{"id":"738edf1b-7b73-40e0-8cdd-5f9d7c1c5b25","arxiv_id":"2608.03405","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A weighted intermediate Ricci curvature lower bound is characterized by tensorial Bochner inequalities and entropy-tensor convexity, yielding intrinsic-dimensional Wasserstein contraction estimates for the heat flow.","lead":"This paper defines a weighted form of intermediate Ricci curvature and proves that bounding it from below is equivalent to tensorial Bochner inequalities and to convexity properties of a Boltzmann entropy tensor along optimal transport. It then derives new heat-flow contraction estimates whose rates depend on a dimension parameter, extending recent work from sectional to intermediate curvature.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's converse directions (4)⇒(1) and (5)⇒(1) depend on an unstated d²/2-concave jet-construction lemma; without it the main equivalence is unsupported.","rationale":"The reader's weakest_assumption identifies the same load-bearing gap: the converse directions of Theorem 3.1 require an unproved d²/2-concave jet construction. I agree that this is the main threat to the central claim, because the forward directions are routine and the equivalence (5)⇔(6) is a direct second-derivative computation. The issue is a missing lemma rather than a demonstrated internal inconsistency: the homogeneity of Ric^N_{k,f} suggests the jet construction can likely be repaired by scaling v, but the paper does not supply the argument or the precise statement from Villani that would justify it. The other concerns noted by the reader, such as the missing π²>K|∇θ|²/N hypothesis in Corollary 3.2 for K>0 and the dropped constant term in Corollary 4.1's proof, are real but secondary: they affect applications and sharpness, not the main equivalence. Since the concern is a gap in supporting proof machinery and not a demonstrated counterexample, the appropriate outcome is to keep the reader's CONDITIONAL verdict, so no change is needed.","tokens_in":39733,"tokens_out":32968,"duration_ms":289640,"concrete_test":"Verify the jet lemma used in §3.1: prove or disprove that for every complete Riemannian manifold (M,g), x∈M, v∈T_xM, and symmetric S with ‖S‖_{op}<1, there exists θ∈C_c^∞(M) such that ∇θ(x)=v, ∇²θ(x)=S, and −θ is d²/2-concave; if the lemma is true, insert its proof into the paper, and if false, exhibit a counterexample on a concrete manifold such as S² and rework the converse directions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equivalence Theorem 3.1 is well structured in its forward directions, but the converse directions (4)⇒(1) and (5)⇒(1) both invoke a 'standard jet construction' that is never stated as a lemma: given x, v, and a symmetric operator S, one must produce θ∈C_c^∞(M) with ∇θ(x)=v, ∇²θ(x)=S, and −θ d²/2-concave, after scaling. The only support is the citation to [Vil09, Theorem 13.5] in §3.1. This is genuinely load-bearing: the contradiction argument evaluates condition (4) or (5) at t=0 on the Wasserstein geodesic F_t=exp(t∇θ), and if such a θ does not exist for the prescribed jets, the converse implications are not established. The issue is not merely cosmetic: d²/2-concavity imposes a one-sided bound on ∇²θ (in the Euclidean model, ∇²θ≥−Id), so the prescribed Hessian S must be compatible; for direction (5), S=−df(v)/(N−k)Id is admissible only after scaling v small, which homogeneity permits but the proof does not state. A self-contained jet lemma, or a precise extraction of the assertion from Villani's theorem, is required for the central claim to be fully supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a weighted version of intermediate Ricci curvature, Ric^N_{k,f} = Ric_k + Hess f - (N-k)^{-1} df^2, and proves Theorem 3.1, an equivalence between the lower bound Ric^N_{k,f} >= K and several conditions: a tensorial Bochner inequality for the operator \\tilde{\\Gamma}^N_{2,k}; an