{"id":"65d03940-a484-4dcb-b06f-d65f11dbb840","arxiv_id":"2608.06344","paper_version":1,"verdict":"UNVERDICTED","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The front matter of a monograph announces that minimal measures, a Mather constant, weak KAM solutions, and an Aubry set can be attached to any Kantorovich operator with finite Mather constant, via a duality with skew-linear entropies on pairs of probability measures.","lead":"This preprint reproduces the preface, overview, and table of contents of a monograph that proposes a general Aubry-Mather theory for a wide class of nonlinear operators called Kantorovich operators. The promised payoff is a unified ergodic theory spanning Hamiltonian dynamics, optimal transport, control theory, and thermodynamics, but the submission contains announcements rather than proofs.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Two-state Bellman operator with finite Mather constant has no weak KAM solution, so the Overview's unrestricted claim is false.","rationale":"The reader's UNVERDICTED was based on the absence of derivations. This stress-test finds a stronger, checkable objection: the announced theorem fails in a finite-state example that is itself one of the advertised applications (a risk-sensitive/robust Bellman operator). The failure is not about compactness versus non-compactness; X is finite. It is about the absence of a normalization or reachability hypothesis ensuring a global additive eigenvector. The reader's weakest_assumption identified finite Mather constant as an entry condition but then worried about non-compact scope; the actual failure is more basic and occurs already in the compact setting. Because the central advertised claim is false as stated, UNCHANGED is not appropriate and the verdict should move to REJECT. The framework may be salvageable with additional hypotheses, which the table of contents suggests the author already knows are needed for many subcases, but the abstract and Overview as submitted overclaim; a corrected statement would need to restrict the class of operators or add the missing hypotheses to Definition 1.","tokens_in":28339,"tokens_out":19463,"duration_ms":243017,"concrete_test":"Run the two-state check: verify Definition 1 for T(g)_1 = max(g_1+1, g_2), T(g)_2 = g_2; compute c(T) = -1 from (50); observe that Th+c=h forces h_2 = h_2-1, a contradiction. If the author's intended statement includes a further hypothesis (e.g. c(T)>0, T0 constant, a communicating/irreducibility condition, or an oscillation-contraction condition), that hypothesis must be added to Definition 1 and to the Overview's theorem statement; otherwise the counterexample stands.","verdict_should_be":"REJECT","load_bearing_attack":"Definition 1 imposes no normalization (e.g. T0 = 0) and no communicating/irreducibility condition, yet the Overview's package (properties 1–6, especially (52)) is announced for every backward Kantorovich operator with finite Mather constant c(T). A two-state operator refutes this. Let X={1,2} and T(g)_1 = max(g_1+1, g_2), T(g)_2 = g_2. This is a Bellman/robust-control operator of the type in (19): at state 1, either collect reward 1 and stay, or collect reward 0 and move to absorbing state 2; state 2 yields reward 0. It satisfies monotonicity, translation invariance, convexity and continuity, hence Definition 1. From (50), for μ=(p,1-p), sup_h ∫(h-Th) dμ = -p, so c(T) = inf_p(-p) = -1, finite. But the weak KAM equation (52) with this c reads u_2 - 1 = u_2 in the second coordinate, impossible. No weak KAM operator T∞ exists. The monograph's own table of contents (Sections 14.3–14.4 and 16.2–16.8: bounded oscillations, cone contraction, oscillation contractions, balanced/power-bounded/amenable hypotheses) indicates that additional hypotheses are needed; the Overview and abstract do not state them. Therefore the 'arbitrary Kantorovich operator' claim is false as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The submission is the front matter of a planned monograph: it contains a preface, a long overview, a table of contents, and a reference list, but no proofs. The paper proposes a class of nonlinear operators called Kantorovich operators and claims that, in duality with skew-linear entropies, every such operator with finite Mather constant carries a complete Aubry–Mather package: minimal measures, a Mather constant, weak KAM solutions, and an Aubry set. The central structural result is stated as Theorem 2, which represents backward Kantorovich operators as suprema over probability measures