{"id":"50c14407-0445-4602-9ed7-6a1912362d75","arxiv_id":"1908.03397","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For reversible mean-field dynamics on finite state spaces, a positive entropic Ricci curvature bound implies modified logarithmic Sobolev, Talagrand transport-entropy and exponential entropy decay inequalities.","lead":"Many-particle systems on discrete state spaces can be studied through a single nonlinear equation for the particle distribution. This paper defines a geometric 'curvature' for such equations and shows that positive curvature guarantees exponentially fast convergence to equilibrium, with explicit constants for the Curie-Weiss magnet and zero-range processes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.7 is the bridge from geodesic-convexity curvature to the Bochner inequality verified in all examples; its proof is delegated to [21] and the replacement Lemma 3.8 contains an unexplained time-rescaling factor, so the central implication is not yet established.","rationale":"The paper has real supporting evidence: the Curie-Weiss computation in Proposition 5.1 is internally consistent (I checked the substitution and the formula for kappa(x)), and the functional-inequality deductions in Section 4 are clean once Theorem 3.7 is granted. However, the proof of Theorem 3.7 is the keystone that connects the abstract definition of curvature to the Bochner-type inequalities actually computed in the examples. Since that proof is delegated to prior work and the one displayed lemma needed for the nonlinear adaptation exhibits an apparent inconsistency in the time parametrization, the central implication Ric >= lambda => MLSI/decay/ET is not fully supported by the text as written. The reader's designated weakest assumption (Assumption 2.1) is a scope restriction on reversible mean-field dynamics; it is explicit and therefore not a correctness flaw. My concern is different: it is about the unproven/possibly mis-stated bridge. Because the issue is potentially fixable (a typo or omitted rescaling), the appropriate verdict remains CONDITIONAL, which is what the reader already issued; hence no verdict change. If the re-derivation shows the lemma is genuinely false, the example bounds would not yield the claimed consequences and the verdict would move toward REJECT.","tokens_in":23043,"tokens_out":13124,"duration_ms":129843,"concrete_test":"Re-derive Lemma 3.8 from scratch. Fix a smooth curve mu_s in P*(X), let mu_s_t be the solution of partial_tau mu = L-hat_mu mu starting from mu_s at tau=0, and compute 1/2 partial_t A(mu_s_t, psi_s_t) + partial_s F(mu_s_t) explicitly for this standard semigroup; then repeat the computation for the rescaled definition mu_s_t = Phi_{s t}(mu_s). Check which definition, if either, makes both equation (3.9) and the lemma's conclusion 1/2 partial_t A + partial_s F = -s B true, and verify that the resulting identity is sufficient for the Daneri-Savare argument in Theorem 3.7. Compare with [21, Lemma 4.6] to determine whether the factor s is a genuine nonlinear effect or a typographical error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.2 / 4.3) follows from Ric >= lambda, but Section 5 verifies only the pointwise Bochner inequality B(mu,psi) >= kappa A(mu,psi) (Theorem 3.7(2)) for the examples. The paper's bridge from that inequality to the geodesic-convexity definition of Ric (Definition 3.1) is Theorem 3.7, whose proof is not supplied: the text says to 'follow verbatim' [21, Thm. 4.5] with Lemma 3.8 in place of [21, Lem. 4.6]. Lemma 3.8 is not secure as printed. It defines mu_s_t as the solution 'at time s+t' of the nonlinear Fokker-Planck equation starting from mu_s, but the proof begins with the identity partial_t mu_s_t = s * L-hat_{mu_s_t} mu_s_t. For the standard semigroup the time derivative is L-hat_{mu_s_t} mu_s_t, with no factor s; the factor s is only correct if the time variable is rescaled, e.g. mu_s_t = Phi_{s t}(mu_s). The text does not state such a rescaling, and the derivation of equation (3.9) explicitly uses the factor s. Therefore the adaptation to the nonlinear setting cannot be verified from the manuscript, and consequently the example