{"id":"66f47f23-34d3-437a-9b1a-bcfc34b88a90","arxiv_id":"2501.19369","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A KL-divergence-penalized transport problem, viewed as a pressure, recovers the Monge-Kantorovich solution and Kantorovich duality in the zero-temperature limit β→∞.","lead":"Optimal transport matches two distributions at minimum cost; its exact solution is often hard to compute. This paper regularizes the problem with a statistical entropy penalty, defines a pressure at inverse temperature β, and proves that as temperature goes to zero the regularized solutions and their dual potentials converge to the classical Monge-Kantorovich optimum and Kantorovich duality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof is sound under the full-support hypothesis, but that hypothesis is load-bearing and the abstract omits it; with smaller supports the limiting Schrödinger potentials need not lie in Φ_c globally.","rationale":"The reader identified exactly the right technical soft spot: the proof's use of supp(μ)=X and supp(ν)=Y via Lemma 3.4. My stress-test confirms that this is not a mere technicality but a genuine boundary of the argument: the abstract and the reader's strongest_claim omit the support condition, so an unwary reader could apply Theorem 1.1 to point masses and obtain a false statement. The theorem itself is internally consistent because the full-support hypothesis is stated at the start of Section 1; but the advertised central claim is too broad. I also examined the 'we can suppose uniformly bounded' normalization in Section 3.3. It is more justifiable than the reader suggests: choosing the representative with max φ_β=0 gives φ_β/β∈[-Lip(A)diam(X),0] by Lipschitz continuity, and the normalization equation then transfers the bounds to ψ_β/β. So that is a sketch rather than a gap. The larger issue is the support scope, which strengthens the case for a conditional acceptance: the authors should state the support hypothesis in the abstract and explicitly discuss the reduction to supp(μ)×supp(ν). Since the reader's verdict is already CONDITIONAL, my analysis does not change the verdict; it refines the reason.","tokens_in":12112,"tokens_out":18586,"duration_ms":180964,"concrete_test":"Set X=Y={0,1}, μ=ν=δ_0, and c(x,y)=|x-y|. Write the Schrödinger system with βA=-βc, normalize by φ_β(0)=0, solve for φ_β and ψ_β explicitly, and pass to the limit φ=lim φ_β/β, ψ=lim ψ_β/β. Check whether c(1,1)≥φ(1)+ψ(1) holds; it does not, confirming that the global Φ_c conclusion fails without full support. Then repeat on X'=supp μ={0}, Y'=supp ν={0} and verify the limit is an optimal dual pair on the reduced spaces.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is proved under the explicit assumptions supp(μ)=X and supp(ν)=Y, but the abstract presents the result for arbitrary probabilities without this qualification. The full-support condition is used in two indispensable places. First, Lemma 3.4 requires supp(ρ)=U to conclude (1/β)log∫e^{βWβ}dρ → sup_U W; without it, the limit is only sup over the support, so the equations in Section 3.3 yield sup over supp(μ)×supp(ν), not over X×Y. Consequently the limiting (φ,ψ) is only shown to satisfy c≥φ+ψ on the support product, and the conclusion (φ,ψ)∈Φ_c can fail. Second, Proposition 3.3 uses μ(B)>0 for every open ball B (and similarly for ν) in the last step of the uniqueness proof, so the uniqueness argument itself relies on full support. A concrete failure mode occurs for X=Y={0,1}, μ=ν=δ_0, c(x,y)=|x-y|: solving the normalized Schrödinger equations gives φ(1)=c(1,0)-c(0,0)=1 and ψ(1)=c(0,1)=1, so c(1,1)=0<φ(1)+ψ(1)=2, violating the global constraint. Thus the central claim, as advertised in the abstract, overstates the theorem's scope; the fix is to restrict the statement to supports or explicitly retain the full-support hypothesis throughout.