{"id":"3b13db67-b08e-4ac9-a0c0-eab813aad71b","arxiv_id":"2411.14083","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For exchange-driven growth, this paper extends global and local existence results and proves finite-time gelation and instantaneous gelation for fast-growing interaction kernels.","lead":"This paper studies the exchange-driven growth model, where clusters trade single units, and proves when solutions exist for all time, for a short time, or not at all because the system gels. It extends earlier existence theorems to a wider class of initial data and gives the exact gelation time for the quadratic interaction rule.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The gelation conclusions depend on an imported lemma ([3, Lemma 9.2.2]) that moment blow-up implies T_gel, and this implication is not proved for exchange-driven dynamics; the paper's central non-existence claims stand or fall on it.","rationale":"The paper aims to settle the exchange-driven growth phase diagram by proving global and local existence and then connecting moment blow-up to gelation. The existence arguments are the best-supported part; the finite-time and instantaneous gelation results are where the proof leans on external machinery. The reader's weakest assumption identifies the exact load-bearing joint: the imported lemma is the only bridge from divergence of M_alpha or M_n to the conclusion that total mass is lost. I found no counterexample to the lemma, and it is a standard tool in coagulation-fragmentation theory, so a REJECT verdict would be too strong. However, the paper does not adapt or prove the lemma for the EDG system, and several secondary issues (the apparent typo in (3.17), the missing endpoint case max{mu,nu}=min{mu,nu}=1 in Theorem 2.2(b), and the lack of an explicit continuation argument in Theorem 2.7) reinforce the need for revision. The CONDITIONAL verdict already given by the reader remains appropriate, so the recommended verdict is UNCHANGED. The proposed concrete check would settle whether the gelation-time conversion is valid for EDG, which is the single most decisive uncertainty in the paper's central claim.","tokens_in":25919,"tokens_out":23084,"duration_ms":230877,"concrete_test":"Re-derive the contrapositive of [3, Lemma 9.2.2] directly for (1.1)-(1.3): assume a mild solution in Y+_2 on [0,T] with K <= Cj^2k^2 and M_1 conserved on [0,T], and prove that M_2(T) is finite. If the proof uses an identity specific to coagulation-fragmentation with no EDG analogue, the lemma is not transferable; if the proof succeeds, Theorems 2.7 and 2.9 are grounded. For a sharper check, take the exact kernel K = Cj^2k^2 and verify analytically whether any mass-conserving Y+_2 solution can be continued past T = 1/(2CM_2(0)), or whether the computed M_2 ODE forces the first loss of mass exactly at that time.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is the transfer of [3, Lemma 9.2.2] to the exchange-driven system (1.1)-(1.3). The paper invokes this lemma in the introduction and in the proofs of Theorems 2.7 and 2.9 to convert divergence of a moment M_m into the bound T_gel <= t0. For coagulation-fragmentation theory the lemma has a proof; for the EDG equation it is asserted without proof. This is not a pedantic point: finiteness of M_1 does not force finiteness of M_2, since a distribution with f_j ~ j^{-3} has finite mass but infinite second moment. Thus divergence of a higher moment does not by itself imply loss of mass; one must prove that a mass-conserving EDG solution cannot continue past the moment singularity. Because Theorems 2.7-2.10 conclude non-existence of mass-conserving solutions from finite-time or instantaneous gelation, and because the exact formula T_gel = 1/(2M_2(f(0))C) for K = Cj^2k^2 uses M_2 blow-up plus this lemma, the central phase-diagram claims are not fully self-contained. A secondary gap is that Theorem 2.7 proves M_alpha blow-up only on the local existence interval of Theorem 2.5, and no written continuation argument reaches the blow-up time; this may be fixable, but it compounds the reliance on the imported gelation-time