with a convex cost c(x, σ). A list of six properties is then announced for the associated weak KAM operator T∞, including idempotence, commutation, the equation T T∞ g + c(T) = T∞ g, and the measure-level Aubry set. The monograph is said to work on compact spaces, with extensions to non-compact settings deferred. No derivations appear in the submitted text.","tokens_in":28556,"tokens_out":9933,"duration_ms":111029,"significance":"If the program were correct, it would unify substantial pieces of Hamiltonian dynamics, optimal transport, risk-sensitive control, potential theory, and ergodic optimization, and the breadth of examples assembled in the overview is genuinely impressive. The explicit formulation of skew-linear entropies and the proposed measure-level Aubry set are potentially valuable organizing ideas. However, the submission contains no proofs of the central theorems, and the main universal claim is false as stated (see major comments). The paper is therefore best read as an extended book prospectus rather than a verifiable research contribution; its current value lies in the catalogue of examples and references, not in established theorems.","major_comments":[{"comment":"The claim that every backward Kantorovich operator with finite Mather constant admits a weak KAM operator is false as stated. Let X = {1, 2} and define T(g)_1 = max(g_1 + 1, g_2), T(g)_2 = g_2. This operator satisfies monotonicity, translation invariance, convexity, and lower semicontinuity, hence all four clauses of Definition 1. From Eq. (50), for μ = (p, 1-p), sup_h ∫ (h - T h) dμ = sup_{h1,h2} p (h1 - max(h1+1, h2)), which equals -p for p > 0 and 0 for p = 0; therefore c(T) = -1, which is finite. But the weak KAM equation (52) at x = 2 reads u_2 - 1 = u_2, so no finite-valued weak KAM solution exists. This example is a Bellman operator of exactly the type displayed in Eq. (19), so it lies within the proposed framework. The announced theorem therefore requires additional hypotheses, such as irreducibility or a communicating/normalization condition, and the abstract's phrase 'arbitrary Kantorovich operator' must be withdrawn.","section":"Overview, Definition 1 and Eq. (52)"},{"comment":"The submission contains no derivations for its load-bearing statements. Theorem 2 is described as crucial, but it is stated without proof; the equality c(T) = inf_μ T(μ, μ) in Eq. (51) is asserted without proof; and the six properties of the weak KAM operator T∞ are announced without proof. Since the abstract states that 'this paper reproduces the front matter' of a monograph, the reader cannot verify any of the central claims from the submitted text. A journal submission would need either complete proofs or a clear statement that the theorems are proved in a cited companion manuscript with precise hypotheses; the current text provides neither.","section":"Theorem 2 and the announced weak KAM properties"},{"comment":"The scope of the advertised claims is internally mismatched. The abstract promises an Aubry–Mather theory for an 'arbitrary Kantorovich operator', but the Overview later restricts the weak KAM construction to compact spaces and to operators with finite Mather constant, and adds that the non-compact cases are only 'expected to hold' with further analysis and suitable hypotheses. Moreover, the table of contents lists many additional assumptions in Sections 14.3–14.4 and 16.2–16.8, including bounded oscillations, cone contraction, oscillation contractions, balanced skew-linear entropies, power-bounded contractions, and amenable entropies; none of these appear in Definition 1 or in the Overview's unconditional list of properties. The main theorem should be stated with its actual hypotheses, and the advertised applications on R^n, Riemannian manifolds, and Sinkhorn operators should be explicitly marked as conjectural extensions rather than consequences of the compact-space theory.","section":"Overview, compactness and additional hypotheses"}],"minor_comments":[{"comment":"The displayed identity T(lim_n ↓ T^n(g+nc)) + c = lim_n ↓ T^{n+1}g + (n+1)c has unbalanced parentheses and is hard to parse; the intended limiting argument should be written with explicit parentheses and indices.","section":"Eq. (29)"},{"comment":"In the large-deviation formula, the right-hand side writes T(x, σ), but the skew-linear entropy is defined on pairs of probability measures; the notation should be T(δ_x, σ) or the corresponding cost c(x, σ) to avoid confusing a point with its Dirac measure.","section":"Eq. (28)"},{"comment":"Reference [182] is missing its title, and references [184] and [186] are listed as personal communications, which cannot be independently checked; these should be completed or removed before publication.","section":"References"}],"recommendation":"reject","confidential_remarks":"The submitted text is essentially monograph front matter, not a self-contained research paper, and the central universality claim is refuted by a simple two-state Bellman operator. The counterexample in my report should be communicated to the author; it also confirms that the extra hypotheses listed in the table of contents (e.g., Sections 14.3–14.4 and 16.2–16.8) are genuinely needed. If the journal is willing to consider a revised version, it would need to narrow the claims to the actual hypotheses, include proofs or a verifiable companion document, and explicitly separate proved results from conjectural non-compact extensions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This submission is the preface, overview, and table of contents of a monograph, and it should be judged as an announcement, not as a proof-bearing preprint. The framework is genuinely ambitious: backward Kantorovich operators, skew-linear entropies, and the proposed extension of Aubry–Mather theory to Bellman operators, Ruelle operators, Sinkhorn iterations, and stochastic transport. The definition is clean, the examples are well chosen, and the representation theorem (Theorem 2) is a plausible and interesting generalization of Bernard–Buffoni. The prose is clear, and the reference list is thorough. Credit where due: this is a serious research program, and the announcement gives a good map of it.\n\nThe soft spots are serious. First, there are no derivations. Theorem 2 is called crucial but is not proved; the six properties of the weak KAM operator are asserted; the duality c(T) = inf_mu T(mu,mu) is asserted. That alone makes the submission unverifiable. Second, the stress-test counterexample is valid: take X={1,2} and T(g)_1 = max(g_1+1, g_2), T(g)_2 = g_2. This satisfies Definition 1, and its Mather constant is -1, finite. But the claimed weak KAM equation (52) for state 2 reads u_2 - 1 = u_2, impossible. So no weak KAM operator exists. The abstract's \"arbitrary Kantorovich operator\" is therefore false; the book's own table of contents suggests that extra hypotheses (bounded oscillations, cone contraction, balanced, amenable) are needed, but those are not stated in the Overview or abstract.\n\nWhere does this leave the work? The framework may well be salvageable under additional conditions, and the counterexample does not kill the entire program. But as submitted, the central claim is wrong as stated, and the absence of proofs means the reader cannot check what actually holds. A serious editor should send the full monograph to a referee, not desk-reject the idea; the counterexample should be brought to the author's attention early. For the arXiv text alone, though, the right verdict is that it overclaims and needs heavy revision or withdrawal until the real hypotheses and proofs are supplied.","headline":"This is the front matter of a monograph, not a research paper, and the central claim it announces is false as stated.","tokens_in":29174,"tokens_out":2203,"would_cite":false,"duration_ms":24101,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37J51","49Q22","35F21"],"pacs":[],"model":"deepseek-v4-flash","headline":"A class of nonlinear operators called Kantorovich operators carries a complete Aubry-Mather package through a duality with skew-linear entropies.","keywords":["Kantorovich operators","skew-linear entropies","Aubry-Mather theory","weak KAM theory","Mather constant","optimal mass transport","minimal measures","large deviations"],"falsifier":"A concrete check: take a two-point space $X$ and the Sinkhorn operator $Tg(x)=\\epsilon\\log\\int_Y e^{(g(y)-c(x,y))/\\epsilon}\\,d\\nu(y)$ with a continuous cost, and compute the normalized iterates $T^n g-nc(T)$. If they fail to converge to a function $u$ with $Tu+c(T)=u$, or if some minimizer of $T(\\mu,\\mu)$ is not in $N(T_\\infty)=\\{\\sigma:T_\\infty(\\sigma,\\sigma)=0\\}$, the claimed package fails in a concrete instance.","tokens_in":27958,"feed_emoji":"📐","tokens_out":16778,"duration_ms":152759,"temperature":0.7,"pith_summary":"This front matter of a monograph claims that many nonlinear operators appearing across analysis, probability, dynamical systems, economics, and finance are instances of one class, called Kantorovich operators: monotone, translation-invariant on