{"id":"6e9d89c7-5763-44ca-a6a0-31f1b2c63e33","arxiv_id":"2608.02435","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under linear constraints on invariant measures, the constrained ergodic optimization problem still admits optimizers, has a unique optimizer for generic and prevalent objective functions, and satisfies a duality formula unifying ergodic optimization and optimal transport.","lead":"This paper builds a general framework for ergodic optimization problems in which the admissible invariant measures must satisfy linear constraints, such as fixed rotation vectors or fixed marginals on a product system. It proves when such problems have solutions, that typical objective functions have a unique maximizer, and that a Kantorovich-style duality holds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 2.14 overstates finite-type: Examples 1.4–1.5 with non-ergodic marginals fail the facial property, narrowing Theorem 1.9's scope.","rationale":"The reader's conditional verdict is appropriate. I agree with the reader's first listed concern (Corollary 2.14 overstates finite-type for non-ergodic marginals) and I find it more concrete and more demonstrably a flaw than the second concern about Proposition 5.6. Proposition 5.6 is an external cited lemma; unless the citation is wrong, relying on it is standard mathematical practice, so I did not make it the primary attack. The central uniqueness result Theorem 1.8 is not affected by the finite-type overstatement: the proofs in Section 4 are standard and internally consistent. However, the overstatement does narrow the advertised scope of the realization theorem and of the paper's claim to unify Examples 1.4 and 1.5. Therefore the manuscript should be corrected or clarified before acceptance, keeping the verdict conditional rather than outright acceptance or rejection.","tokens_in":25093,"tokens_out":23317,"duration_ms":230606,"concrete_test":"Test Corollary 2.14 on X={0,1} with T=id, μ1=μ2=(δ0+δ1)/2 in Example 1.5. The feasible set is the continuum of couplings of μ1 with itself. Show that for any finite C'⊂C(X×X), M_C' strictly contains this set (e.g., two distinct couplings agree on all finitely many prescribed moment functions). If so, finite-type fails and Corollary 2.14's claim for non-ergodic marginals is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central uniqueness theorem (Theorem 1.8) appears sound: the upper semicontinuity of the argmax correspondence, the Fort-theorem argument identifying continuity points with unique maximizers, and the Christensen-based prevalence argument are internally consistent. The paper's real soft spot is the finite-type claim used to advertise the realization theorem. Example 1.4 and Example 1.5 are stated for arbitrary invariant measures, but Corollary 2.14's proof asserts that these examples have the facial property and therefore finite type. This is false when the prescribed measure/marginal is non-ergodic: any factor of an ergodic measure is ergodic, so M_C contains no ergodic measures when π_* μ = ν for non-ergodic ν (and similarly for joinings with non-ergodic marginals). By Remark 2.7, the facial property would force M_C to contain an ergodic measure. Thus the facial-property proof in Corollary 2.14 silently relies on an ergodicity hypothesis that Example 2.8 explicitly restricts to but the corollary does not. Moreover finite-type itself generally fails in these cases: a non-ergodic measure is not determined by finitely many moment constraints, so no finite C' can cut out exactly the feasible set. Consequently Theorem 1.9 does not apply to the full scope claimed for Examples 1.4–1.5; the realization result is only supported for ergodic marginals unless an additional argument is supplied.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a constrained version of ergodic optimization. Given a continuous map T on a compact metrizable space X and a constraint set C ⊂ C(X), the feasible set M_C(X,T) consists of T-invariant measures that vanish on C. The main results are: (i) Theorem 1.7, a characterization of nonemptiness of M_C(X,T) and existence of optimizing measures; (ii) Theorem 1.8, asserting that if M_C(X,T) is nonempty, then the set U_C = {φ ∈ C(X) : the constrained maximization has a unique maximizer} is both residual and prevalent in C(X); (iii) Theorem 1.9, a realization theorem stating that for finite-type constraint sets every closed face of M_C(X,T) is the solution set of some continuous objective; and (iv) Theorem 1.10, a duality formula that generalizes Kantorovich duality and known ergodic-optimization duality. The paper positions this framework as a common generalization of unconstrained ergodic optimization, rotations-vector constrained optimization, relative ergodic optimization, and ergodic optimal transport.","tokens_in":25458,"tokens_out":18316,"duration_ms":176522,"significance":"If