{"id":"63667963-f8f8-4c25-b5f3-2a5e03a54550","arxiv_id":"2505.10582","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For scale-free percolation in the finite-variance regime γ>2, the contact process extinction time on an n-vertex box is at least exp(c n (log n)^{-A}) with high probability.","lead":"A simple model of infection spreading on a network survives for exponentially long times on scale-free percolation graphs, including the finite-variance small-world regime where prior results did not apply. The proof builds a multiscale constellation of high-degree hubs and paths that keep the infection alive for exp(c n (log n)^{-A}) time with high probability.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The growing-parameter constellation estimate in Prop. 2.1 is the load-bearing step for Theorem 1.1(ii), but its proof is only asserted via 'minor changes' to Mountford et al. and is not carried out.","rationale":"The reader's weakest-assumption pick (Proposition 2.1) matches my own reading. I checked the other flagged issues in the manuscript: the FKG step in Lemma 3.4 conditions on the non-monotone event E1; however the same estimates can likely be obtained by bounding P(A cap E1) and using P(E1 cap E2) -> 1, so this is a technical gap. The wrong-direction inequality in Lemma 3.3 appears to be a notational slip: the proof actually bounds the complement event (bxi not in the largest component), and the union bound becomes correct once E2_2(i) is read as the bad event. Neither of these threatens the architecture of the proof as much as Proposition 2.1. The constellation construction depends on the new polylogarithmic scaling S_n,D_n; no detailed proof is supplied for the contact-process estimate at that scaling, and the cited lemmas are not stated in that regime. This is a load-bearing concern, but there is no indication of a fundamental obstruction: the condition (10) is the natural analogue of the fixed-parameter condition and the proof strategy of Mountford et al. is likely to adapt. The paper should therefore remain CONDITIONAL, pending a full derivation of Lemma 2.2 with explicit constants.","tokens_in":28938,"tokens_out":25093,"duration_ms":241467,"concrete_test":"Perform an independent, self-contained proof of Lemma 2.2: for a path of D_n edges joining two stars of size S_n, starting with N(x) infested, bound the probability that N(y) is infested within time kappa = exp(c1 lambda^2 S_n). Track the dependence on D_n explicitly and verify that, under S_n >= C lambda^{-2} log(1/lambda) D_n, the failure probability is at most exp(-c lambda^2 S_n) with constants uniform in n. If the exponent instead contains S_n/D_n, or if the implicit constant in Lemma 3.2 depends on D_n, then condition (10) is insufficient and Theorem 1.1(ii) is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1(ii) rests on two pillars: Proposition 3.1, which produces a polylogarithmic constellation in SFP, and Proposition 2.1, which converts any such constellation into an exponentially long survival time. The second pillar is the less secure. Its proof (Section 2.2) delegates the key estimate, Lemma 2.2, to 'a direct application' of [Mountford et al., 2013, Lemmas 3.1(ii) and 3.2] and to [Mountford et al., 2016, Lemma 6.2], adding only that 'careful analysis of the proof shows' the required infested-star estimate and that 'the proof of this generalisation only requires minor changes'. The original statements are for fixed S and D, whereas the theorem needs S_n and D_n growing polylogarithmically with n. In particular, condition (10), S_n >= C lambda^{-2} log(1/lambda) D_n, is asserted to be sufficient for the infection to cross any inter-star path of length D_n with failure probability at most exp(-c lambda^2 S_n). The text does not show how D_n enters the failure exponent. If the correct bound is exp(-c lambda^2 S_n / D_n), or if the constant in the exponent depends on D_n in any way not absorbed by (10), then the union bound in Lemma 2.3 fails and the lower bound on tau_{G_n} collapses. Internal gaps elsewhere (FKG after conditioning on non-monotone E1 in Lemma 3.4, and a wrong-direction inequality in Lemma 3.3) are likely repairable; the missing generalization of Proposition 2.1 is the deepest unproved