conversion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the discrete exchange-driven growth (EDG) model (1.1)--(1.3) and claims a comprehensive existence and gelation phase diagram. The main results are: global classical solutions for symmetric kernels satisfying K_{j,k} \\le C(j^\\mu k^\\nu + j^\\nu k^\\mu) with \\mu,\\nu \\le 2 and \\mu+\\nu \\le 3, for initial data in Y_\\lambda (\\lambda=\\max\\{\\mu,\\nu\\}>1) or in Y_1 in the sublinear case; local existence for K_{j,k} \\le C j^2 k^2 with data in Y_2; finite-time gelation for kernels bounded below by C_1(j^2 k^\\alpha + j^\\alpha k^2), 1<\\alpha\\le2, with the explicit value T_{\\rm gel}=(2M_2(f(0))C)^{-1} for K=C j^2 k^2; and instantaneous gelation, T_{\\rm gel}=0, for kernels growing superquadratically, K_{j,k} \\ge C(j^\\beta+k^\\beta), \\beta>2. The proofs use truncation, uniform moment estimates based on de la Vall\\'ee-Poussin functions, Arzel\\`a-Ascoli compactness, and Gronwall-type inequalities.","tokens_in":26251,"tokens_out":17141,"duration_ms":144882,"significance":"If the results are correct, the paper would settle the exchange-driven growth phase diagram conjectured in [10] and meaningfully extend the global and local existence results of [7, 10, 23]. The explicit gelation-time formula for the multiplicative kernel K=C j^2 k^2 is a crisp, falsifiable prediction, and the use of refined de la Vall\\'ee-Poussin moments to relax the initial-data assumptions is a promising technique. The paper is ambitious and addresses an active topic. However, several load-bearing technical steps in the gelation and non-existence proofs are currently not rigorous, so the central claims are not yet fully established.","major_comments":[{"comment":"The proof of Lemma 4.1 ends with the algebraic inequality X_N(t) \\le M_r(0) + \\tilde C_* X_N(t), and the authors then invoke Gronwall's lemma to conclude X_N(t) \\le M_r(0) e^{\\tilde C_* t}. This is not a Gronwall argument: if \\tilde C_* > 1 the algebraic inequality yields no bound at all. One needs a differential inequality of the form (d/dt) M_r(t) \\le C M_r(t), or an integrated form with the moment on the right inside the time integral. Since Lemma 4.1 is used to justify that the local solution of Theorem 2.5 remains in Y_{2+\\alpha}, this gap directly affects the proof of Theorem 2.7.","section":"This comment concerns Section 4, Lemma 4.1."},{"comment":"The conversion of moment blow-up into a bound on the gelation time is based entirely on the imported statement [3, Lemma 9.2.2], which is proved in the coagulation-fragmentation literature and is not proved for the exchange-driven system. The paper asserts this lemma as 'well-known' in the introduction and then uses it to conclude T_{\\rm gel} \\le t_0 from divergence of a moment. Finiteness of M_1 does not imply finiteness of higher moments, so divergence of M_\\alpha at t_0 does not by itself establish loss of mass. The equality T_{\\rm gel}=(2M_2(f(0))C)^{-1} and the conclusion T_{\\rm gel}=0 in Theorem 2.9 are therefore not self-contained. The authors should either prove the EDG analogue of the lemma or reformulate the non-existence statements so that they follow directly from failure of Y_2 or Y_r regularity, without invoking mass loss.","section":"This comment concerns Section 1 and the proofs of Theorems 2.7, 2.9, and Corollary 2.10."},{"comment":"The proof supposes a solution on [0,T) and then invokes Lemma 5.3, which requires 0<T<T_{\\rm gel}. But Theorem 2.9 gives T_{\\rm gel}=0, so the hypothesis of Lemma 5.3 is not satisfied for any positive T. The contradiction is therefore not obtained. The non-existence of any Y_2 solution on an interval—even one that may have undergone gelation—requires a separate argument that does not rely on pre-gelation moment finiteness.","section":"This comment concerns the proof of Corollary 2.10 in Section 5."