >= 2, under a deathtime bound that makes connected clusters small, they obtain joint convergence for both Cech and Vietoris-Rips filtrations, with the limiting intensity factorized as $\\alpha$ * kappa(y)^m * q * u^(q-1) * H_q(v). If the Poisson description is correct, it turns extreme persistence into a null model that can be used for significance testing and for the asymptotic law of the largest lifetime.","feed_headline":"Long-lived cycles in random clouds converge to a Poisson pattern","feed_subtitle":"Joint limit for centers, lifetimes and deathtimes turns extreme persistence into a testable null model.","key_machinery":"The argument is carried by the deterministic maximal lifetime ell_max (the supremum of lifetimes over all m-point clusters with deathtime at most 1) and by the integrals g(u) and h(u,v) that count, up to scaling, the volume of cluster configurations whose lifetime is within u of ell_max. The threshold u_{n,$\\alpha$} is defined through $g^{{-1}}$, and the limiting density is obtained from the Taylor behavior of g and h: the exponent q and the function H_q come from the first nonvanishing derivatives. In the sparse regime the proof approximates the point process of large-lifetime cycles by a Poisson U-statistic built from cycles inside m-clusters, bounds the error by the expected number of (m+1)-clusters, and applies the functional Poisson approximation for U-statistics; Assumption U, which requires at most one near-maximal cycle per cluster, is what turns the U-statistic into a simple marked point process. In the unbounded regime the proof instead truncates to mortal cycles with a deathtime bound, stabilizes the functional to a ball around the cycle center, and controls simultaneous large cycles through the continuous disjoint-occurrence inequality and the planar separation theorem.","core_discovery":"The central claim is Theorem 2.5: for a Poisson point cloud with intensity n * kappa on a region W in R^d, under assumptions F, M, P, T, U, and H, the marked point process of large-lifetime cycles—recording each cycle's center, its scaled deviation (ell_max - ell)/u_{n,$\\alpha$} from the deterministic maximal lifetime, and the scaled deathtime deviation (r_n - r_x)/u_{n,$\\alpha$}—converges in Kantorovich-Rubinstein distance to a Poisson point process with intensity $\\alpha$ * kappa(y)^m * q * u^(q-1) * H_q(v), with rate O(rho_{n,m+1} + rho_{n,m}^{-1/q}). Here rho_{n,m} = n (n r_n^d)^(m-1) is the expected number of m-clusters, and the threshold u_{n,$\\alpha$} is set so that the expected number of large-lifetime features is $\\alpha$. In the unbounded regime on the flat torus, Theorem 2.1 establishes Poisson convergence of the centers alone for Cech complexes with multiplicative lifetimes, with rate O($n^{{-1/36+epsilon}}$). A consequence is that the largest lifetime's deviation follows a Weibull law in the sparse regime, while the transformed largest lifetime is asymptotically exponential on the torus.","pith_inferences":["If the uniqueness assumption U fails for some (d,k,m) combination, clusters could contribute several near-maximal cycles; the limiting point process would then have a clustered, non-Poisson structure, so locating such configurations would mark the boundary of the theorem's applicability.","The product form of the limiting intensity suggests a diagnostic: normalized deathtime deviations should look independent of spatial location and lifetime, so a correlation or goodness-of-fit test on that independence could be used before trusting the Poisson model.","The methods may extend to the flat torus in the sparse regime and to multiplicative lifetimes in d >= 3, since the limiting formulas are scale-invariant; the paper states the torus extension as a conjecture.","The simulation results hint that the Taylor order q for Cech complexes with m>3 may differ from the naive 2m-3 guess; if confirmed, it would change the Weibull shape used in statistical tests."],"forward_implications":["The scaled deviation of the largest lifetime from ell_max is asymptotically Weibull with shape q; on the torus, the transformed largest lifetime n^3 v(ell) is asymptotically Exponential(1).","Deathtime deviations of near-maximal cycles are asymptotically i.i.d. with density H_q, so the rightmost strip of the persistence diagram has a tractable product structure for model checking.","Inhomogeneous intensity functions kappa are handled: the limiting center intensity is alpha * kappa(y)^m, meaning large-lifetime cycles appear in regions where kappa is high.","The convergence rate O(rho_{n,m+1} + rho_{n,m}^{-1/q}) tells practitioners how sparse the deathtime bound must be for the Poisson approximation to be accurate.","When the model assumptions are met, extreme persistent homology ceases to need Monte Carlo or bootstrap