step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the extinction time of the contact process on scale-free percolation (SFP) restricted to a box of volume n in R^d. In the ultra-small-world regime γ=α(τ−1)/d∈(1,2), it claims an exponential lower bound exp(cn) on the extinction time, adapting the hyperbolic-random-graph method of Linker et al. In the small-world finite-variance regime γ>2 with α∈(d,2d), the main contribution, it claims a lower bound of order exp(c n (log n)^{−A}) for every A>2γ/(2−α/d), valid for every λ>0. The proof combines a generalized constellation theorem (Proposition 2.1), a multi-scale construction of a polylogarithmic constellation in SFP (Proposition 3.1), and percolation-type estimates for component sizes and connecting paths. The paper also states, as Theorem 1.2, asymptotic results for the non-extinction probability Γ(λ) in the regime γ∈(1,2), with only a proof sketch in Section 5.","tokens_in":29191,"tokens_out":11002,"duration_ms":103350,"significance":"If the main theorem is correct, it is a substantial contribution to the metastability theory of the contact process on spatial random graphs. The finite-variance, small-world regime has not been treated before for SFP, and the polylogarithmic correction to the exponential rate is a genuinely new feature relative to the ultra-small-world case. The multi-scale construction in Section 3 is technically ambitious and goes well beyond a direct extension of earlier work. The paper is also transparent about the limits of its method, notably in Remark 1.3. However, the central mechanism Proposition 2.1 is not proved in full; it is delegated to a 'minor changes' generalization of cited results, and this is the main bottleneck for verifying the paper's headline claim.","major_comments":[{"comment":"Proposition 2.1 is the mechanism that converts the polylogarithmic constellation of Proposition 3.1 into the survival lower bound of Theorem 1.1(ii), but its proof is not self-contained at the critical point. Lemma 2.2 asserts (17) and (18) as 'a direct application' of [Mountford et al., 2013, Lemmas 3.1(ii) and 3.2] and [Mountford et al., 2016, Lemma 6.2], adding only that 'careful analysis of the proof shows' the infested-star estimate and that 'the proof of this generalisation only requires minor changes'. The cited statements are for fixed star size and fixed inter-star distance, whereas Theorem 1.1(ii) requires S_n and D_n to grow polylogarithmically with n. In particular, the text does not show how D_n enters the failure exponent in (18) or why the union bound in Lemma 2.3 continues to hold with a constant independent of n once D_n grows. Since condition (10) is only S_n ≥ C λ^{-2} log(1/λ) D_n, a hidden dependence of the failure probability on D_n would destroy the argument. This missing generalization is load-bearing and must be supplied in full.","section":"Section 2.2, Lemma 2.2"},{"comment":"The inequality P(R_i and R_{i+1} are not connected | E1∩E2) ≤ P(R_i and R_{i+1} are not connected | E1) is justified by FKG, with the argument that the first event is decreasing and E2 is increasing. However, the conditioning is on E1∩E2, not on E2, and E1 is the intersection of a lower-tail event (increasing) and an upper-tail event (decreasing) for component sizes, so E1 is not monotone. The FKG inequality for increasing events therefore does not directly give the displayed bound. The same issue appears in the derivation of (43). Since (45) and the subsequent union bound control E0, this is a substantive gap in Lemma 3.4 and needs an additional argument, for example a conditional FKG statement or replacement of E1 by a monotone event.","section":"Section 3.6, Eq. (44)"},{"comment":"The proof of Lemma 3.3 bounds the probability of the success event E_2^2(i) = {bx_i ∈ C^f_{f(i)}} rather than its complement. Equation (36) gives an upper bound for P(E_2^2(i) | E_1^2∩E_1), and the estimate (38) makes this upper bound small; this would show that bx_i belongs to the largest component with small probability, which is the opposite of what Lemma 3.3 needs. The final union-bound display in the lemma then uses (36) as if it controlled P(E_2^2(i)^c | ...). One can repair the argument by applying the same union-bound reasoning to the complement