{"id":"eead9f76-bc22-47ae-a7fb-cbdfd3cb1a88","arxiv_id":"2506.13429","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A central limit theorem is established for Betti numbers in the marked random connection model, and as a corollary for the Boolean model with general convex grains.","lead":"This paper proves a central limit theorem for topological counts, including Betti numbers, in a marked random connection model for higher-dimensional simplicial complexes. The result transfers to the Boolean model with arbitrary convex grains.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1's variance formula appears valid only when W is a union of lattice cubes, because Δ_W depends on coordinates affected by points in the same cube outside W. The proof of Lemma 4.5 applies this identity to non-aligned cubes W_n, leaving the variance normalization unproved.","rationale":"The paper's strongest claim is a CLT for weakly stabilizing functionals, applied to Betti numbers and to the Boolean model. The entire proof chain depends on Lemma 3.1, which transfers a variance representation from Theorem B.1 of [18] to the difference operator (5). The reader correctly identified this as the weakest assumption. My stress-test sharpens it: Lemma 3.1 is stated for arbitrary bounded measurable W, but the construction of Δ_W is only local, in the sense of depending only on Ψ_W, when W is a union of the fixed unit cubes Q_i. For the cubes W_n of side length n^{1/d}, this condition fails for d>1 and for most n in d=1. Then Λ(x,a,t,u)f(Δ_W) does not vanish for x outside W, and the variance representation should include an integral over R^d\\W (or W must be cube-aligned). Since Lemma 4.5 and Theorem 4.6 use the exact identity with integration over W_n, the proof as written has a real gap. This is an internal consistency issue, not a matter of disagreement with existing consensus. The main theorem may well be true and fixable, because the missing contribution involves only boundary layers of W_n and may vanish in the normalization, but that estimate is absent. I also note a secondary gap: the proof of Theorem 4.6 asserts it suffices to treat centered cubes with |W_n|=n, without providing the usual blocking or approximation argument for an arbitrary cube sequence. The paper has real strengths: the abstract framework is natural, the example functionals are worked out in detail, and the Boolean model transfer uses the Nerve Theorem appropriately. Those strengths do not remove the need to repair the variance representation. For these reasons I would change the verdict from ACCEPT to CONDITIONAL: accept if Lemmas 3.1 and 4.5 are corrected to handle non-aligned cubes (or the theorem is restricted to cube-aligned windows), with a check as described above.","tokens_in":25457,"tokens_out":11960,"duration_ms":132965,"concrete_test":"Test the variance identity directly in d=1 with W=[−√n/2,√n/2), single-point mark space, constant φ_1≡1/2, and f(Δ_W)=number of edges. Since W cuts through the unit lattice, points outside W in the same unit intervals affect the ranks used to assign u-components. Compare Var(f(Δ_W)) with the RHS of Lemma 3.1 integrated only over W, e.g. by Monte Carlo at several n or by a closed-form Poisson calculation. If the discrepancy divided by |W| does not tend to 0, the variance normalization used in Lemma 4.5 is wrong; if it is o(|W|), the theorem may survive but requires an explicit boundary estimate not present in the paper.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central CLT relies on Lemma 3.1, the variance representation Var(f(Δ_W)) = γ∫_0^1∫_W∫_A∫_M E[E[Λ f(Δ_W)|Ψ_t]^2] d... . In the proof the authors claim Λ vanishes for x∉W, 'obviously'. But this is false for a general cube W that cuts through the unit cubes Q_i. The coordinates of a vertex in W are (k,i), where i is its t-rank among all points of Ψ in the whole cube Q_k, not just in W. Thus inclusion of a simplex in Δ_W depends on points of Ψ outside W that lie in the same unit cubes. The paper itself notes (Section 3) that Δ_W is fully determined by Ψ_W only 'whenever W is a union of cubes'. For W_n = [-n^{1/d}/2, n^{1/d}/2)^d, the side length n^{1/d} is generally not an integer, so W_n is not a union of unit cubes. Consequently Lemma 3.1 cannot be applied as written to the cubes used in Lemma 4.5; the missing contribution from points outside W_n is not accounted for. Lemma 4.5 then identifies σ² via the exact identity Var = β∫_{W_n} h(W_n−x)dx, and Theorem 4.6 inherits this gap. A separate issue compounds this: the proof of Theorem 4.6 reduces to the special sequence with |W_n|=n without justifying why this implies the result for arbitrary cube sequences W_n→R^d.