inequality for the second derivative of the weighted k-Boltzmann entropy tensor along Wasserstein geodesics; and a concavity condition for a modified exponential of that tensor. It then derives displacement convexity of the weighted k-Boltzmann entropy, an evolution variational inequality, and Wasserstein contraction estimates for the heat flow under Ric^N_{k,f} >= 0, and compares the framework with the Ketterer-Mondino approach via lower-dimensional optimal transport.","tokens_in":40021,"tokens_out":37149,"duration_ms":319615,"significance":"If the main theorem is correct, the paper gives a genuine multi-way characterization of weighted intermediate Ricci curvature lower bounds by tensorial Bochner inequalities, tensorial entropy convexity, and concavity of an exponential entropy functional, extending the sectional-curvature results of Aishwarya, Rotem, and Shenfeld to all intermediate k and to arbitrary curvature bounds K in the weighted setting. The exterior-power reformulation is natural and well executed, and the applications produce intrinsic-dimensional EVI and Wasserstein contraction estimates. The paper also includes careful ODE comparison lemmas and an explicit comparison with Ketterer-Mondino. The forward directions of the main equivalence are clean and largely check out; the main reservation concerns the converse directions, which depend on an unstated jet-construction lemma.","major_comments":[{"comment":"The converse directions of the central equivalence both invoke a 'standard jet construction': for any x, v, and symmetric operator S, one must produce a smooth compactly supported function theta with gradient(theta)(x)=v, Hess(theta)(x)=S, and such that, after scaling, -theta is d^2/2-concave. This is asserted twice with citation to [Vil09, Theorem 13.5], but no statement or proof is given. The d^2/2-concavity condition is not a trivial consequence of prescribing a Hessian: in the Euclidean model it forces a one-sided bound on Hess(theta), and the prescribed Hessians S=0 and S= -df(v)/(N-k)Id need explicit compatibility checks. This is load-bearing because the contradiction argument evaluates condition (4) or (5) at t=0 on the Wasserstein geodesic generated by theta, and without such theta the converse implications are not established. Please state and prove a self-contained jet lemma, or give a precise extraction of the assertion from [Vil09, Theorem 13.5].","section":"Section 3.1, proof of Theorem 3.1, directions (4)=> (1) and (5)=> (1)"}],"minor_comments":[{"comment":"In the displayed chain after summing inequalities (4.18) and (4.19), the equality replacing 4nN/k - 2NC( sigma_k(e^{-A/N}) + sigma_k(e^{A/N}) ) by -8NC tr_{V^k} sinh^2(A^{[k]}/(2N)) is not exact: it drops the constant term 4N(n/k)(1 - binom(n-1,k-1)^2). For 1<k<n this term is negative, so the line should be an inequality rather than an equality. The final estimate is unaffected, since the omitted term makes the stated differential inequality weaker, but the proof should be corrected.","section":"Section 4.1, proof of Corollary 4.1"},{"comment":"The comparison lemma is stated under the hypothesis pi^2 > ab, but in Corollary 3.2 and Corollary 3.3 the quantity ab = (K/N)|grad theta|^2 is not checked. It is true that any finite C^2 solution of h'' >= a(h')^2 + b with a>0 forces ab < pi^2, but the paper should say this explicitly, since otherwise the reader must wonder whether the distortion formula is applicable when K>0 and W2 is large.","section":"Section 2.2, Lemma 2.3"},{"comment":"The text says that when f is constant the parameter N=k is allowed 'as usual', but Definition 1.1 requires N>k. Please clarify how the term 1/(N-k) in the definition of Ric^N_{k,f} and in Theorem 3.1 is interpreted in the constant-weight case, e.g. by taking the limit or by noting that the df^2 term vanishes.","section":"Section 3, conventions after the introduction of N=k"},{"comment":"There