bounds in Section 5 are not yet shown to imply Ric >= kappa or the functional inequalities of Theorem 4.3. This is a load-bearing gap in the central implication, not merely a scope restriction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a notion of entropic Ricci curvature lower bounds for nonlinear mean-field Markov dynamics on finite discrete state spaces, based on geodesic convexity of a free energy F with respect to a discrete transportation distance W coming from the authors' earlier gradient-flow work. Under a positive curvature bound, it derives uniqueness of the stationary state and three quantitative consequences: a modified logarithmic Sobolev inequality, exponential decay of the free energy along solutions, and a transport-entropy inequality. The technical core is a Hessian formula B(mu,psi) for F along W-geodesics, with curvature characterized by a Bochner-type inequality B >= kappa A. Explicit curvature bounds are given for the Curie-Weiss model with Glauber rates (kappa = 2(1-beta) for beta in [0,1]) and, perturbatively, for separable zero-range and misanthrope rates.","tokens_in":23175,"tokens_out":14640,"duration_ms":138373,"significance":"If the bridge theorem (Theorem 3.7) is fully established, this is a valuable extension of the Erbar-Maas curvature theory to nonlinear mean-field systems on discrete spaces. The proofs of the functional-inequality consequences (Theorems 4.2 and 4.3) are clean and self-contained conditional on that bridge, and the Curie-Weiss computation is internally consistent: the regime beta in [0,1] is exactly where the estimate 1/(1-x^2) >= beta validates the lower bound. The perturbative zero-range/misanthrope result provides explicit, parameter-free curvature constants and goes beyond the linear Markov-chain setting. These strengths make the paper potentially publishable, but the central implication currently rests on an unproven and internally inconsistent lemma.","major_comments":[{"comment":"The statement of Lemma 3.8 is internally inconsistent in a way that affects the central implication. The lemma says that mu^s_t is the solution 'at time s+t' starting from mu^s, but the proof begins by asserting that partial_t mu^s_t = s * hat{L}_{mu^s_t} mu^s_t and uses this factor s in equation (3.9). For the standard (unscaled) nonlinear semigroup the time derivative is hat{L}_{mu^s_t} mu^s_t with no factor s; the factor s is correct for the rescaled curve mu^s_t = Phi_{s t}(mu^s). Since Theorem 3.7 is the bridge between the geodesic-convexity definition of curvature (Definition 3.1) and the Bochner inequality B(mu,psi) >= kappa A(mu,psi) that is verified in all examples, and since its proof is delegated to [21, Thm. 4.5] with Lemma 3.8 as the replacement for [21, Lem. 4.6], the main results of Section 4 and the example bounds of Section 5 are not currently supported as written. Please correct the time rescaling in the statement and supply the full argument, or an explicit dictionary that verifies the nonlinear adaptation.","section":"Section 3, Lemma 3.8"},{"comment":"The proof of Theorem 3.7 is not supplied; the text says to follow verbatim the proof of [21, Thm. 4.5] with Lemma 3.8 replacing [21, Lem. 4.6]. The nonlinear setting is not a notational variant of the linear one: the geodesic equation (3.1) contains the extra derivative term partial_{mu(z)} of Lambda, the Hessian B in (3.7) contains the new terms R_Lambda and M, and the text itself notes that the equivalence of (1) and (2) is nontrivial because the metric degenerates at the boundary. Without a proof that the Daneri-Savare argument goes through with these modifications, the example computations in Section 5 only verify the Bochner inequality, not the geodesic convexity that Definition 3.1 requires. Please include a complete proof or a detailed step-by-step reduction to the linear case.","section":"Section 3, Theorem 3.7"}],"minor_comments":[{"comment":"The set P*(X) is used in (2.9), Lemma 3.6, and Theorem 3.7 but is never defined; please define it explicitly as the interior of the simplex, i.e. the set of strictly positive probability measures.","section":"Section 2.1"},{"comment":"The text calls d = |X| the