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies a marked stationary generalization of the random connection model (RCM) for higher-dimensional simplicial complexes, introduced in a companion paper [21]. The model places a Poisson process on R^d × A, where A is a Borel mark space, and adds simplices of dimension up to α with probabilities determined by translation-invariant connection functions. The main results are: (i) a central limit theorem (Theorem 4.6) for weakly stabilizing functionals of the induced complexes on growing cubes, under a moment condition; (ii) a demonstration that Betti numbers and other functionals such as the Euler characteristic, subcomplex counts, and d_l^m are weakly stabilizing under appropriate integrability assumptions (Section 5); (iii) a corollary on positivity of the asymptotic variance (Section 6); and (iv) an application to Betti numbers of the Boolean model with compact convex grains (Section 7). The argument follows the stabilization approach of [4], using a variance representation (Lemma 3.1) and a cube decomposition.","tokens_in":25864,"tokens_out":15292,"duration_ms":149306,"significance":"If the proof were completed, the results would be a significant advance in stochastic topology: they would provide CLTs for Betti numbers in a general marked RCM, including marked graphs, and extend known Boolean model results to grains of arbitrary compact convex shape (earlier work [27] handled only equal-radius balls). The paper is methodical in identifying the class of weakly stabilizing functionals and provides a general framework that includes several existing models. The companion paper [21] supplies the model definition and simplex-count asymptotics, and the present paper contributes the CLT machinery for Betti numbers. The proof of the stabilization class is detailed, and the moment conditions are explicit, yielding an explicit asymptotic variance formula.","major_comments":[{"comment":"The variance representation is asserted for every bounded measurable W, but the construction of Δ_W via the coordinate system (k,i) means that Δ_W is determined by Ψ_W only when W is a union of the unit cubes Q_i, as the paper itself notes in Section 3. For a general cube, adding a point x ∉ W that shares a unit cube with points of W can change the coordinates (and hence the simplices) of points in W, so Λ_{x,a,t,u} f(Δ_W) does not vanish for x ∉ W. The proof's statement that Λ vanishes outside W is therefore false in the generic case, and Lemma 3.1 is not established for the cubes W_n = [-n^{1/d}/2, n^{1/d}/2)^d used in Lemma 4.5. Since Lemma 4.5 and Lemma 3.2 depend on Lemma 3.1, the variance normalization and the Poincaré inequality used in Theorem 4.6 are not justified.","section":"Section 3, Lemma 3.1"},{"comment":"The proof reduces at the outset to the special sequence W_n = [-n^{1/d}/2, n^{1/d}/2)^d, claiming that this is permissible by translation invariance. Translation invariance only allows shifting cubes in space, not changing their side lengths; an arbitrary cube sequence has side lengths a_n → ∞ and cannot be mapped to the special sequence. No approximation or subsequence argument is provided to pass from the special sequence to all cube sequences, so the theorem's statement for any sequence of cubes is not proven. This gap is separate from, and compounds, the Lemma 3.1 issue.","section":"Theorem 4.6 proof"}],"minor_comments":[{"comment":"In the proof of Lemma 4.5, 'Theorem 3.1' should be 'Lemma 3.1', and equation (11) has the same expression on both sides; the right-hand side should presumably contain Λ_{(0,V,1,U)} f(Δ_W).","section":"Lemma 4.5 proof"},{"comment":"Remark 4.3 is deferred to the author's thesis [20] but is used in Theorem 4.6 to ensure finite variance of f(Δ_{G_1}); please include a proof or at least a precise statement of the result from [20] for self-containedness.","section":"Remark 4.3"},{"comment":"Theorem 4.6 contains a typo in the definition of W_n ('d√n' should be n^{1/d}), and the displayed cube should be half-open consistently.","section":"Theorem 4.6"},{"comment":"Equation (25) uses the symbol ∫_{K_d} ambiguously; the integral over the mark space should be written with a variable name (e.g., Θ(dK)) to avoid confusion.","section":"Section 7, equation (25)"}],"recommendation":"major_revision","confidential_remarks":"The central proof currently has a significant gap concerning the variance representation for non-lattice-aligned cubes. This is likely fixable by either restricting to cubes that are unions of unit cubes, by adding a boundary-approximation argument, or by modifying the coordinate construction. The reduction to the special sequence in Theorem 4.6 also needs a rigorous argument. Given the otherwise careful presentation and the novelty of the Boolean model application, I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a clean CLT for weakly stabilizing functionals in the marked stationary random connection model and applies it to Betti numbers and Boolean models with general convex grains. That is a genuinely useful extension of [4] and [27], and the verification that Betti numbers fall into the class is well done. The mixed-Poisson moment condition is a nice touch, and the nerve-theorem passage to the Boolean model is elegant.