},{"comment":"Theorem 2.2(b) claims global existence whenever max{\\mu,\\nu} \\le 1, which includes the endpoint \\mu=\\nu=1. However, Lemma 3.5(2) and the proof of the uniform convergence (3.55) explicitly require min{\\mu,\\nu}<1. The case \\mu=\\nu=1 (the linearly growing kernel K_{j,k}\\le Cjk) is covered neither by part (a), since \\lambda=\\max\\{\\mu,\\nu\\}=1 is excluded by \\lambda>1, nor by the proof of part (b). This is a genuine omission in the theorem as stated; the endpoint should either be proved or handled by citing the known results of [10, 23].","section":"This comment concerns Theorem 2.2(b) and Lemma 3.5(2)."},{"comment":"The step from the tail estimate for \\sum_{j\\ge m} j^2 f_j(\\sigma) to the p-th moment bound (5.5) is not justified. The preceding line provides a bound with the exponential factor e^{-2CC_2 m^{\\beta-2}(t-\\sigma)} depending on the lower cutoff m. In (5.5) the exponential is evaluated with the summation index j, which would require a separate summation-by-parts or layer-cake argument. As written, the inequality does not follow. Since Lemma 5.3 underpins the proof of Theorem 2.9, the instantaneous gelation result needs a corrected tail estimate.","section":"This comment concerns Lemma 5.3, in particular inequality (5.5)."},{"comment":"The lower bound on M_\\alpha is derived only on the local existence interval of Theorem 2.5, i.e., for t<T_0<1/(2M_2(0)C). If the claimed blow-up time T^* = 1/(C_1\\alpha(\\alpha-1)2^{\\alpha-2}M_\\alpha(0)) is larger than 1/(2M_2(0)C), the solution need not exist up to T^*, and the displayed differential inequality does not show that M_\\alpha actually reaches infinity. The proof needs a continuation argument or an explicit comparison with the M_2 blow-up time so that T_{\\rm gel} is bounded by the first time at which the solution leaves Y_2.","section":"This comment concerns the proof of Theorem 2.7 in Section 4."}],"minor_comments":[{"comment":"The second term on the right-hand side of (3.17) should read j^{\\mu+\\nu-2}k^\\mu (equivalently j^{\\lambda+\\nu-2}k^\\lambda after setting \\lambda=\\mu), not j^{\\mu+\\nu-2}k^\\nu. The subsequent estimate (3.20) uses the correct k^\\lambda, so this appears to be a typo, but it should be corrected for consistency.","section":"This comment concerns equation (3.17) in Section 3."},{"comment":"The abstract says 'classical solutions' are established, whereas the theorems construct mild solutions and Corollary 2.4 upgrades them to continuously differentiable solutions. Please align the terminology so that the claims match the proofs.","section":"This comment concerns the abstract and the statements of Theorems 2.2 and 2.5."},{"comment":"The affiliation text contains the typo 'Roor kee'; it should presumably read 'Roorkee'.","section":"This comment concerns the affiliation on page 1."},{"comment":"The display after (5.4) has several sign and index conventions that are hard to follow (for example, the lower limit of the integral changes from \\sigma to t, and the rearrangement of I_j terms is not fully transparent). Please rewrite this part to make the estimates easier to verify.","section":"This comment concerns Section 5, around the derivation of (5.4)."}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious contribution to an active area and the overall strategy is credible, but the gelation and non-existence sections contain several technical gaps that are load-bearing. The existence part is closer to being acceptable, though Theorem 2.2(b) needs the endpoint \\mu=\\nu=1 addressed. I recommend major revision rather than rejection because the issues appear repairable within the manuscript's scope, provided the authors either prove the moment-blow-up-to-gelation lemma for the EDG system or restructure the conclusions to avoid it, and correct the flaws in Lemma 4.1, Lemma 5.3, and Corollary 2.10."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends the existence theory for exchange-driven growth in a way that matters: it relaxes Esenturk's p>2 moment condition to lambda=max{mu,nu}, proves finite-time gelation in the intermediate regime conjectured in [10], computes the exact gelation time for K=C j^2 k^2, and proves instantaneous gelation for super-quadratic kernels. The truncation-plus-De-la-Vallee-Poussin strategy is standard but applied carefully, the moment propagation lemma is structurally sound, and none of the conclusions come from fitted quantities. The exact gelation time is a genuinely useful anchor for numerical and stochastic-particle work.