constant degree of the complete graph; with p(x,y) = 1 for x != y the degree is |X|-1. Please clarify the normalization, since the numerical constants in Theorem 5.3 depend on this convention.","section":"Section 5.2"},{"comment":"The symbol lambda is used both for the bound defined in (5.5) and for the splitting parameter in the proof's conclusion; please use different notation for the splitting parameter to avoid the appearance of a circular definition.","section":"Section 5.3 / Theorem 5.3"},{"comment":"There are several typographical errors, including 'analize' in Section 1.1, 'orrepsond' in Section 1.1, 'correponds' in Remark 3.3, and 'F WI' in Theorem 4.2; a careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a promising extension of Erbar-Maas curvature to nonlinear mean-field systems. The conditional acceptance should hinge on the authors supplying a complete and consistent proof of Theorem 3.7, in particular fixing Lemma 3.8. If that bridge is repaired, the paper is a strong contribution: the functional-inequality part is clean and the examples are convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. This paper extends Erbar–Maas entropic Ricci curvature from linear Markov chains to non-linear mean-field dynamics on finite spaces, and it delivers the discrete analogue of Carrillo–McCann–Villani: positive curvature yields MLSI, exponential free-energy decay, and a Talagrand-type transport-entropy inequality, plus explicit curvature bounds for Curie–Weiss and separable zero-range/misanthrope models. That is the right program, and the payoff is significant.\n\nWhat is genuinely new and good: the Hessian formula B(µ,ψ) in (3.7) with the two µ-dependent terms RΛ and M, the two-point curvature formula (3.10), the non-linear Bochner-type characterization (Theorem 3.7), and the worked example bounds. Section 4 is clean and self-contained: once you accept Ric ≥ λ, the FWI → MLSI → exponential decay → ET chain is airtight. The Curie–Weiss computation is internally consistent; the regime β ∈ [0,1] is exactly where 1/(1−x²) ≥ β lets the estimate close. The paper is honest about Assumption 2.1: the detailed-balance/local-Gibbs structure is restrictive and is imported from the authors' own earlier gradient-flow work, but it is explicit, not hidden.\n\nThe soft spot is the load-bearing bridge. Theorem 3.7's equivalence is the central characterization, and the examples verify only the pointwise Bochner inequality B ≥ κ A. The proof of Theorem 3.7 is delegated to [21], with Lemma 3.8 replacing [21, Lem. 4.6]. But Lemma 3.8 as printed is not right. It defines µs_t as the solution at time s+t starting from µs, then the proof uses ∂t µs_t = s µs_t. For the standard semigroup the derivative is µs_t, no factor s; the factor s only appears under a time rescaling µs_t = Φ_{s t}(µs), which the statement does not give and which would alter the accompanying s-continuity equation. So the proof of Theorem 3.7 cannot be verified from the manuscript, and the example bounds are not yet logically connected to Ric ≥ κ or to Theorem 4.3. This is not a cosmetic typo; it is the central implication. It looks fixable, but it needs to actually be fixed.\n\nSecondary issues: Theorem 5.3's assumption (5.5) is typeset ambiguously, so I cannot independently verify the constant (5.6); the citation pattern is sane, and the comparison to [22], [25], [26], [39] is fair. No fits, no hidden parameters.\n\nBottom line: this deserves serious peer review. A referee should insist on a corrected Lemma 3.8 and a complete proof of Theorem 3.7 (or a directly proved EVI for the examples). I'd bring the paper to reading group and would cite the framework once the bridge is standing.","headline":"A serious and mostly right paper, but the bridge from the Bochner check to geodesic convexity rests on a Lemma 3.8 that does not parse as written; fix that and this becomes the standard reference for nonlinear discrete entropic curvature.","tokens_in":23947,"tokens_out":8828,"would_cite":true,"duration_ms":86083,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J27","60K35","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Positive entropic Ricci curvature