the main theorems hold, Theorem 1.8 is a valuable generalization of the classical generic/prevalent uniqueness results of Jenkinson and Morris, and it applies to a broad class of constrained problems, including relative ergodic optimization and ergodic optimal transport. The proof of Theorem 1.8 is carefully executed: the upper semicontinuity of the argmax correspondence, the Fort-theorem argument, and the Christensen-based prevalence argument are internally consistent and appear correct. The duality framework (Section 6) is also a useful unification. However, the advertised scope of the realization theorem (Theorem 1.9) is overstated. The claim that Examples 1.4–1.5 always have finite type is not justified and is generally false for non-ergodic prescribed marginals, and the proof of Theorem 1.9 relies on an imported lemma (Proposition 5.6) that is not proved and contains undefined notation. These issues are load-bearing for the realization part of the paper, though they do not affect the soundness of the central uniqueness theorem.","major_comments":[{"comment":"The claim that Examples 1.4 and 1.5 have finite type is not supported. Example 2.8 establishes the facial property only for ergodic ν (respectively ergodic μ1, μ2); for non-ergodic marginals the feasible set M_C contains no ergodic measures (Remark 2.7), so the proof via Proposition 2.13 fails. Moreover finite type itself generally fails: if π is the identity and ν has infinitely many ergodic components, no finite set C' can cut out the singleton {ν} in the infinite-dimensional simplex M(X,T). Thus Theorem 1.9 does not apply to the full scope claimed for Examples 1.4–1.5 unless ergodicity assumptions are added or a direct proof of finite type in the non-ergodic cases is supplied.","section":"Corollary 2.14"},{"comment":"The realization theorem (Theorem 1.9) depends critically on Proposition 5.6, which is quoted from [33] without proof and contains undefined notation ('H1'). The authors should provide a proof or a precise reference with the exact hypotheses, and verify that M_C(X,T), as a finite-codimensional slice of the simplex M(X,T), satisfies those hypotheses. As written, the proof of Theorem 1.9 cannot be checked from the paper's own arguments.","section":"Section 5, Proposition 5.6"},{"comment":"In the proof of Lemma 3.2, the measure μ is described as 'a Borel probability measure on C(X)' but it is used as a measure on X. This appears to be a typo, but it is confusing in a lemma that is used to prove the existence theorem. Please correct to 'on X'.","section":"Theorem 1.7 / Lemma 3.2"}],"minor_comments":[{"comment":"The hypotheses state 'Let (X,T) be a topological vector space'; this should be 'topological dynamical system'.","section":"Theorems 4.4, 4.7, 4.9"},{"comment":"The undefined symbol 'H1' appears in the statement of Proposition 5.6. It should be replaced with the intended symbol (likely 'M' or the slice's ambient subspace).","section":"Proposition 5.6"},{"comment":"The derivation of (6.8) from Theorem 6.1 requires replacing φ by −φ. This sign change should be made explicit to avoid confusion.","section":"Example 6.4"},{"comment":"There is a typographical error: 'the we call' should be 'then we call'.","section":"Definition 5.3"}],"recommendation":"major_revision","confidential_remarks":"This is a solid paper with a correct and important uniqueness theorem. The main issue is that the realization theorem is advertised for a wider class of examples than the proofs support; the finite-type claims for non-ergodic marginals in Examples 1.4–1.5 appear false in general. The reliance on an unproved external lemma (Proposition 5.6) should be addressed, perhaps by including a proof or a more detailed citation. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my quick take for your file. This paper does something genuinely useful: it sets up a single constrained problem — maximize ∫φ dμ subject to ∫g dμ = 0 for g in a subset of C(X) — and shows that unconstrained ergodic optimization, subsystem/rotation-vector problems, relative optimization, and ergodic optimal transport all fit as special cases. The existence criterion (Theorem 1.7) looks right; the duality theorem (Theorem 1.10) is a nice unification; and the centerpiece, Theorem 1.8, generalizes the known generic and prevalent uniqueness results of Jenkinson and Morris to every non-empty feasible set of this linear-constraint type. I read the proof carefully and it appears sound: the upper semicontinuity argument, Fort's theorem for residual continuity, and Christensen's theorem for prevalence all line up. There are minor typos — Theorems 4.4 and 4.7 say 'topological vector space' where they mean 'topological dynamical system', and Lemma 3.2 says 'measure on C(X)' instead of 'on X' — but those are cosmetic.