\n\nThe soft spots are real but mostly repairable. The biggest one is the imported [3, Lemma 9.2.2]: the paper repeatedly converts divergence of a moment into a bound on T_gel without proving that implication for the exchange-driven system. The stress-test note is right that this is not pedantic, since a distribution with finite mass can have infinite higher moments. Theorems 2.7 and 2.9, and hence the non-existence corollaries, currently lean on this unproved transfer. A referee should ask for either a proof of the lemma in the EDG setting or a direct mass-loss argument. Also, the displayed identity (3.17) has a wrong index on the second term, and inequality (3.18) is not pointwise as written; both are likely typos but they make the already technical estimates harder to verify. The proof of Theorem 2.2(b) omits the endpoint mu=nu=1 despite the statement covering it, and Theorem 2.7 proves moment blow-up only on the local existence interval without a continuation argument reaching the blow-up time.\n\nNone of this sinks the paper. The central claims are plausible, the methods are recognizable, and the gaps look fixable rather than fatal. For specialists in coagulation and exchange-driven growth, this is a useful, citable contribution that deserves a serious referee. Send it to review, and ask for the imported lemma to be justified or replaced.","headline":"A serious EDG paper that likely settles the existence/gelation phase diagram if the imported moment-blow-up-to-gelation lemma is proved for the exchange-driven system; worth refereeing, needs revision.","tokens_in":26779,"tokens_out":4333,"would_cite":true,"duration_ms":44592,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34A35","34A12","46B50","34G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For exchange-driven growth, global solutions stop existing once the interaction kernel grows past a sharp threshold: below $\\mu+\\nu\\le 3$ they exist for all time, while faster kernels cause gelation or instantaneous gelation.","keywords":["exchange-driven growth","interaction kernels","existence","mild solutions","non-existence","gelation","instantaneous gelation","moment blow-up"],"falsifier":"Find a symmetric kernel with $C_1(j^2k^\\alpha+j^\\alpha k^2)\\le K_{j,k}\\le Cj^2k^2$, $1<\\alpha\\le 2$, and initial data in $Y^+_{2+\\alpha}$ with $M_\\alpha(f(0))>0$, for which a global mass-conserving solution with all moments finite exists; this would refute Theorem 2.7 and Corollary 2.8. A more direct check is to test the imported lemma on the EDG system by looking for a solution with $\\sum_j j^m f_j(t_0)=\\infty$ while $\\sum_j j f_j(t_0)=\\sum_j j f_j(0)$, which would show that moment blow-up need not imply gelation.","tokens_in":25727,"feed_emoji":"📈","tokens_out":8835,"duration_ms":75782,"temperature":0.7,"pith_summary":"Exchange-driven growth is an infinite system of ordinary differential equations in which clusters exchange monomers; the paper asks when this system has a global solution and when it does not. It claims that symmetric kernels with $K_{j,k}\\le C(j^\\mu k^\\nu+j^\\nu k^\\mu)$, $\\mu,\\nu\\le 2$, $\\mu+\\nu\\le 3$, admit global classical solutions for initial data in $Y^+_\\lambda$ with $\\lambda=\\max\\{\\mu,\\nu\\}>1$, and for data in $Y^+_1$ when $\\max\\{\\mu,\\nu\\}\\le 1$, relaxing earlier requirements of finite higher moments. In the intermediate range $3<\\mu+\\nu\\le 4$, for kernels satisfying $C_1(j^2k^\\alpha+j^\\alpha k^2)\\le K_{j,k}\\le Cj^2k^2$ with $1<\\alpha\\le 2$, it establishes finite-time gelation by blowing up the moment $M_\\alpha$, with the exact formula $T_{\\mathrm{gel}}=(2M_2(f(0))C)^{-1}$ for $K_{j,k}=Cj^2k^2$. For kernels with $K_{j,k}\\ge C(j^\\beta+k^\\beta)$, $\\beta>2$, it proves instantaneous gelation, $T_{\\mathrm{gel}}=0$, so no solution exists on any positive time interval. If these results are correct, they settle the existence/non-existence phase diagram that earlier work had conjectured.","feed_headline":"Exchange-driven growth gels at a sharp kernel-growth threshold","feed_subtitle":"Global