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a 'pressure' functional P(βA) for the Monge-Kantorovich problem, defined as the supremum over transference plans of ∫βA dπ + H(π), where A = −c and H is the negative Kullback-Leibler divergence relative to μ×ν. The main result, Theorem 1.1, shows that the associated Schrödinger-type normalization equations admit Lipschitz solutions (φβ, ψβ), that the Gibbs-type measures dπβ = e^{βA+φβ+ψβ}dμdν are transference plans attaining P(βA), and that as β→∞ any uniform limit of (φβ/β, ψβ/β) solves the Kantorovich dual problem while any weak-* limit of πβ solves the Monge-Kantorovich problem. The paper also derives an entropy-selection result for the limits (Proposition 1.4), a large-deviation principle for πβ (Proposition 1.3), and a Kantorovich-Rubinstein corollary when c is a distance. The proofs use contraction arguments, Arzelà-Ascoli compactness, a Laplace-principle lemma, and weak-* compactness. The paper explicitly assumes supp(μ)=X and supp(ν)=Y at the start of Section 1.","tokens_in":12398,"tokens_out":8661,"duration_ms":88758,"significance":"If the main theorem is correct under the stated full-support hypothesis, it gives a clean zero-temperature variational route to Kantorovich duality, connecting optimal transport with thermodynamic formalism and Sinkhorn-type normalization. The paper is largely self-contained in its central estimate, and the strategy of deriving the dual potentials and optimal plans from a single convex pressure functional is attractive. The strengths are the explicit construction of (φβ, ψβ), the relatively elementary proof of convergence, and the additional results on entropy selection and large deviations. The main concern is that the abstract states the result for arbitrary probability measures while the proof and the theorem rely essentially on full support; without it the central conclusion can fail, as the counterexample below shows. This mismatch is fixable by restating the scope, but it is load-bearing.","major_comments":[{"comment":"The abstract presents the main result for arbitrary probabilities μ∈P(X) and ν∈P(Y), but the proof is carried out under the full-support hypothesis supp(μ)=X and supp(ν)=Y, which appears only in the first paragraph of §1. This hypothesis is load-bearing: Lemma 3.4 requires supp(ρ)=U to conclude (1/β)log∫e^{βWβ}dρ → sup_U W, and in §3.3 it is applied to μ and ν to obtain equations (8) over all X and Y. If the supports are proper, the limiting equations hold only on supp(μ)×Y and X×supp(ν), so the conclusion (φ,ψ)∈Φ_c does not follow. A concrete failure is X=Y={0,1}, μ=ν=δ_0, c(x,y)=|x−y|; the normalized Schrödinger equations force φ(1)/β→1 and ψ(1)/β→1, so c(1,1)=0<φ(1)+ψ(1)=2. The abstract should either state the full-support assumption or the theorem should be reformulated for the supports of μ and ν.","section":"Abstract and §1/§3.3"},{"comment":"The uniqueness part of Theorem 1.1 also depends on full support. In the proof that p is constant, the paper assumes p(x̃)>p0 and then uses ∫_{B(x̃,δ)} e^{A+φ1+ψ1}(e^ε−1)dμ>0; this is justified only if μ(B)>0 for every open ball, i.e., supp(μ)=X, and analogously for ν. Without full support the uniqueness claim can fail: in the δ_0 example above the normalization equations have a one-parameter family (φ(0)=t, ψ(0)=−t, φ(1)=β+t, ψ(1)=β−t), so item 1's 'unique up to a constant' statement is false. The full-support hypothesis should be recorded beside Theorem 1.1 and Proposition 3.3, not only in the opening paragraph.","section":"§3.1 (Proposition 3.3)"},{"comment":"The extraction of uniform limits for φβ/β and ψβ/β needs an explicit normalization argument. The text says 'we can suppose uniformly bounded' and cites the proof of Proposition 3.1, but the bounds displayed there are for the differences φ_s − max φ_s and ψ_s − max ψ_s together with the constant l_s; they do not directly give bounds for the particular pair (φβ,ψβ) from Theorem 1.1 unless one fixes the normalization (for instance max φβ=0 and max of the auxiliary ψ before adding lβ). Please spell out this normalization and the resulting bounds, since the Arzelà-Ascoli step in item 3 depends on it. This is a local gap in presentation rather than an error in the underlying argument.","section":"§3.3"}],"minor_comments":[{"comment":"The second Schrödinger equation is written with e^{A(x,y)+φβ(x)+ψβ(y)}, but it should be e^{βA(x,y)+φβ(x)+ψβ(y)}; without the β the statement is inconsistent with the proof and with item 2.","section":"Theorem 1.1, item 1"},{"comment":"In the statement of Proposition 3.3, the second equation has the quantifier '∀x∈Y'; it should be '∀x∈X'.","section":"Proposition 3.3"},{"comment":"After the Arzelà-Ascoli extraction, the sentence 'Particularly we get max(φβ)=max(ψβ)=0' is a typo; it should refer to the limiting φ and