is a small typographical error in the displayed computation: 'tr((nabla^2 theta)^2(x)|_Sigma)(x)' has an extra '(x)' after the trace. This does not affect the argument but should be cleaned up.","section":"Section 3.1, proof of Theorem 3.1, equation (3.9)"}],"recommendation":"major_revision","confidential_remarks":"The central equivalence is appealing and the forward directions are well argued, but the converse directions rest on a jet-construction lemma that is neither stated nor proved. This is a genuine gap in the main theorem, though it appears fixable by adding a standard lemma (likely available in the ARS25 framework or provable by a local c-concavity perturbation argument). Once that lemma is supplied, I see no other obstacle to publication. The Corollary 4.1 constant issue is a presentation error, not a fatal one."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper is a genuine generalization of ARS25 to weighted intermediate Ricci curvature, and the main equivalence theorem is the real content. It deserves a careful reading, but there are a couple of gaps that need filling before I'd rely on it.\n\nThe new definition — N-Bakry-Emery intermediate k-Ricci curvature — is natural, and the equivalence between this lower bound and the tensorial Bochner inequality (condition 2) and the entropy-tensor inequalities (conditions 4-6) is a solid piece of work. The Riccati calculus in Section 2 is correct, and the exterior-power reformulation is a nice touch that makes the dimensional constants transparent. I also appreciate the trace identity connecting the entropy functional to the tensor; it's clean and useful.\n\nThe soft spots are real but mostly fixable. The biggest one is in the proof of Theorem 3.1: both (4)⇒(1) and (5)⇒(1) rely on a 'standard jet construction' producing a compactly supported θ with prescribed gradient and Hessian at a point, such that -θ is d²/2-concave after scaling. This is never stated as a lemma or proved. The assertion is plausible — in local coordinates you can take a quadratic-plus-linear function and localize, then scale to make the semiconvexity constant arbitrarily small — but on a general complete manifold it needs a proof or a precise extraction from Villani. As written, the converse directions are not fully supported. This is load-bearing, not cosmetic.\n\nCorollary 3.2 has a different issue: it applies the ODE lemma with a=1/(C N) and b=C K|∇θ|², but the lemma requires ab < π². For K>0 and large |∇θ| this fails, so the inequality (3.19) is not established in that regime. The K=0 case, which is what Section 4 uses, is fine.\n\nThere's also a minor algebraic slip in the proof of Corollary 4.1 where an equality is written while dropping a nonzero constant term; it should be an inequality. It doesn't affect the final estimate.\n\nFinally, the abstract says the paper 'compares' with Ketterer-Mondino, but the text claims more: 'prove that the two formulations are equivalent in the unweighted setting.' Section 4.2 only shows that the traced Riccati equation coincides with KM18's equation; that's a comparison, not an equivalence. The claim should be softened.\n\nBottom line: the core is good, the gaps are fixable, and a serious referee should engage with it. I'd accept it for review and ask for the jet lemma, the K>0 hypothesis, and the KM18 equivalence claim to be addressed.","headline":"Solid generalization of ARS25 to weighted intermediate Ricci curvature, with a fixable gap in the converse directions and an overclaimed comparison with Ketterer–Mondino.","tokens_in":40580,"tokens_out":4927,"would_cite":true,"duration_ms":42742,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","53C24","49Q22","58J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"On weighted Riemannian manifolds, the lower bound $\\mathrm{Ric}^N_{k,f}\\geq K$ is characterized exactly by a tensorial Bochner inequality and by convexity of the weighted $k$-Boltzmann entropy tensor along Wasserstein geodesics.","keywords":["weighted intermediate Ricci curvature","Bakry–Émery curvature","tensorial entropy convexity","Wasserstein geodesics","displacement convexity","Bochner inequality","heat flow contraction","optimal transport"],"falsifier":"Check whether the theorem cited as [Vil09, Theorem 13.5] actually produces compactly supported $d^2/2$-concave functions with arbitrarily prescribed gradient and Hessian at a point; if it does not, find a compact weighted manifold and a point where $\\mathrm{Ric}^N_{k,f}(\\Sigma,v)<K|v|^2$ yet every admissible $d^2/2$-concave potential with the required jets fails to exist, which would make condition (5) hold without the curvature bound and refute Theorem 3.1.","tokens_in":39475,"feed_emoji":"📐","tokens_out":18787,"duration_ms":147745,"temperature":0.7,"pith_summary":"The paper proposes a weighted version of intermediate Ricci curvature, a family that interpolates between sectional and Ricci curvature, and claims that a lower bound $\\mathrm{Ric}^N_{k,f}\\geq K$ is exactly equivalent to a tensorial Bochner inequality, a second-order differential inequality for the trace of a new weighted Boltzmann entropy tensor along Wasserstein geodesics, and a concavity condition built from that tensor. If the claim is right, it completes a dictionary in which curvature bounds, Bochner identities, and entropy convexity coincide at the level of tensors rather than scalars, letting information finer than Ricci curvature be read off from optimal transport. The authors use this dictionary to prove dimension-dependent evolution variational inequalities and Wasserstein contraction estimates for the heat flow, generalizing the tensorial approach to sectional curvature to arbitrary lower bounds and weighted manifolds, and they compare their formulation with an optimal-transport approach to intermediate Ricci curvature via lower-dimensional measures.","feed_headline":"Intermediate curvature bound equals tensor entropy convexity","feed_subtitle":"Heat-flow contraction with sharp dimension dependence follows for weighted intermediate Ricci bounds.","key_machinery":"The load-bearing object is the weighted $k$-Boltzmann entropy tensor $H^{\\mu_0\\to\\mu_1}_{t;k,f}(x)=-\\int_0^t U_{s;k,f}(x)\\,ds$, where $U_{s;k,f}(x)=\\dot{J}_s(x)J_s(x)^{-1}-\\frac{1}{k}\\langle\\nabla f,\\dot{\\gamma}_s(x)\\rangle\\mathrm{Id}$ is built from the matrix of Jacobi fields along the Wasserstein geodesic $\\gamma_s(x)=\\exp_x(s\\nabla\\theta(x))$ and from the weight $f$. Its role is to translate curvature into transport: the weighted matrix Riccati inequality (Lemma 2.7) says $\\mathrm{tr}(\\dot{U}_{s;k,f}|_\\Sigma)\\leq -\\frac{1}{N}\\mathrm{tr}(U_{s;k,f}|_\\Sigma)^2-\\mathrm{Ric}^N_{k,f}(\\Sigma_s,\\dot{\\gamma})$, and integrating this along the geodesic turns the pointwise curvature bound into the second-order differential inequality for $\\mathrm{tr}(H_{t;k,f}|_\\Sigma)$ that appears in Theorem 3.1. The companion analytic object is the tensorial Bakry–Émery operator $\\tilde{\\Gamma}^N_{2;k,f}(\\psi)=\\frac{1}{2}\\nabla^2|\\nabla\\psi|^2-\\nabla_{\\nabla\\psi}\\nabla^2\\psi+\\frac{1}{k}(\\mathrm{Hess}\\,f(\\nabla\\psi,\\nabla\\psi)-\\frac{df(\\nabla\\psi)^2}{N-k})\\mathrm{Id}$, whose trace over a $k$-plane isolates exactly $\\mathrm{Ric}^N_{k,f}$ via the tensorial Bochner identity.","core_discovery":"The central claim, Theorem 3.1, is that on a smooth complete weighted Riemannian manifold $(M^n,g,e^{-f}d\\mathrm{Vol})$, the condition $\\mathrm{Ric}^N_{k,f}\\geq K$, where $\\mathrm{Ric}^N_{k,f}(\\Sigma,v)=\\mathrm{Ric}_k(\\Sigma,v)+\\mathrm{Hess}\\,f(v,v)-\\frac{df(v)^2}{N-k}$ and $\\mathrm{Ric}_k$ is the sum of sectional curvatures over a $k$-plane $\\Sigma$, is equivalent to six conditions. Among them are the tensorial Bochner inequality $\\mathrm{tr}(\\tilde{\\Gamma}^N_{2;k,f}(\\psi)|_\\Sigma)\\geq K|\\nabla\\psi|^2$ for every smooth compactly supported $\\psi$ and every $k$-plane $\\Sigma$, the entropy-tensor differential inequality $\\mathrm{tr}(\\ddot{H}^{\\mu_0\\to\\mu_1}_{t;k,f}|_\\Sigma)\\geq \\frac{1}{N}(\\mathrm{tr}(\\dot{H}^{\\mu_0\\to\\mu_1}_{t;k,f}|_\\Sigma))^2+K|\\nabla\\theta|^2$ along every Wasserstein geodesic, and the concavity of $t\\mapsto e^{-\\frac{1}{N}\\mathrm{tr}(H^{\\mu_0\\to\\mu_1}_{t;k,f}|_\\Sigma)}+\\frac{K}{N}|\\nabla\\theta|^2\\int_0^t (t-s)e^{-\\frac{1}{N}\\mathrm{tr}(H^{\\mu_0\\to\\mu_1}_{s;k,f}|_\\Sigma)}\\,ds$. The proof runs through a weighted matrix Riccati inequality for the Jacobi-field matrix $U_{s;k,f}=\\dot{J}J^{-1}-\\frac{1}{k}\\langle\\nabla f,\\dot{\\gamma}\\rangle\\mathrm{Id}$, which converts the curvature lower bound into the second-order inequality for the entropy tensor, and through the tensorial Bochner identity $\\tilde{\\Gamma}^N_{2;k,f}(\\psi)=(\\nabla^2\\psi)^2+\\mathrm{Riem}(\\bullet,\\nabla\\psi)\\nabla\\psi+\\frac{1}{k}(\\mathrm{Hess}\\,f(\\nabla\\psi,\\nabla\\psi)-\\frac{df(\\nabla\\psi)^2}{N-k})\\mathrm{Id}$.","pith_inferences":["Inference: The equivalence in Theorem 3.1 suggests a synthetic definition of weighted intermediate Ricci bounds in non-smooth metric-measure spaces, obtained by requiring the entropy-tensor differential inequality or the concavity condition along all Wasserstein geodesics; this would parallel how synthetic Ricci bounds are defined from entropy convexity. This is an extension the authors do not sta","Inference: The one-dimensional comparison estimates with sinh/cos/sin distortion coefficients indicate the same machinery could produce sharp Brunn–Minkowski-type inequalities for intermediate curvature, since the curvature bound controls the concavity of the logarithm of the entropy tensor trace.","Inference: The heat-flow contraction bound's exponential rate depends on $N$ rather than $k$, suggesting the effective dimension for contraction under $\\mathrm{Ric}^N_{k,f}\\geq 0$ is $N$; this is testable by checking whether the estimate fails when $N$ is replaced by $k$ on a manifold with $k<n$.","Inference: Because the converse proof leans on a cited but unproved jet construction, a counterexample to the existence of compactly supported $d^2/2$-concave potentials with prescribed jets would restrict the equivalence to a smaller class of potentials; verifying the cited theorem in this form is a concrete next step."],"forward_implications":["Under $\\mathrm{Ric}^N_{k,f}\\geq 0$ on a compact weighted manifold, the heat flow satisfies the intrinsic-dimensional evolution variational inequality of Theorem 4.1, with dissipation governed by the $k$-th elementary symmetric polynomial of $\\exp(-H^{P_\\tau\\mu_0\\to\\mu_1}_{1;k,f}/N)$.","The Wasserstein distance between two heat flows obeys $W_2^2(P_T\\mu_0,P_T\\mu_1)-[W_2(\\mu_0,\\mu_1)+\\frac{2nT}{k}\\|\\nabla f\\|_{L^\\infty}]^2\\leq -8N\\binom{n-1}{k-1}\\int_0^T\\int_M \\mathrm{tr}_{\\wedge^k}\\left(\\sinh^2\\left(\\frac{(H^{P_\\tau\\mu_0\\to P_\\tau\\mu_1}_{1;k,f})^{[k]}}{2N}\\right)\\right)dP_\\tau\\mu_0\\,d\\tau$, reducing in the constant-weight case to a nonincreasing-distance estimate (Corollary 4.1).","For the unweighted case $N=k$, condition (5) becomes $\\mathrm{tr}(\\ddot{H}_t|_\\Sigma)\\geq \\frac{1}{k}(\\mathrm{tr}(\\dot{H}_t|_\\Sigma))^2$, so nonnegative intermediate $k$-Ricci curvature is characterized by this quadratic tensor inequality; for $k=n$ this recovers the classical equivalence between nonnegative Ricci curvature and displacement convexity of Boltzmann entropy.","The weighted $k$-Boltzmann entropy functional $H_{k,f}$ is displacement convex along Wasserstein geodesics with explicit distortion coefficients, giving the quadratic interpolation bound $H_{k,f}(\\mu_t)\\leq(1-t)H_{k,f}(\\mu_0)+tH_{k,f}(\\mu_1)-\\frac{nK}{2k}t(1-t)W_2^2(\\mu_0,\\mu_1)$ (Corollary 3.3).","A rigidity result (Proposition 4.1): if heat flow preserves the Wasserstein