constants, convex (or concave, for the forward version) maps between function spaces. Because such an operator has no adjoint, the book replaces measure duality with a convex functional on pairs of probability distributions, a skew-linear entropy $T(\\mu,\\nu)$ whose value is the optimal cost of linking source $\\mu$ to target $\\nu$. The central claim is that every backward Kantorovich operator with finite Mather constant (the least diagonal cost $\\inf_\\mu T(\\mu,\\mu)$ of the associated entropy) on a compact space carries a full Aubry-Mather package: minimal measures, a Mather constant, weak KAM solutions (fixed points up to that constant), and a measure-level Aubry set. If correct, this extends the classical weak KAM theory of Hamiltonian dynamics to Hopf-Lax-Oleinik semigroups, Sinkhorn and Ruelle operators, risk-sensitive Bellman equations, envelopes, and stochastic transports alike.","feed_headline":"One duality gives weak KAM structure to every finite-Mather operator","feed_subtitle":"A unified framework gives weak KAM solutions and an Aubry set to Hopf-Lax, Sinkhorn, Bellman, Ruelle operators alike.","key_machinery":"The central object is a backward Kantorovich operator: a map $T:C(Y)\\to USC(X)$ that is monotone, translation invariant on constants, convex, and lower semicontinuous, with a forward analogue that is concave and upper semicontinuous. The load-bearing identity is the duality $T(\\mu,\\nu)=\\sup_g\\{\\int_Y g\\,d\\nu-\\int_X Tg\\,d\\mu\\}$, which attaches to each operator a skew-linear entropy on pairs of probability distributions, together with the representation $Tg(x)=\\sup_\\sigma\\{\\int_Y g\\,d\\sigma-c(x,\\sigma)\\}$ for a cost $c$ convex in $\\sigma$. This entropy plays the role the adjoint plays for a linear Markov operator: its diagonal value $T(\\mu,\\mu)$ defines the Mather constant $c(T)=\\inf_\\mu T(\\mu,\\mu)$ and the minimal measures, while monotone limits of iterates $T^n(g+nc)$ produce the idempotent weak KAM operator $T_\\infty$ whose diagonal $T_\\infty(\\sigma,\\sigma)$ defines the measure-level Aubry set $N(T_\\infty)$. The construction works because Kantorovich operators extend to Choquet functional capacities, which justifies the monotone limits and the iteration of the operator.","core_discovery":"The core discovery is a duality that makes nonlinear operators amenable to ergodic theory. Theorem 2 states that a map $T:C(Y)\\to B_b(X)$ is a backward Kantorovich operator exactly when there is a proper lower semicontinuous cost $c:X\\times P(Y)\\to \\mathbb{R}\\cup\\{+\\infty\\}$, convex in its second argument, such that $Tg(x)=\\sup_\\sigma\\{\\int_Y g\\,d\\sigma-c(x,\\sigma)\\}$. Equivalently, $T$ admits a skew-linear entropy $T(\\mu,\\nu)=\\sup_g\\{\\int_Y g\\,d\\nu-\\int_X Tg\\,d\\mu\\}$ on pairs of probability measures. From this duality the book defines the Mather constant $c(T)=\\inf_\\mu\\sup_h\\int_X(h-Th)\\,d\\mu$, proves $c(T)=\\inf_\\mu T(\\mu,\\mu)$, calls the minimizers minimal measures, and then constructs, whenever $c(T)$ is finite, an idempotent weak KAM operator $T_\\infty$ satisfying $T\\circ T_\\infty+c(T)=T_\\infty$. Its measure-level Aubry set $N(T_\\infty)=\\{\\sigma:T_\\infty(\\sigma,\\sigma)=0\\}$ contains all minimal measures, and every weak KAM solution is recovered from its integrals against $N(T_\\infty)$, making the Aubry set a uniqueness set; forward and backward versions of the same package are developed. This is the claim that Aubry-Mather theory is not specific to Lagrangian dynamics but is the ergodic theory of any Kantorovich operator.","pith_inferences":["A concrete finite-state test is available: on probability simplices the paper's formulas, for example $T(\\beta,\\alpha)=\\sum_i \\beta_i G_i(\\alpha_i/\\beta_i)$, make $c(T)$ and $T_\\infty$ computable, and working out explicit examples would show what the Aubry set looks like outside mechanics.","If the non-compact extension goes through, the Mather constant of entropic Sinkhorn or Schr\\\"odinger transport would become a large-deviation rate for empirical occupation measures, turning the large-deviation reading sketched in the overview into a quantitative statement.","The framework suggests defining 'Mather measures' for optimal stopping and Skorokhod embedding as diagonal minimizers of Brownian stopping entropies, giving a variational selection principle for optimally stopped distributions.","A natural cross-check with dynamic programming: for a finite-state risk-sensitive Bellman operator, the weak KAM solution produced by $T_\\infty$ should coincide with