event, but as written the lemma does not establish P(E_2^2|E1∩E1_2)→1.","section":"Section 3.5, Eqs. (36)-(38)"},{"comment":"Theorem 1.2 is stated as a theorem, but the section is explicitly a sketch. The adaptation of [Linker et al., 2021, Theorem 1.1] to SFP is described qualitatively, and the four lemmas behind the upper and lower bounds in the two regimes are not proved; in particular, the treatment of soft edges via (54) and (56) is not accompanied by the path-counting estimates or the star-construction lemma needed for the upper bounds. If Theorem 1.2 is to remain part of the paper, its proof must be completed; alternatively, the statement should be reclassified as a conjecture or clearly separated from the main proved theorem.","section":"Section 5"}],"minor_comments":[{"comment":"The display 'P(E0|E1∩E2) = 1 − P(E0|E1∩E2)' should read 'P(E0^c|E1∩E2)', since the printed formula is a tautology and does not express the intended complement probability.","section":"Section 3.6, before Eq. (44)"},{"comment":"The phrase 'For,r >1' is a typo and should read 'For r>1'.","section":"Section 5"},{"comment":"The abstract says β≥3, while Theorem 1.1(ii) is stated for γ>2, i.e. β>3; the boundary case β=3 is not covered and the wording should be aligned.","section":"Abstract and Theorem 1.1"},{"comment":"The Schapira–Valesin bound is missing a closing parenthesis in the displayed inequality; as printed it is not a well-formed formula.","section":"Remark 1.3"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to acceptance is the unproved generalization in Proposition 2.1; I would ask the authors to write out the proof of Lemma 2.2 in full, including how D_n affects all constants and failure probabilities. The FKG and complement-event issues in Lemmas 3.3-3.4 are fixable but need correction. The sketch-only status of Theorem 1.2 is also below the standard for a stated theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First: the genuinely new result is Theorem 1.1(ii): the contact process on scale-free percolation survives exponentially long, up to a polylogarithmic correction in the exponent, when gamma>2 (finite variance, small-world regime). Previous work on spatial scale-free graphs stopped at beta in (2,3), so this is a real step. The multiscale construction of constellations in Section 3 is nontrivial, and the paper is honest that the log correction may be an artifact of the approach.\n\nWhat the paper does well: the overall architecture is sound. The coarse/fine partition, the choice of scales, and the percolation estimates are laid out in enough detail that a patient reader can follow the main argument. Part (i), the gamma in (1,2) case, is a clean adaptation of Linker et al. and Gracar-Grauer. The paper cites prior work appropriately and does not oversell the novelty.\n\nSoft spots, in order of importance. (1) Proposition 2.1 is load-bearing for part (ii), but its proof is only asserted. Lemma 2.2 says the key estimate follows by 'a direct application' of Mountford et al. and that 'the proof of this generalisation only requires minor changes'. The text never shows how the growing inter-star distance D_n enters the failure exponent. If the correct bound from star to neighbouring star is exp(-c lambda^2 S_n / D_n) rather than exp(-c lambda^2 S_n), then condition (10) only gives a constant exponent and the stochastic domination in Lemma 2.3 collapses. This is not a cosmetic gap; it needs to be written out fully before Theorem 1.1(ii) is fully supported. (2) Theorem 1.2 is stated as a theorem but Section 5 is only a sketch. That is acceptable for a secondary result only if it is explicitly labelled as such or moved to a clearly marked outline. (3) Smaller issues: FKG is applied after conditioning on the non-monotone event E1 in Lemma 3.4 without justification, and at least one inequality in Lemma 3.3 appears to have the wrong direction. Both look repairable. The tree property of the constructed constellation is asserted reasonably but deserves a line of verification.