solutions hold up to a kernel-growth threshold; beyond it, moments blow up and mass conservation fails.","key_machinery":"The proof is carried by three mechanisms. Finite truncations of the infinite ODE system produce approximating solutions; uniform moment bounds come from testing the truncated equations with convex functions $G\\in\\mathcal G_{1,\\infty}$ selected by a refined de la Vallée-Poussin theorem, and Arzelà–Ascoli plus the convex-function tail control let the interaction sums pass to the limit. Gelation is detected through the exact evolution of a selected moment: writing $(j+1)^\\alpha-2j^\\alpha+(j-1)^\\alpha$ by the mean value theorem turns the kernel lower bound into a closed differential inequality for $M_\\alpha$ whose solution blows up at a computable time. Instantaneous gelation uses the criterion that blow-up of $\\sum_j j^m f_j(t_0)$ implies $T_{\\mathrm{gel}}\\le t_0$, imported from coagulation–fragmentation theory, combined with Jensen's inequality applied to $M_{n-2+\\beta}$ to show that for every $n$ the blow-up time is at most $O(1/n)$; letting $n\\to\\infty$ gives $T_{\\mathrm{gel}}=0$.","core_discovery":"The paper's central claim is that the exchange-driven growth system has a kernel-growth threshold with three regimes. For kernels with $\\mu+\\nu\\le 3$ and $\\mu,\\nu\\le 2$, the truncated-system limit is a mild solution that conserves total mass and particle number; the de la Vallée-Poussin machinery upgrades it to a classical solution while removing the earlier need for finite moments of order $p>2$. For kernels between $C_1(j^2k^\\alpha+j^\\alpha k^2)$ and $Cj^2k^2$ with $1<\\alpha\\le 2$, the $\\alpha$-th moment obeys $M_\\alpha(t)\\ge (1/M_\\alpha(f(0))-C_1\\alpha(\\alpha-1)2^{\\alpha-2}t)^{-1}$, blowing up in finite time and thereby preventing any global mass-conserving solution. For the multiplicative kernel $Cj^2k^2$, the second moment solves $M_2'(t)=2CM_2(t)^2$, so $T_{\\mathrm{gel}}=(2M_2(f(0))C)^{-1}$ exactly. Finally, if $K_{j,k}\\ge C(j^\\beta+k^\\beta)$ with $\\beta>2$, Jensen's inequality forces every higher moment to blow up at times shrinking to zero, so the gelation time is zero and no $Y^+_2$ solution exists on $[0,T)$ for any $T>0$.","pith_inferences":["The exact formula for $K_{j,k}=Cj^2k^2$ suggests a numerical check: integrating the truncated system, $M_2(t)$ should track $(1/M_2(f(0))-2Ct)^{-1}$ up to the blow-up time; any systematic deviation would indicate that finite-size effects change the gelation time.","Because the proof of finite-time gelation uses only the lower bound on the kernel and the moment equation, the same mechanism should hold for non-symmetric kernels with comparable growth, extending the non-existence regime beyond symmetric interactions.","The instantaneous-gelation theorem implies that for $\\beta>2$ the EDG equation is ill-posed in $Y^+_2$ from time zero; any worthwhile regularisation must alter the kernel or impose a cutoff, and it would be informative to compare the resulting limiting solutions as the cutoff is removed."],"forward_implications":["If correct, initial data need only lie in $Y^+_\\lambda$ (or $Y^+_1$ for slower kernels), not in spaces with finite moments of order $p>2$, so the existence theory covers a strictly larger class of physical cluster distributions.","For kernels sandwiched between $C_1(j^2k^\\alpha+j^\\alpha k^2)$ and $Cj^2k^2$ with $1<\\alpha\\le 2$, no global mass-conserving solution can exist; the system must lose mass at the finite gelation time.","For $K_{j,k}=Cj^2k^2$, the gelation time is exactly $(2M_2(f(0))C)^{-1}$, an explicit, initial-data-dependent formula that can be tested numerically.","For kernels growing faster than quadratically in either argument, solutions with finite second moment do not exist on any positive time interval, making the non-existence immediate rather than asymptotic."],"supporting_citations":[{"why":"Establishes the earlier existence theory and states the gelation conjecture that the paper extends; supplies the truncation scheme and kernel classes.","marker":"[10]"},{"why":"Provides the lemma that moment blow-up implies an upper