distance between two measures and the Hessians of the Kantorovich potentials act nonpositively on exterior $k$-vectors, then $\\mathrm{Ric}_k$ vanishes along the transport."],"supporting_citations":[{"why":"Supplies the tensorial Bochner and matrix displacement convexity template that this paper extends to arbitrary lower bounds and weighted manifolds.","marker":"[ARS25]"},{"why":"Provides the cited 'standard jet construction' of $d^2/2$-concave potentials and the general Wasserstein-space background for the converse directions.","marker":"[Vil09]"},{"why":"Gives the Brenier–McCann theorem that produces the optimal transport map and Kantorovich potential used to build the entropy tensor.","marker":"[McC01]"},{"why":"Introduces displacement interpolation and the almost-everywhere Hessian of the Kantorovich potential, which underpin the definition of the Jacobi-field matrix.","marker":"[CMS01]"},{"why":"Defines the Bakry–Émery weighted Ricci tensor and $\\Gamma_2$ operators that the paper generalizes to intermediate $k$-Ricci curvature.","marker":"[BE85]"},{"why":"Provides the lower-dimensional optimal transport approach to intermediate Ricci curvature that the paper compares with and recovers.","marker":"[KM18]"},{"why":"Establishes the classical equivalence between nonnegative Ricci curvature and displacement convexity that the $k=n$ case reproduces.","marker":"[VRS05]"},{"why":"Supplies the dimensional evolution variational inequality framework that Theorem 4.1 generalizes to weighted intermediate curvature.","marker":"[EKS15]"},{"why":"Provides the equivalence between dimensional Wasserstein contraction and curvature-dimension conditions used as the benchmark for Corollary 4.1.","marker":"[BGGK15]"}],"fun_headline_variants":["Entropy convexity tied to weighted intermediate Ricci bound","Weighted Ricci bound yields sharp heat-flow contraction","Curvature lower bound equals tensor entropy convexity","Dimension-dependent heat contraction from weighted Ricci","Weighted intermediate Ricci equals tensorial entropy convexity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the converse directions $(4)\\Rightarrow(1)$ and $(5)\\Rightarrow(1)$ assumes that for any tangent vector and any symmetric operator at a point there exists a smooth compactly supported function with those prescribed first and second derivatives whose negative is, after rescaling, a $d^2/2$-concave Kantorovich potential; this 'standard jet construction' is cited to a standard optimal-transport reference but not proved in the paper, and if such potentials do not exist then the entropy-tensor inequalities could hold while the curvature bound fails.","fun_headline_variants_meta":{"raw":{"variants":["Entropy convexity tied to weighted intermediate Ricci bound","Weighted Ricci bound yields sharp heat-flow contraction","Curvature lower bound equals tensor entropy convexity","Dimension-dependent heat contraction from weighted Ricci","Weighted intermediate Ricci equals tensorial entropy convexity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000757,"raw_usage":{"total_tokens":3436,"prompt_tokens":1091,"completion_tokens":2345,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":2273}},"tokens_in":707,"tokens_out":2345,"duration_ms":16228,"temperature":1.0,"reasoning_tokens":2273,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:54:18.066759+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether the theorem cited as [Vil09, Theorem 13.5] actually produces compactly supported $d^2/2$-concave functions with arbitrarily prescribed gradient and Hessian at a point; if it does not, find a compact weighted manifold and a point where $\\mathrm{Ric}^N_{k,f}(\\Sigma,v)<K|v|^2$ yet every admissible $d^2/2$-concave potential with the required jets fails to exist, which would make condition (5) hold without the curvature bound and refute Theorem 3.1.","supporting_citations":[],"review_version":2}