\n\nWho this is for: researchers working on contact processes on spatial scale-free random graphs, and anyone interested in metastability on small-world graphs. The paper deserves a serious referee. The gaps are identifiable and local, not hidden or load-bearing beyond Proposition 2.1. I would send it out, with the expectation that the referee asks for a complete proof of Proposition 2.1 and a cleanup of the FKG step.","headline":"First exponential-extinction result for contact process on scale-free percolation in the finite-variance small-world regime; the main theorem is plausible but rests on an unproved growing-parameter generalization of the constellation lemma.","tokens_in":29845,"tokens_out":2663,"would_cite":true,"duration_ms":27513,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","05C80","60D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Even sparse power-law networks with finite-variance degrees keep the contact process alive for exponentially long times.","keywords":["contact process","scale-free percolation","extinction time","metastability","small-world graph","power-law degree distribution","constellation graph","interacting particle systems"],"falsifier":"For fixed $d=1$, $\\alpha=1.5$, $\\tau=4$ (so $\\gamma=4.5>2$, $\\alpha\\in(d,2d)$) and a small $\\lambda$, estimate the median extinction time from full occupancy on scale-free percolation boxes for increasing $n$. The theorem predicts $\\log\\tau_{G_n}\\ge c n(\\log n)^{-A}$ for every $A>2\\gamma/(2-\\alpha/d)$, so if $\\log\\tau_{G_n}/(n(\\log n)^{-A})$ is observed to go to $0$ for a range of $A$, the claimed rate is false.","tokens_in":28663,"feed_emoji":"🦠","tokens_out":10100,"duration_ms":93198,"temperature":0.7,"pith_summary":"This paper studies how long an infection survives on scale-free percolation, a spatial random graph with power-law degrees, when the graph is confined to a box of volume $n$ and initially every vertex is infected. It claims that in the ultra-small-world regime (degree tail exponent $\\beta\\in(2,3)$), the extinction time grows like $e^{c n}$, and that in the finite-variance small-world regime ($\\beta>3$), it still grows like $\\exp(c n (\\log n)^{-A})$ for every infection rate $\\lambda>0$. This matters because the finite-variance regime is much sparser, and the star-construction arguments that worked for heavier tails fail there; the paper supplies a multiscale construction of many high-degree stars connected by short paths instead. If the theorem is right, spatially embedded power-law networks with only mild degree heterogeneity still keep the contact process alive for times exponential in network size, no matter how weak the infection rate.","feed_headline":"Even weak infections survive exponentially long on small-world graphs","feed_subtitle":"On finite-variance scale-free percolation, extinction time grows like exp(c n/(log n)^A) for every λ>0.","key_machinery":"The load-bearing object is the $(S,D,\\Delta)$-constellation: a tree subgraph with distinguished vertices (stars) of degree at least $S/2$, consecutive stars at graph distance at most $D$, and the star-to-star skeleton itself a tree of degree at most $\\Delta$. Proposition 2.1 extends a theorem of Mountford, Mourrat, Valesin and Yao by allowing $S$ and $D$ to grow with $n$; it asserts that if $S_n\\ge C\\lambda^{-2}\\log(1/\\lambda)D_n$, then the contact process on the constellation survives for time at least $e^{c(\\lambda^2 S_n+|J_n|)}$ with high probability. The proof of part (ii) uses a multiscale partition of the box into nested cubes of side $(\\log n)^{\\theta^k A}$, choosing the highest-weight vertex in each coarse box as a potential star, controlling the largest connected components of fine subcubes via supercritical percolation estimates, and connecting the stars by paths of length $O((\\log n)^{\\nu_p})$ using a red/blue chessboard colouring to keep the paths disjoint. The constellation has $|J_n|\\ge c n (\\log n)^{-A}$ stars, which yields the exponential-with-log-correction survival time; in the ultra-small-world regime, heavier tails allow a simpler construction with $\\Theta(n)$ stars at distance $1$, giving survival time $e^{c n}$.","core_discovery":"In the model, vertices come from a unit-intensity Poisson process on $[0,n^{1/d})^d$, each vertex gets a Pareto weight with tail index $\\tau-1$, and two vertices $x,y$ are joined independently with probability $1-\\exp(-\\rho W_x W_y/\\|x-y\\|^\\alpha)$. Writing $\\gamma=\\alpha(\\tau-1)/d$, the