\n\nThe soft spot is serious. Lemma 3.1, the exact variance representation, is transferred from [18] with the claim that the add-one operator vanishes for x outside W. That is only true when W is a union of the unit cubes Q_i from the construction. The paper itself says exactly this: Δ_W is fully determined by Ψ_W only when W is a union of cubes. Lemma 4.5 then applies Lemma 3.1 to arbitrary cubes W_n with side length n^{1/d}, which is generally not an integer, so W_n cuts through the unit cubes. Adding a point x ∉ W_n but in the same unit cube changes the coordinates of points inside W_n and therefore can change Δ_{W_n}; the operator does not vanish. The exact variance identity used in Lemma 4.5 is thus unsupported, and the convergence of the normalized variance to σ² is not proved as written.\n\nA second, lesser gap: Theorem 4.6 reduces to the special cube sequence W_n = [-n^{1/d}/2, n^{1/d}/2)^d and asserts that translation invariance makes this sufficient for arbitrary cube sequences. No argument is provided. Even if the variance issue were fixed, this step needs a separate justification.\n\nThe theorem might still be true. The mistake is in the range of application of a transferred lemma, not obviously in the stabilization concept or the moment estimates. A repair could come from restricting the theorem to cube sequences that are unions of the Q_i, or by adding boundary correction terms for the cubes that cut through the lattice. But as it stands, the central normalization argument has a hole.\n\nThis paper deserves peer review — the idea is good, the examples are meaningful, and the gap is the kind a careful referee can help close. It should not be desk-rejected, but it also should not be accepted without revision. For a reading group it would be a good case study in how transferred lemmas can fail when geometry shifts.","headline":"Useful CLT for Betti numbers in marked RCM and Boolean models, but the variance normalization in Lemma 4.5 rests on an invalid application of Lemma 3.1 to cubes that are not unions of the coordinate unit cubes.","tokens_in":26312,"tokens_out":4962,"would_cite":false,"duration_ms":53470,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60D05","60B99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a central limit theorem for Betti numbers in a broad class of random simplicial complexes, including the Boolean model with arbitrary convex grains, with an explicit asymptotic variance.","keywords":["Random Connection Model","Betti numbers","central limit theorems","Boolean model","simplicial complexes","weakly stabilizing functionals","marked Poisson processes","stochastic geometry"],"falsifier":"For the unmarked random connection model on the line with an explicit connection function, take $f$ to be the number of isolated vertices, compute the limit $\\sigma^2$ from Lemma 4.5 directly, and compare it with $\\operatorname{Var}(f(\\Delta_{W_n}))/|W_n|$ obtained by exact finite-window computation or high-precision simulation; agreement at growing $n$ supports the variance identity, while systematic mismatch would refute Lemma 4.5 and with it the central limit theorem's normalization.","tokens_in":25292,"feed_emoji":"🔺","tokens_out":8063,"duration_ms":73769,"temperature":0.7,"pith_summary":"This paper establishes a central limit theorem for functionals of a marked, stationary random simplicial complex built from a Poisson process on $\\mathbb{R}^d\\times\\mathbb{A}$, where each potential simplex appears with a probability given by a translation-invariant connection function. The main result says that for any weakly stabilizing functional satisfying a moment condition, the fluctuation around its mean, scaled by the square root of the window volume, converges to a normal distribution with an explicit variance. The paper proves that $p$-th Betti numbers are such functionals provided $\\mathbb{E}[\\pi(V)^{3(p+1)}]<\\infty$, where $\\pi(V)$ is a mixed-Poisson edge-degree parameter. It then transfers the theorem to Betti numbers of the Boolean model with arbitrary compact convex grains via the nerve theorem, removing a restriction to equal-radius grains that earlier results needed. What makes the statement useful is that it covers higher-dimensional cohomology, such as loops, voids, and higher holes, rather than only component counts.","feed_headline":"Higher Betti numbers get a central limit theorem","feed_subtitle":"Applies to higher-dimensional