ψ before the shift by l, not to the β-indexed functions.","section":"Proof of Proposition 3.1"},{"comment":"In the final displayed conclusion, 'H(μ∞)' should be 'H(π∞)'.","section":"Proof of Proposition 1.4"}],"recommendation":"major_revision","confidential_remarks":"The paper appears mathematically sound under the full-support assumption stated in Section 1, and the main theorem is a nice contribution. The revision should primarily align the abstract and theorem statements with the hypotheses actually used, and clarify the normalization in §3.3. The counterexample with δ_0 shows that the advertised scope in the abstract is not a harmless omission. I do not see grounds for rejection, but the statement must be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Candid take: the paper is a clean, self-contained proof that the KL-entropy regularized optimal transport problem converges to the classical Monge-Kantorovich problem as the regularization vanishes, dressed up as a zero-temperature limit. The proofs check out under the full-support assumption. The trouble is that the abstract drops that assumption, and the theorem as advertised is false without it. The full-support hypothesis is load-bearing: Lemma 3.4 needs it to turn the Schrödinger equations into sup equations over X and Y, and Proposition 3.3 needs it for uniqueness. The stress-test counterexample (μ=ν=δ_0 on a two-point space) shows the limiting potentials can fail the global constraint c≥φ+ψ. So the paper has a genuine scope error, not just a missing qualification.\n\nWhat is actually new? Very little. The Schrödinger system in Theorem 1.1 is Sinkhorn's theorem, and the β→∞ limit to the Kantorovich solution is the established convergence of entropically regularized OT (Cominetti-San Martín, Léonard, Carlier et al.), none of which is cited. The paper re-proves these results with a pressure notation from ergodic optimization. That recasting is not a new mathematical result, though it may be a useful bridge for that community. The paper does some things well: the contraction argument for 0<s<1 is clean, the Lip bounds are explicit, and the LDP proposition is a nice touch. Minor typo: the second integral in Theorem 1.1 should have βA, not A.\n\nSoft spots beyond the scope issue: the uniform boundedness step in Section 3.3 is asserted rather than proved, and the missing prior-work citations are a serious omission for a paper claiming novelty. The self-citations to the author's prior work are not problematic per se, but they are used to import the entropy representation and Lemma 3.4, so the paper is not fully self-contained.\n\nWho is this for? People in ergodic optimization who want an accessible derivation of the OT connection. A reader already familiar with entropic OT will not learn anything new beyond the thermodynamic packaging. As a serious referee, I would engage because the core mathematics is correct and the scope error is fixable, but I would require a major revision: correct the statement to reflect the full-support assumption, add the missing references, and tone down the novelty claims.","headline":"Clean re-derivation of known entropic OT convergence, but the abstract overstates the theorem: without full-support assumptions the central claim fails.","tokens_in":12959,"tokens_out":2200,"would_cite":false,"duration_ms":22167,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A50","28A33","28D20","46E27","60B10","60F10","47H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A zero-temperature pressure limit recovers optimal transport and its Kantorovich duality.","keywords":["Monge-Kantorovich problem","Kantorovich duality","Kullback-Leibler divergence","entropy","pressure","zero temperature limit","thermodynamic formalism"],"falsifier":"Take the finite full-support example $X=Y=\\{1,2\\}$, $μ=ν$ uniform, and $c(1,1)=c(2,2)=0$, $c(1,2)=c(2,1)=2$, solve the fixed-point equations for $φ_β,ψ_β$ at increasing $β$, and check whether every convergent subsequence of $(φ_β/β,ψ_β/β)$ satisfies $φ+ψ≤c$ with equality of $α(c)$ and whether every weak-* limit of $π_β$ has cost $α(c)$; a single subsequence violating the dual constraint or achieving a strictly larger cost would refute Theorem 1.1.","tokens_in":11884,"feed_emoji":"❄️","tokens_out":10906,"duration_ms":101529,"temperature":0.7,"pith_summary":"This paper