degree distribution has power-law exponent $\\beta=\\gamma+1$. The paper's main theorem states that for $\\alpha>d$ and $\\rho$ above the percolation threshold: (i) if $\\gamma\\in(1,2)$, then for every $\\lambda>0$ there is $c>0$ with $\\mathbb{P}(\\tau_{G_n}\\ge e^{c n})\\to 1$; (ii) if $\\gamma>2$ and $\\alpha\\in(d,2d)$, then for every $\\lambda>0$ and every $A>2\\gamma/(2-\\alpha/d)$ there is $c>0$ with $\\mathbb{P}(\\tau_{G_n}\\ge \\exp(c n (\\log n)^{-A}))\\to 1$. Part (ii) is the genuinely new result: the graph is supercritical but sparse, with finite-variance degrees and only logarithmic graph distances, and the proof finds a constellation subgraph containing polylogarithmically many stars that lets the infection persist. The paper also states a matching-order result for the non-extinction probability $\\Gamma(\\lambda)$ as $\\lambda\\to 0$ in the regime $\\gamma\\in(1,2)$, recorded as Theorem 1.2.","pith_inferences":["If the logarithmic correction is genuinely necessary rather than a proof artefact, the phase transition at $\\beta=3$ (the finite-variance threshold) would mark a qualitative change in metastability rates: survival time would drop from $e^{c n}$ to $e^{c n/(\\log n)^A}$, a slower but still exponential scale. This prediction is not tested in the paper.","A natural testable extension is to simulate the contact process on scale-free percolation for parameters $d=1$, $\\alpha=1.5$, $\\tau=4$ (so $\\gamma=4.5>2$) and check whether $\\log\\tau_{G_n}\\cdot(\\log n)^A/n$ settles at a positive constant; if it tends to zero, the lower bound is not tight.","The same constellation construction might carry over to geometric inhomogeneous random graphs and other kernel-based spatial graphs in their finite-variance regime, because the proof uses only the Pareto weight tail and the polynomial connection kernel, though the paper does not state this.","The Theorem 1.2 asymptotics imply that in the ultra-small-world regime, the survival probability from a single vertex vanishes polynomially in $\\lambda$, with the same exponents as non-spatial power-law configuration models; this suggests the spatial embedding does not change the critical exponent there, a comparison the paper sketches but does not develop."],"forward_implications":["For every $\\lambda>0$, the contact process on finite scale-free percolation boxes is metastable: starting from full occupancy, extinction is exponentially unlikely in both the infinite-variance and finite-variance degree regimes.","In the finite-variance small-world regime, the lower bound is $\\exp(c n (\\log n)^{-A})$ with $A>2\\gamma/(2-\\alpha/d)$; the logarithmic factor is a residue of the star-counting construction, and the authors leave open whether the true rate is fully exponential.","In the ultra-small-world regime ($\\gamma\\in(1,2)$), the theorem holds for all $\\rho>0$, since $\\rho_c=0$, so no supercritical percolation assumption is needed there.","The proof identifies an explicit random substructure — many high-degree stars joined by disjoint short paths — that supports the infection; any graph containing such a constellation inherits the exponential survival lower bound.","The non-extinction probability $\\Gamma(\\lambda)$ from a single infected vertex obeys the same power-law order as configuration models and hyperbolic graphs in the $\\gamma\\in(1,2)$ range, with a logarithmic correction when $\\gamma\\in(3/2,2)$."],"supporting_citations":[{"why":"Supplies the constellation contact-process machinery (Lemmas 6.2 and 6.3, Proposition 5.2) that Proposition 2.1 generalises.","marker":"[Mountford et al., 2016]"},{"why":"Provides Lemmas 3.1 and 3.2 on persistence and spread of infection on large-degree stars, the engine of the star-to-star estimate.","marker":"[Mountford et al., 2013]"},{"why":"Gives the hyperbolic-random-graph metastability proof that Section 4 adapts to ultra-small-world SFP and that Theorem 1.2 follows.","marker":"[Linker et al., 2021]"},{"why":"Supplies the annulus-based construction used to handle the soft connection kernel when adapting the non-extinction probability proof to SFP.","marker":"[Gracar and Grauer, 2024]"},{"why":"Defines continuum SFP, determines $\\rho_c$, and supplies the supercritical