holes and the Boolean model with arbitrary convex grains.","key_machinery":"The load-bearing object is the difference operator from (5), $$\\Lambda_{(x,a,t,u)}f(\\Delta_W)=f(\\$Delta_W^{{(x,a,t,u)}}$)-f(\\$Delta_W^{{(x,a,t,u)\\setminus\\{x\\}}$}),$$ which inserts a vertex and compares the functional on the enlarged and the unenlarged complex; unlike the classical add-one-cost operator, it keeps the rest of the configuration's coordinates unchanged when $t=1$, which makes conditional-expectation computations tractable. Lemma 3.1 gives the identity $$\\operatorname{Var}(f(\\Delta_W))=\\gamma\\$int_0^{1}$\\int_W\\int_{\\mathbb{A}}\\int_{\\mathbb{M}}\\mathbb{E}\\big[\\mathbb{E}[\\Lambda_{(x,a,t,u)}f(\\Delta_W)\\mid\\Psi_t]^2\\big]\\,\\mathbb{Q}(\\mathrm{d}u)\\Theta(\\mathrm{d}a)\\,\\mathrm{d}x\\,\\mathrm{d}t,$$ and Lemma 4.5 turns this into the variance asymptotics $\\operatorname{Var}(f(\\Delta_{W_n}))/|W_n|\\to\\sigma^2$ by showing the integrand stabilizes. The weak-stabilization condition in Definition 4.2 is what lets the difference operator converge to a limiting random variable $Z_t$, and the moment condition (7) upgrades this to $L^2$-convergence, which is what the two-sequence normal-approximation theorem in Theorem 4.1 needs.","core_discovery":"The paper's central claim is Theorem 4.6: for the marked stationary random simplicial complex built from a Poisson process on $\\mathbb{R}^d\\times\\mathbb{A}$ with translation-invariant connection functions, every weakly stabilizing functional $f$, meaning that the local effect at the origin measured by the difference operator $\\Lambda_{(\\mathbf{0},V,1,U)}f(\\Delta_{W_n})$ converges in probability as the window $W_n$ grows, that also satisfies the moment bound $\\sup_{W\\ni \\mathbf{0}}\\mathbb{E}[|\\Lambda_{(\\mathbf{0},V,1,U)}f(\\Delta_W)|^{2+\\varepsilon}]<\\infty$ obeys $$\\frac{f(\\Delta_{W_n})-\\mathbb{E}[f(\\Delta_{W_n})]}{\\sqrt{|W_n|}}\\xrightarrow{d}N_{\\$sigma^{2}$},$$ with $\\sigma^2=\\gamma\\int_0^1\\mathbb{E}[\\tilde{\\mathbb{E}}[Z_t\\mid\\Psi_t]^2]\\,\\mathrm{d}t$ and with strict positivity of $\\sigma^2$ whenever the stabilizing limit $Z=Z_1$ is nonzero with positive probability. The paper then shows that the $p$-th Betti number $\\beta_p$ is weakly stabilizing whenever $\\mathbb{E}[\\pi(V)^{3(p+1)}]<\\infty$, where $\\pi(V)$ is the mean number of edges an added vertex attaches to; the proof uses the stability bound $|\\beta_p(K)-\\beta_p(L)|\\le (f_p(K)-f_p(L))+(f_{p+1}(K)-f_{p+1}(L))$ to control the difference operator by simplex degrees, and mixed-Poisson moment calculus to control those degrees. In the Boolean model with arbitrary compact convex grains, the nerve theorem identifies the Betti numbers of the union set with Betti numbers of the model's $\\alpha$-skeleton, so the central limit theorem carries over to the Boolean model without the equal-radius restriction of earlier results.","pith_inferences":["The paper does not state this, but the same stabilization machinery should prove central limit theorems for the Betti numbers of sublevel sets in a filtration, because the difference operator records only local homological changes; this would require the moment condition to hold for the filtration's local contribution.","A testable extension suggested by the variance formula is to use it for statistical inference in topological data analysis: asymptotic confidence intervals for estimated Betti numbers follow whenever the limit variance is positive and the stabilization condition is checked.","Because the mark space $\\mathbb{A}$ is an arbitrary Borel space, the theorem is not tied to geometric marks; one could apply it to marked network models in which connection probabilities depend on both spatial distance and categorical attributes, with only the spatial component required to be translation invariant.","The Boolean-model section hints that the equal-radius condition is an artifact of the Cech-complex identification; for bounded grains the mixing parameter is bounded, and for unbounded grains the paper's integrability condition on the Minkowski difference moment appears to be the natural replacement."],"forward_implications":["A multivariate central limit theorem holds for any finite family of weakly stabilizing functionals satisfying the moment condition, with a covariance matrix given by the corresponding limits of the difference operators.","Under a boundedness condition on the connection function, Betti numbers satisfy a law of large numbers: $\\beta_p(\\Delta_{W_n})/|W_n|$ converges in probability to a constant.","The theorem applies to the Euler characteristic, counts of induced subcomplexes, counts of connected components isomorphic to a fixed complex, and vertex degree counts, under explicit moment assumptions.","For the Boolean model with arbitrary