builds a pressure function from the Kullback-Leibler divergence between a transport plan and the product of its marginals, and proves that its zero-temperature limit (β→∞) recovers the classical Monge-Kantorovich optimal transport problem together with Kantorovich duality. At each finite temperature the optimizing plan has a density of the form $e^{βA+φ_β+ψ_β}$ with respect to $μ×ν$, where $A=-c$; the paper shows such potentials exist, are essentially unique, and are Lipschitz. The main result is that the rescaled potentials $φ_β/β$ and $ψ_β/β$ converge, along subsequences, to a solution of the Kantorovich dual problem, while the plans themselves accumulate only on optimal transport plans. This matters because it connects entropic transport with thermodynamic-formalism pressure in a setting with no underlying dynamics, and it shows that the zero-temperature limit selects optimal plans of maximal relative entropy.","feed_headline":"Zero-temperature pressure recovers optimal transport duality","feed_subtitle":"Cooling a KL-entropy pressure to infinity recovers the optimal transport plan and its dual potentials.","key_machinery":"Two devices carry the argument. First, the relative entropy $H(π)=-D_{\\mathrm{KL}}(π|μ×ν)$ is a concave, upper semi-continuous functional on the compact convex set of transference plans, so the pressure supremum is attained; the entropy's variational representation converts optimality into the normalization equations, whose solution is reduced to the fixed points of the contractions $T^s_μ$ and $T^s_ν$. Second, the passage to zero temperature uses the log-sum-exp limit: Lemma 3.4 states that $(1/β)\\log∫ e^{βW_β}dρ$ converges uniformly to $\\sup W$ when $W_β→W$ and $ρ$ has full support, and this turns the normalization equations into the dual constraints $\\sup_x(A+φ+ψ)=0$ for every $y$ and $\\sup_y(A+φ+ψ)=0$ for every $x$, which are exactly the Kantorovich optimality conditions.","core_discovery":"The central claim is Theorem 1.1: for a Lipschitz cost $c$ and probabilities $μ,ν$ with full support, the pressure $P(βA)=\\sup_{π∈Π(μ,ν)}[\\int βA\\,dπ + H(π)]$ with $A=-c$ and $H(π)=-D_{\\mathrm{KL}}(π|μ×ν)$ has, for every $β>0$, a unique (up to an additive constant) pair $(φ_β,ψ_β)$ solving the normalization equations $\\int e^{βA(x,y)+φ_β(x)+ψ_β(y)}dμ(x)=1$ for every $y$ and $\\int e^{βA(x,y)+φ_β(x)+ψ_β(y)}dν(y)=1$ for every $x$. The density $dπ_β=e^{βA+φ_β+ψ_β}dμ\\,dν$ is a transference plan and attains the pressure. As $β→∞$, every uniform limit $(φ,ψ)$ of $(φ_β/β,ψ_β/β)$ lies in the cone $Φ_c=\\{φ(x)+ψ(y)≤c(x,y)\\}$ and saturates Kantorovich duality, $α(c)=\\int φ\\,dμ+\\int ψ\\,dν$; every weak-* cluster point of $π_β$ is an optimal plan. The paper further shows that under convergence assumptions the family satisfies a large-deviation principle with rate function $c-φ-ψ$, and that the limiting plans are precisely the optimal plans with maximal relative entropy.","pith_inferences":["One can expect the construction to extend to costs that are merely continuous or to non-compact spaces if the log-sum-exp limit is replaced by a sup over the support, although the Lipschitz bounds used for compactness would need a substitute.","The maximal-entropy selection rule identifies a canonical optimal plan among many, which is the same spirit as entropic optimal transport regularization but obtained here as a zero-temperature limit rather than as a fixed small temperature.","If a marginal's support is a proper subset, the dual constraints are only enforced on that support; testing whether the theorem's conclusions survive with a modified normalization would clarify the essential role of the full-support assumption.","The rate function $c-φ-ψ$ has the form of a calibration in optimal transport, so the large-deviation result may carry over to settings such as martingale or causal transport where comparable dual potentials appear."],"forward_implications":["For the distance cost on $X=Y$, any uniform limit of $φ_β/β$ solves the Kantorovich-Rubinstein dual problem with $α(c)=\\int φ\\,d(μ-ν)$.","When the limits exist, the entropic plans $π_β$ obey a large-deviation principle with rate function $c-φ-ψ$, quantifying concentration onto the optimal set.","All zero-temperature accumulation points of $π_β$ lie in the set of optimal plans and have the largest possible relative