forces exponential convergence to a unique equilibrium for mean-field dynamics on finite discrete state spaces.","keywords":["entropic Ricci curvature","mean-field dynamics","discrete state spaces","gradient flow","modified logarithmic Sobolev inequality","transport-entropy inequality","Curie-Weiss model","zero-range process"],"falsifier":"Directly compute the infimum defining the optimal curvature, $\\kappa_{\\mathrm{opt}} = \\inf_{\\mu,\\psi} B(\\mu,\\psi)/A(\\mu,\\psi)$, for a candidate non-linear Markov triple; the theorem's key condition is exactly that this infimum is positive. The claim would be falsified by a positive-infimum example that nonetheless has two stationary points or a trajectory whose free energy decays slower than $e^{-2\\kappa_{\\mathrm{opt}}t}$; such an example could be sought among separable zero-range kernels by solving the two-point system, where the formula in Lemma 3.9 gives $\\kappa_{\\mathrm{opt}}$ explicitly.","tokens_in":105,"feed_emoji":"📉","tokens_out":18683,"duration_ms":220597,"temperature":0.7,"pith_summary":"This paper establishes a curvature-based route to quantitative long-time behavior for mean-field dynamics on finite discrete state spaces. The central claim is that a lower bound $\\mathrm{Ric}(X,Q,\\pi) \\ge \\lambda > 0$ on the entropic Ricci curvature—defined as geodesic convexity of the free energy in a specially adapted discrete transport distance—has strong consequences: the system has a unique stationary state, it satisfies a modified logarithmic Sobolev inequality, and every solution decays exponentially to equilibrium in free energy. The result matters because it transfers the successful gradient-flow strategy via quadratic transport distances for continuous mean-field equations to the discrete setting, where the usual Wasserstein distance degenerates. The paper backs the abstract theorem with explicit curvature computations for the Curie–Weiss model and for separable zero-range/misanthrope-type rates, showing the hypotheses are satisfied by classical statistical-mechanics examples.","feed_headline":"Positive curvature pins mean-field systems to one equilibrium","feed_subtitle":"A discrete transport-distance curvature bound upgrades qualitative convergence to explicit exponential rates.","key_machinery":"The load-bearing structure is the non-linear Markov triple of Assumption 2.1: for every probability measure $\\mu$ on the finite state space $X$, the rate matrix $Q(\\mu)$ is reversible with respect to a Gibbs measure $\\pi(\\mu) = Z(\\mu)^{-1}\\exp(-H(\\mu))$, where $H_x(\\mu) = \\partial_{\\mu_x} U(\\mu)$ and $U(\\mu)=\\sum_x \\mu_x K_x(\\mu)$. This detailed-balance condition produces the Onsager operator built from the logarithmic mean $\\Lambda(a,b)=(a-b)/(\\log a-\\log b)$, and hence the discrete transport distance $W$ defined through the continuity equation $\\partial_t\\mu + \\nabla\\cdot(\\Lambda(\\mu)\\nabla\\psi)=0$ with action $A(\\mu,\\psi)=\\langle\\nabla\\psi,\\Lambda(\\mu)\\nabla\\psi\\rangle$. The free energy $F(\\mu)=\\sum_x \\mu_x\\log\\mu_x+U(\\mu)$ is then the driving functional of the gradient flow, and the paper defines Ricci curvature $\\mathrm{Ric}(X,Q,\\pi)\\ge\\kappa$ by geodesic convexity of $F$: $F(\\mu_t) \\le (1-t)F(\\mu_0)+tF(\\mu_1) - (\\kappa/2)t(1-t)W(\\mu_0,\\mu_1)^2$. The central technical engine is Theorem 3.7, which equates this geodesic-convexity condition with the Bochner-type Hessian inequality $B(\\mu,\\psi) \\ge \\kappa A(\\mu,\\psi)$ and with the evolution-variational inequality $\\mathrm{EVI}_\\kappa$; all subsequent functional inequalities flow from that equivalence.","core_discovery":"The paper's own claim, stated as Theorem 4.3 (and announced in Theorem 1.2), is that every non-linear Markov triple $(X,Q,\\pi)$ satisfying Assumption 2.1 with $\\mathrm{Ric}(X,Q,\\pi) \\ge \\lambda > 0$ has exactly one stationary point $\\pi_*$, which is the unique minimizer of the free energy, and the following quantitative controls hold: the modified logarithmic Sobolev inequality $F_*(\\mu) \\le I(\\mu)/(2\\lambda)$ for all $\\mu$; exponential free-energy decay $F_*(\\mu_t) \\le e^{-2\\lambda t}F_*(\\mu_0)$ along every solution of the