the known value function of the ergodic control problem, connecting the abstract package to textbook results."],"forward_implications":["Every operator in the class, including the Hopf-Lax-Oleinik semigroup, Sinkhorn entropic regularization, risk-sensitive Bellman operators, Ruelle free-energy operators, concavification, and pluri-superharmonic envelopes, carries minimal measures, a Mather constant, weak KAM solutions, and an Aubry set whenever the Mather constant is finite.","For a linear Markov operator the package collapses to the classical one: the Mather constant is $0$, the minimal measures are exactly the invariant measures, and the weak KAM solutions are the invariant functions.","The identity $c(T)=\\inf_\\mu\\sup_h\\int_X(h-Th)\\,d\\mu=\\inf_\\mu T(\\mu,\\mu)$ gives two faces of the same constant, one computable through functions and one through the diagonal entropy of measures.","The measure-level Aubry set $N(T_\\infty)$ is a uniqueness set: weak KAM solutions are determined, and reconstructed, by their integrals against $N(T_\\infty)$ via the Peierls-barrier formula.","When the operator is both backward and forward skew-linear, a symmetric pair of weak KAM operators exists, with forward and backward solutions related by the same Aubry set."],"supporting_citations":[{"why":"The dual formulation of optimal transport from which the operator class takes its name supplies the original representation of costs by suprema over measures, which Theorem 2 generalizes.","marker":"[172]"},{"why":"The action-minimizing invariant measures of Lagrangian systems are the original minimal measures that the theory extends.","marker":"[190]"},{"why":"Supplies the critical value in Lagrangian dynamics that the Mather constant $c(T)$ generalizes.","marker":"[185]"},{"why":"Frames the Hamiltonian background and the PDE description of what survives of invariant tori.","marker":"[116]"},{"why":"Makes explicit the duality between Lax-Oleinik-type operators and optimal mass transport, the prototype for Kantorovich operators.","marker":"[24]"},{"why":"Connects weak KAM pairs to Monge-Kantorovich duality, the pattern behind the weak KAM operator and its entropy.","marker":"[25]"},{"why":"Develops weak KAM theory for Lagrangian systems, identifying calibrating functions as fixed points up to a constant.","marker":"[119]"},{"why":"Introduces weak transport costs with general costs over distributions, used to represent skew-linear entropies as optimal costs of balayage.","marker":"[161]"}],"fun_headline_variants":["Aubry-Mather beyond dynamics: a general operator theory","One duality unifies Aubry-Mather for all operators","Weak KAM structure for every Kantorovich operator","Ergodic Aubry-Mather theory for nonlinear operators","Generalized weak KAM: from Hamiltonians to Kantorovich"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction of weak KAM solutions assumes the Mather constant is finite and the underlying space is compact; the overview states that results on complete metric spaces, manifolds, or $\\mathbb{R}^n$ are only expected to hold with additional hypotheses, even though many advertised applications live there.","fun_headline_variants_meta":{"raw":{"variants":["Aubry-Mather beyond dynamics: a general operator theory","One duality unifies Aubry-Mather for all operators","Weak KAM structure for every Kantorovich operator","Ergodic Aubry-Mather theory for nonlinear operators","Generalized weak KAM: from Hamiltonians to Kantorovich"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00027,"raw_usage":{"total_tokens":1690,"prompt_tokens":1078,"completion_tokens":612,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":694,"completion_tokens_details":{"reasoning_tokens":529}},"tokens_in":694,"tokens_out":612,"duration_ms":6696,"temperature":1.0,"reasoning_tokens":529,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:50:14.695037+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: take a two-point space $X$ and the Sinkhorn operator $Tg(x)=\\epsilon\\log\\int_Y e^{(g(y)-c(x,y))/\\epsilon}\\,d\\nu(y)$ with a continuous cost, and compute the normalized iterates $T^n g-nc(T)$. If they fail to converge to a function $u$ with $Tu+c(T)=u$, or if some minimizer of $T(\\mu,\\mu)$ is not in $N(T_\\infty)=\\{\\sigma:T_\\infty(\\sigma,\\sigma)=0\\}$, the claimed package fails in a concrete instance.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The action-minimizing invariant measures of Lagrangian systems are the original minimal measures that the theory extends."}],"review_version":1}