bound on the gelation time, the link between the two halves of the paper.","marker":"[3]"},{"why":"The physics study whose critical-exponent predictions about gelation and instantaneous gelation are made rigorous.","marker":"[5]"},{"why":"Earlier global-existence result for linearly growing kernels whose restrictive initial-data assumption this paper relaxes.","marker":"[23]"},{"why":"Earlier local and global existence for product-type kernels, generalised here to non-homogeneous symmetric kernels.","marker":"[7]"},{"why":"Source of the convex-function inequalities and the de la Vallée-Poussin variant used to control higher moments.","marker":"[17]"},{"why":"Classical treatment of gelation in cluster equations that motivates the instantaneous-gelation argument.","marker":"[2]"}],"fun_headline_variants":["Kernel threshold dictates gelation time in exchange-driven growth","Sharp kernel threshold flips exchange-driven growth to gelation","Exchange-driven growth: finite-time gelation beyond sharp kernel bound","Exchange-driven growth: existence to instantaneous gelation at kernel threshold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported lemma from coagulation–fragmentation theory that if $\\sum_j j^m f_j(t_0)=\\infty$ for some $m$, then the gelation time is at most $t_0$; the paper does not prove this implication for the exchange-driven system, and both finite-time and instantaneous gelation conclusions depend on it.","fun_headline_variants_meta":{"raw":{"variants":["Kernel threshold dictates gelation time in exchange-driven growth","Sharp kernel threshold flips exchange-driven growth to gelation","Exchange-driven growth: finite-time gelation beyond sharp kernel bound","Exchange-driven growth: existence to instantaneous gelation at kernel threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000936,"raw_usage":{"total_tokens":4144,"prompt_tokens":1227,"completion_tokens":2917,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":843,"completion_tokens_details":{"reasoning_tokens":2848}},"tokens_in":843,"tokens_out":2917,"duration_ms":21619,"temperature":1.0,"reasoning_tokens":2848,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:34:58.697130+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a symmetric kernel with $C_1(j^2k^\\alpha+j^\\alpha k^2)\\le K_{j,k}\\le Cj^2k^2$, $1<\\alpha\\le 2$, and initial data in $Y^+_{2+\\alpha}$ with $M_\\alpha(f(0))>0$, for which a global mass-conserving solution with all moments finite exists; this would refute Theorem 2.7 and Corollary 2.8. A more direct check is to test the imported lemma on the EDG system by looking for a solution with $\\sum_j j^m f_j(t_0)=\\infty$ while $\\sum_j j f_j(t_0)=\\sum_j j f_j(0)$, which would show that moment blow-up need not imply gelation.","supporting_citations":[{"cited_title":"Esenturk","cited_arxiv_id":null,"evidence_quote":"Establishes the earlier existence theory and states the gelation conjecture that the paper extends; supplies the truncation scheme and kernel classes."},{"cited_title":"Banasiak, W","cited_arxiv_id":null,"evidence_quote":"Provides the lemma that moment blow-up implies an upper bound on the gelation time, the link between the two halves of the paper."},{"cited_title":"Ben-Naim and P","cited_arxiv_id":null,"evidence_quote":"The physics study whose critical-exponent predictions about gelation and instantaneous gelation are made rigorous."},{"cited_title":"Schlichting","cited_arxiv_id":null,"evidence_quote":"Earlier global-existence result for linearly growing kernels whose restrictive initial-data assumption this paper relaxes."},{"cited_title":"Eichenberg and A","cited_arxiv_id":null,"evidence_quote":"Earlier local and global existence for product-type kernels, generalised here to non-homogeneous symmetric kernels."},{"cited_title":"Lauren¸ cot","cited_arxiv_id":null,"evidence_quote":"Source of the convex-function inequalities and the de la Vallée-Poussin variant used to control higher moments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classical treatment of gelation in cluster equations that motivates the instantaneous-gelation argument."}],"review_version":1}