entropy with respect to $μ×ν$.","The gap $P(βA)-βm(A)$ decreases to $H_{\\max}$, giving a measure of how the pressure value approaches the optimal transport value as $β→∞$.","The positive-temperature optimizer is an explicitly positive plan $e^{-βc+φ_β+ψ_β}μ×ν$, so the result supplies a built-in smoothing approximation to an optimal plan."],"supporting_citations":[{"why":"Supplies Lemma 3.4, the log-sum-exp limit that carries the normalization equations to the Kantorovich dual constraints.","marker":"[11]"},{"why":"Provides the matrix-scaling method that yields the normalization pair in the discrete uniform case, which the proof extends.","marker":"[12]"},{"why":"Companion to the matrix-scaling result used for the same normalization construction.","marker":"[13]"},{"why":"Source for the Kantorovich duality and c-concave conjugation statements used to identify the limit pair.","marker":"[15]"},{"why":"Defines the relative entropy H_μ and gives the variational formula for H used in proving duality of the pressure.","marker":"[8]"},{"why":"Supplies the inverse Varadhan lemma used to derive the large-deviation principle.","marker":"[6]"}],"fun_headline_variants":["Cooling entropic pressure to zero T recovers optimal transport plans","KL-pressure at absolute zero yields transport duality and potentials","Zero-temperature limit of pressure solves optimal transport problem","Entropic pressure zero-T limit: optimal plans and dual certificates","Pressure at beta to infinity recovers optimal transport duality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes both marginals have full support on their compact metric spaces, because the log-sum-exp limit that converts the normalization equations into the Kantorovich dual constraints is taken over the whole of $X$ and $Y$; if a marginal's support is smaller, the limiting constraint holds only on that support and an extra argument would be needed to conclude the limit pair belongs to $Φ_c$.","fun_headline_variants_meta":{"raw":{"variants":["Cooling entropic pressure to zero T recovers optimal transport plans","KL-pressure at absolute zero yields transport duality and potentials","Zero-temperature limit of pressure solves optimal transport problem","Entropic pressure zero-T limit: optimal plans and dual certificates","Pressure at beta to infinity recovers optimal transport duality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001024,"raw_usage":{"total_tokens":4405,"prompt_tokens":1117,"completion_tokens":3288,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":733,"completion_tokens_details":{"reasoning_tokens":3208}},"tokens_in":733,"tokens_out":3288,"duration_ms":25653,"temperature":1.0,"reasoning_tokens":3208,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T20:21:26.659258+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the finite full-support example $X=Y=\\{1,2\\}$, $μ=ν$ uniform, and $c(1,1)=c(2,2)=0$, $c(1,2)=c(2,1)=2$, solve the fixed-point equations for $φ_β,ψ_β$ at increasing $β$, and check whether every convergent subsequence of $(φ_β/β,ψ_β/β)$ satisfies $φ+ψ≤c$ with equality of $α(c)$ and whether every weak-* limit of $π_β$ has cost $α(c)$; a single subsequence violating the dual constraint or achieving a strictly larger cost would refute Theorem 1.1.","supporting_citations":[{"cited_title":"Mengue and E","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.4, the log-sum-exp limit that carries the normalization equations to the Kantorovich dual constraints."},{"cited_title":"Sinkhorn","cited_arxiv_id":null,"evidence_quote":"Provides the matrix-scaling method that yields the normalization pair in the discrete uniform case, which the proof extends."},{"cited_title":"Sinkhorn and P","cited_arxiv_id":null,"evidence_quote":"Companion to the matrix-scaling result used for the same normalization construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source for the Kantorovich duality and c-concave conjugation statements used to identify the limit pair."},{"cited_title":"Lopes and J","cited_arxiv_id":null,"evidence_quote":"Defines the relative entropy H_μ and gives the variational formula for H used in proving duality of the pressure."},{"cited_title":"Dembo and O","cited_arxiv_id":null,"evidence_quote":"Supplies the inverse Varadhan lemma used to derive the large-deviation principle."}],"review_version":1}