compact convex grains, the $p$-th Betti number of the union set satisfies a central limit theorem; choosing the dimension parameter $\\alpha$ large enough removes any skeleton-dimension restriction.","A combinatorial positivity criterion is given: if there is a connected complex $K$ with $|f(K)-f(L)|>0$ for every induced subcomplex on one fewer vertex, and if such a $K$ appears with positive probability, then the asymptotic variance is strictly positive."],"supporting_citations":[{"why":"Supplies the variance representation and the difference-operator construction that Lemma 3.1 transfers to this model.","marker":"[18]"},{"why":"Provides the proof template for the central limit theorem for weakly stabilizing functionals and the characterization of weak stabilization used in Proposition 5.1.","marker":"[4]"},{"why":"Introduces the marked stationary random connection model for simplicial complexes and supplies the simplex-function bounds used to verify integrability.","marker":"[21]"},{"why":"Provides the stability estimate for Betti numbers used to bound the difference operator, and the equal-radius Boolean/Cech central limit theorem that this paper extends.","marker":"[27]"},{"why":"Supplies the Poisson-process existence, Mecke's formula, and Borel-space facts used throughout the proofs.","marker":"[19]"},{"why":"Gives the two-sequence normal-approximation criterion that Theorem 4.1 restates and Theorem 4.6 applies.","marker":"[25]"},{"why":"Provides the Boolean-model central limit theorem for additive functionals, which the paper notes does not cover Betti numbers and is therefore extended.","marker":"[10]"}],"fun_headline_variants":["Betti numbers get CLT in random connection model","CLT for higher Betti numbers in Boolean model","Stabilizing functionals: Betti numbers obey CLT","Holes in random complexes get central limit theorem","Boolean model: Betti numbers satisfy CLT"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the variance representation of Lemma 3.1, carried over from the random-graph setting: if that identity does not survive the cube-coordinate construction of the simplicial complex, the variance normalization and the central limit theorem would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Betti numbers get CLT in random connection model","CLT for higher Betti numbers in Boolean model","Stabilizing functionals: Betti numbers obey CLT","Holes in random complexes get central limit theorem","Boolean model: Betti numbers satisfy CLT"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000462,"raw_usage":{"total_tokens":2389,"prompt_tokens":1100,"completion_tokens":1289,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":716,"completion_tokens_details":{"reasoning_tokens":1213}},"tokens_in":716,"tokens_out":1289,"duration_ms":9519,"temperature":1.0,"reasoning_tokens":1213,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:02:01.054364+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the unmarked random connection model on the line with an explicit connection function, take $f$ to be the number of isolated vertices, compute the limit $\\sigma^2$ from Lemma 4.5 directly, and compare it with $\\operatorname{Var}(f(\\Delta_{W_n}))/|W_n|$ obtained by exact finite-window computation or high-precision simulation; agreement at growing $n$ supports the variance identity, while systematic mismatch would refute Lemma 4.5 and with it the central limit theorem's normalization.","supporting_citations":[{"cited_title":"Therandomconnectionmodelandfunctionsofedge-markedPois- son processes: Second order properties and normal approximation.The Annals of Applied Probability, 31(1):128–168, 2021","cited_arxiv_id":null,"evidence_quote":"Supplies the variance representation and the difference-operator construction that Lemma 3.1 transfers to this model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the proof template for the central limit theorem for weakly stabilizing functionals and the characterization of weak stabilization used in Proposition 5.1."},{"cited_title":"Yogeshwaran, E","cited_arxiv_id":null,"evidence_quote":"Provides the stability estimate for Betti numbers used to bound the difference operator, and the equal-radius Boolean/Cech central limit theorem that this paper extends."},{"cited_title":"Last and M","cited_arxiv_id":null,"evidence_quote":"Supplies the Poisson-process existence, Mecke's formula, and Borel-space facts used throughout the proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the two-sequence normal-approximation criterion that Theorem 4.1 restates and Theorem 4.6 applies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Boolean-model central limit theorem for additive functionals, which the paper notes does not cover Betti numbers and is therefore extended."}],"review_version":2}