mean-field equation; and the transport-entropy inequality $W(\\mu,\\pi_*) \\le \\sqrt{2/\\lambda}\\,F_*(\\mu)$. These are the discrete, non-linear analogues of the functional-inequality route that works for continuous mean-field equations and for linear reversible Markov chains, and they are derived by a short chain: curvature gives an evolution-variational inequality, which gives a free-energy–Fisher-information–distance inequality, which gives the modified log-Sobolev bound and then the decay and transport bounds. The paper also computes curvature for representative models, obtaining $\\kappa = 2(1-\\beta)$ for the Curie–Weiss model with Glauber rates (positive curvature on the entire high-temperature phase $\\beta < 1$) and a perturbative positive bound for separable zero-range and misanthrope kernels on the complete graph.","pith_inferences":["An implication the paper leaves implicit: because the Bochner-type inequality $B(\\mu,\\psi)\\ge\\kappa A(\\mu,\\psi)$ is a finite-dimensional condition in $(\\mu,\\psi)$ for each fixed state space, the optimal curvature of any detailed-balance mean-field model can be computed or bounded by a finite-dimensional infimum problem, giving a direct recipe for other models.","The zero-range/misanthrope result is perturbative in the size of the rate perturbation; a natural extension is to ask whether positivity of the same curvature functional, rather than smallness of the perturbation, is the actual threshold for exponential decay in those models.","The framework is restricted to reversible, Gibbs-structured mean-field dynamics by Assumption 2.1; for non-reversible mean-field limits the Onsager operator and the distance $W$ are not available, so a different metric or curvature notion would be needed to obtain analogous quantitative rates.","Because the consequences are derived from convexity in $W$ rather than from spectral data, the method could yield transport-entropy inequalities for models whose equilibrium measure is not known explicitly, as long as the Hessian functional can be bounded."],"forward_implications":["A positive curvature bound $\\lambda > 0$ rules out multiple equilibria: $\\pi_*$ is the unique stationary state and the unique minimizer of the free energy.","The modified logarithmic Sobolev inequality $F_*(\\mu) \\le I(\\mu)/(2\\lambda)$ holds, so entropy dissipation controls free energy uniformly over the state space.","Every solution of the mean-field equation satisfies $F_*(\\mu_t) \\le e^{-2\\lambda t}F_*(\\mu_0)$, so convergence to equilibrium is exponential with explicit rate $2\\lambda$.","The transport-entropy inequality $W(\\mu,\\pi_*) \\le \\sqrt{2/\\lambda}\\,F_*(\\mu)$ holds, linking the discrete transport distance to free energy; in particular $W(\\mu_t,\\pi_*)$ decays at rate $\\lambda$.","Two solutions starting from different initial data contract exponentially in $W$: $W(\\mu^1_t,\\mu^2_t) \\le e^{-\\kappa t}W(\\mu^1_0,\\mu^2_0)$, and the Curie–Weiss example shows the whole high-temperature regime $\\beta<1$ is covered, with $\\kappa = 2(1-\\beta)$."],"supporting_citations":[{"why":"Supplies the gradient-flow formulation: the Onsager operator, the discrete transport distance W, and the free-energy dissipation relation on which the whole curvature theory rests.","marker":"[19]"},{"why":"Gives the linear-Markov-chain entropic curvature theory, the Bochner-type formula, and the EVI-equivalence proof that Theorem 3.7 extends to the non-linear case.","marker":"[21]"},{"why":"Provides the continuous mean-field convexity-to-equilibration strategy that Theorem 4.3 transfers to discrete mean-field dynamics.","marker":"[9]"},{"why":"Constructs the discrete transport distance and gradient flow for finite Markov chains, the linear ancestor of the distance W used here.","marker":"[29]"},{"why":"Establishes the free-energy-Fisher-information-transport-distance inequalities in the Wasserstein setting that Theorem 4.2 mimics.","marker":"[33]"},{"why":"Supplies the logarithmic-mean estimates and perturbative bounding techniques used for the zero-range and misanthrope examples.","marker":"[22]"},{"why":"Provides the weakly-interacting-Markov-chain curvature bounds