component-size decay used to size the fine-subcube components.","marker":"[Deprez and Wüthrich, 2019]"},{"why":"Establishes the power-law degree tail exponent $\\gamma$ and the quenched degree estimates used to translate weight thresholds into degree thresholds.","marker":"[Dalmau and Salvi, 2021]"},{"why":"Establishes the basic principle that a high-degree node retains the infection for a long time, motivating the star construction.","marker":"[Berger et al., 2005]"},{"why":"Inspires the multiscale box partition and path-construction technique used to connect stars in the finite-variance regime.","marker":"[Coppersmith et al., 2002]"}],"fun_headline_variants":["Finite-variance scale-free graphs still trap infections exponentially long","Sparse small-world networks keep infections alive exponentially","Contact process persists on sparse scale-free percolation","Exponential survival despite sparse finite-variance degrees","Small-world graphs keep infections alive even with finite variance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The generalized star-to-star infection estimate of Proposition 2.1 is the load-bearing step: it assumes that a star of size $S_n$ can keep its neighbourhood sufficiently occupied and pass the infection to stars at distance $D_n$, with $S_n$ and $D_n$ growing polylogarithmically, even though the lemmas it is built on were originally proved for bounded sizes and distances; the paper says only that the generalization requires minor changes and does not spell them out.","fun_headline_variants_meta":{"raw":{"variants":["Finite-variance scale-free graphs still trap infections exponentially long","Sparse small-world networks keep infections alive exponentially","Contact process persists on sparse scale-free percolation","Exponential survival despite sparse finite-variance degrees","Small-world graphs keep infections alive even with finite variance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000566,"raw_usage":{"total_tokens":2748,"prompt_tokens":1075,"completion_tokens":1673,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":1597}},"tokens_in":691,"tokens_out":1673,"duration_ms":11941,"temperature":1.0,"reasoning_tokens":1597,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:39:09.248948+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For fixed $d=1$, $\\alpha=1.5$, $\\tau=4$ (so $\\gamma=4.5>2$, $\\alpha\\in(d,2d)$) and a small $\\lambda$, estimate the median extinction time from full occupancy on scale-free percolation boxes for increasing $n$. The theorem predicts $\\log\\tau_{G_n}\\ge c n(\\log n)^{-A}$ for every $A>2\\gamma/(2-\\alpha/d)$, so if $\\log\\tau_{G_n}/(n(\\log n)^{-A})$ is observed to go to $0$ for a range of $A$, the claimed rate is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the constellation contact-process machinery (Lemmas 6.2 and 6.3, Proposition 5.2) that Proposition 2.1 generalises."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Lemmas 3.1 and 3.2 on persistence and spread of infection on large-degree stars, the engine of the star-to-star estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the hyperbolic-random-graph metastability proof that Section 4 adapts to ultra-small-world SFP and that Theorem 1.2 follows."},{"cited_title":"and Grauer, A","cited_arxiv_id":null,"evidence_quote":"Supplies the annulus-based construction used to handle the soft connection kernel when adapting the non-extinction probability proof to SFP."},{"cited_title":"and W \\\"u thrich, M","cited_arxiv_id":null,"evidence_quote":"Defines continuum SFP, determines $\\rho_c$, and supplies the supercritical component-size decay used to size the fine-subcube components."},{"cited_title":"and Salvi, M","cited_arxiv_id":null,"evidence_quote":"Establishes the power-law degree tail exponent $\\gamma$ and the quenched degree estimates used to translate weight thresholds into degree thresholds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the basic principle that a high-degree node retains the infection for a long time, motivating the star construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Inspires the multiscale box partition and path-construction technique used to connect stars in the finite-variance regime."}],"review_version":1}