calibration for a Poisson null."],"supporting_citations":[{"why":"Supplies the high-probability order of the largest multiplicative lifetime that fixes the threshold scale in the unbounded regime.","marker":"[3]"},{"why":"Provides the stabilization-based Poisson approximation theorem used to prove convergence on the torus.","marker":"[4]"},{"why":"Additive-lifetime Poisson approximation for codimension-one features whose null-set and inversion ideas are reused for thresholds and Vietoris-Rips continuity.","marker":"[9]"},{"why":"Functional Poisson approximation for U-statistics in Kantorovich-Rubinstein distance, the core engine for the sparse-regime theorems.","marker":"[10]"},{"why":"Supplies the negative-simplex identification, stability of persistence, and Alpha complex facts used throughout.","marker":"[7]"},{"why":"Continuous disjoint-occurrence inequality used to rule out multiple near-maximal cycles in the unbounded regime.","marker":"[14]"},{"why":"Contractibility of the Cech complex at the deathtime bound, justifying the mortal-cycle truncation.","marker":"[17]"},{"why":"Sparse-regime cluster analysis and the lower bound on vertices of nontrivial Vietoris-Rips cycles used to verify assumption U.","marker":"[18]"}],"fun_headline_variants":["Poisson law for centers and lifetimes of extreme cycles","Extreme cycles in Poisson clouds follow Poisson distribution","Long-lived topological features converge to Poisson process","Random point clouds yield Poisson for large-lifetime cycles","Poisson convergence for large-lifetime cycles in point clouds"],"cache_read_input_tokens":49536,"weakest_assumption_plain":"Assumption U says that in any m-point cluster, at most one subset of points can form a cycle whose lifetime is close to the maximal possible lifetime; without this uniqueness, a single cluster could hold several large-lifetime cycles and the limiting process would no longer be Poisson.","fun_headline_variants_meta":{"raw":{"variants":["Poisson law for centers and lifetimes of extreme cycles","Extreme cycles in Poisson clouds follow Poisson distribution","Long-lived topological features converge to Poisson process","Random point clouds yield Poisson for large-lifetime cycles","Poisson convergence for large-lifetime cycles in point clouds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000577,"raw_usage":{"total_tokens":2736,"prompt_tokens":971,"completion_tokens":1765,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":1691}},"tokens_in":587,"tokens_out":1765,"duration_ms":12335,"temperature":1.0,"reasoning_tokens":1691,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:27:04.534858+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a parameter triple (d,k,m) not covered by Proposition 5.4, search numerically for an m-point set containing two different negative simplices with deathtime at most 1 and lifetime within epsilon of ell_max(m). Such a configuration would violate Assumption U and remove the Poisson limit in Theorems 2.3-2.5, while a proof that no such configuration exists would extend the theorems to that parameter triple.","supporting_citations":[{"cited_title":"Bobrowski, M","cited_arxiv_id":null,"evidence_quote":"Supplies the high-probability order of the largest multiplicative lifetime that fixes the threshold scale in the unbounded regime."},{"cited_title":"Bobrowski, M","cited_arxiv_id":null,"evidence_quote":"Provides the stabilization-based Poisson approximation theorem used to prove convergence on the torus."},{"cited_title":"Chenavier and C","cited_arxiv_id":null,"evidence_quote":"Additive-lifetime Poisson approximation for codimension-one features whose null-set and inversion ideas are reused for thresholds and Vietoris-Rips continuity."},{"cited_title":"Decreusefond, M","cited_arxiv_id":null,"evidence_quote":"Functional Poisson approximation for U-statistics in Kantorovich-Rubinstein distance, the core engine for the sparse-regime theorems."},{"cited_title":"Boissonnat, F","cited_arxiv_id":null,"evidence_quote":"Supplies the negative-simplex identification, stability of persistence, and Alpha complex facts used throughout."},{"cited_title":"Lace Expansion and Mean-Field Behavior for the Random Connection Model","cited_arxiv_id":"1908.11356","evidence_quote":"Continuous disjoint-occurrence inequality used to rule out multiple near-maximal cycles in the unbounded regime."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contractibility of the Cech complex at the deathtime bound, justifying the mortal-cycle truncation."},{"cited_title":"Limit theorems for Betti numbers of random simplicial complexes","cited_arxiv_id":"1009.4130","evidence_quote":"Sparse-regime cluster analysis and the lower bound on vertices of nontrivial Vietoris-Rips cycles used to verify assumption U."}],"review_version":1}