and the template for the perturbative lower bounds in Section 5.","marker":"[20]"}],"fun_headline_variants":["Curvature forces unique equilibrium for mean-field dynamics","Exponential decay to unique state from positive curvature","Discrete curvature ensures unique equilibrium and exponential rates","Curvature bound pins down steady state in mean-field systems","Positive curvature yields unique steady state for mean-field systems"],"cache_read_input_tokens":25728,"weakest_assumption_plain":"Everything depends on Assumption 2.1: for every possible state $\\mu$ of the system, the jump rates $Q(\\mu)$ must be reversible with respect to the Gibbs measure $\\pi(\\mu)$ whose Hamiltonian is derived from the potential $U$, so the dynamics has a detailed-balance structure; if a mean-field dynamics lacks this structure, the transport distance, the free energy, and the curvature criterion are not defined.","fun_headline_variants_meta":{"raw":{"variants":["Curvature forces unique equilibrium for mean-field dynamics","Exponential decay to unique state from positive curvature","Discrete curvature ensures unique equilibrium and exponential rates","Curvature bound pins down steady state in mean-field systems","Positive curvature yields unique steady state for mean-field systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000945,"raw_usage":{"total_tokens":4035,"prompt_tokens":942,"completion_tokens":3093,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":3017}},"tokens_in":558,"tokens_out":3093,"duration_ms":19644,"temperature":1.0,"reasoning_tokens":3017,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:16:56.702881+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly compute the infimum defining the optimal curvature, $\\kappa_{\\mathrm{opt}} = \\inf_{\\mu,\\psi} B(\\mu,\\psi)/A(\\mu,\\psi)$, for a candidate non-linear Markov triple; the theorem's key condition is exactly that this infimum is positive. The claim would be falsified by a positive-infimum example that nonetheless has two stationary points or a trajectory whose free energy decays slower than $e^{-2\\kappa_{\\mathrm{opt}}t}$; such an example could be sought among separable zero-range kernels by solving the two-point system, where the formula in Lemma 3.9 gives $\\kappa_{\\mathrm{opt}}$ explicitly.","supporting_citations":[{"cited_title":"Gradient flow structure for McKean-Vlasov equations on discrete spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the gradient-flow formulation: the Onsager operator, the discrete transport distance W, and the free-energy dissipation relation on which the whole curvature theory rests."},{"cited_title":"Ricci Curvature of Finite M arkov Chains via Convexity of the Entropy","cited_arxiv_id":null,"evidence_quote":"Gives the linear-Markov-chain entropic curvature theory, the Bochner-type formula, and the EVI-equivalence proof that Theorem 3.7 extends to the non-linear case."},{"cited_title":"Carrillo, Robert J","cited_arxiv_id":null,"evidence_quote":"Provides the continuous mean-field convexity-to-equilibration strategy that Theorem 4.3 transfers to discrete mean-field dynamics."},{"cited_title":"Gradient flows of the entropy for finite Markov c hains","cited_arxiv_id":null,"evidence_quote":"Constructs the discrete transport distance and gradient flow for finite Markov chains, the linear ancestor of the distance W used here."},{"cited_title":"Generalization of an In equality by Talagrand and Links with the Logarithmic Sobolev Inequality","cited_arxiv_id":null,"evidence_quote":"Establishes the free-energy-Fisher-information-transport-distance inequalities in the Wasserstein setting that Theorem 4.2 mimics."},{"cited_title":"Entropic Ricci curvature bounds for discrete interacting systems","cited_arxiv_id":null,"evidence_quote":"Supplies the logarithmic-mean estimates and perturbative bounding techniques used for the zero-range and misanthrope examples."},{"cited_title":"Ricci curvature bounds for weakly interacting Markov chains","cited_arxiv_id":null,"evidence_quote":"Provides the weakly-interacting-Markov-chain curvature bounds and the template for the perturbative lower bounds in Section 5."}],"review_version":1}