\n\nThe real soft spots are two, and both are narrower than they sound. First, Corollary 2.14 claims every example in the introduction has finite type, but the proof leans on the facial property, which only holds for relative ergodic optimization and ergodic optimal transport when the prescribed measure/marginal is ergodic. For non-ergodic ν, the feasible set typically contains no ergodic measures, and finite type generally fails — no finite set of constraints can pin down a non-ergodic marginal. So the realization theorem (1.9) does not cover those cases as advertised. This doesn't touch Theorems 1.7, 1.8, or 1.10, but it does need a correction or a carefully stated hypothesis.\n\nSecond, Theorem 1.9 rests on Proposition 5.6, a slice lemma credited to Lazar and cited from Lau without proof. The paper does not verify that the lemma's hypotheses hold for the finite-type slices that arise. The lemma may be true, but the proof of Theorem 1.9 is incomplete as written. A referee should ask the authors to supply a proof or a precise reference with the hypotheses explicitly checked.\n\nBottom line: the central results are solid and the framework is worth having. The overclaim in Corollary 2.14 and the unproved slice lemma are fixable. I'd send it to a good referee, and I'd expect a revision that tightens the scope of the realization results and supplies the missing lemma. Worth bringing to the group.","headline":"A useful unifying framework for constrained ergodic optimization; the main uniqueness theorem is sound, but the advertised scope of the realization theorem is too wide and one key lemma is unproved.","tokens_in":25899,"tokens_out":4438,"would_cite":true,"duration_ms":38606,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A05","37D20","49Q20","49N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Ergodic optimization with linear constraints has unique optimizers for typical objectives, plus a dual formula.","keywords":["ergodic optimization","linear constraints","invariant measures","generic uniqueness","prevalence","duality","optimal transport","realization"],"falsifier":"Find a continuous map T and a constraint set C with non-empty M_C(X,T) such that the objectives with more than one maximizer form a set with non-empty interior, or a set that is not negligible in the prevalent sense; such a system would directly contradict the generic/prevalent uniqueness theorem.","tokens_in":24996,"feed_emoji":"🎯","tokens_out":8663,"duration_ms":80227,"temperature":0.7,"pith_summary":"Ergodic optimization looks for the invariant probability measure that maximizes the average of a given continuous function. This paper adds linear constraints—the measure must give zero average to every function in a specified family—and studies the same maximization over the resulting feasible set. The central claim is that whenever the constrained feasible set is non-empty, a topologically generic and measure-theoretically prevalent set of objective functions has a unique maximizing measure. The paper also gives a precise non-emptiness criterion, a realization result describing which subsets of the feasible set can be optimal, and a duality formula that unifies classical ergodic optimization duality with optimal-transport duality.","feed_headline":"For typical objectives, constrained ergodic optimizers are unique","feed_subtitle":"Zero-average constraints don't break typical uniqueness: a single optimal measure wins for almost every objective.","key_machinery":"The central object is the feasible set M_C(X,T), defined as the intersection of the compact convex simplex M(X,T) of invariant measures with the common zero sets of the linear conditions ∫g dμ = 0. The main mechanism is the solution-set correspondence that assigns to each objective φ the non-empty set of its maximizers. This correspondence has a closed graph, so by a standard semicontinuity theorem its continuity points form a residual set; continuity of the correspondence is shown to be equivalent to uniqueness of the maximizer. For the measure-theoretic statement, the paper studies the maximal-value functional F_C(φ) = sup∫φ dμ, which is Lipschitz; its points of directional differentiabili","core_discovery":"The paper establishes that the constrained optimization problem—maximize ∫φ dμ over T-invariant measures satisfying ∫g dμ = 0 for all g in the constraint set—has a unique solution for a residual and prevalent set of objective functions φ, provided only that the feasible set M_C(X,T) is non-empty. It also proves that non-emptiness is equivalent to the absence of a strictly positive function in the smallest closed T-invariant subspace generated by the constraints, and that optimizers exist whenever the feasible set is non-empty. For constraints of finite type (those definable by finitely many functions), every closed face of the feasible set is realized as the optimizer set of some continuous","pith_inferences":["The same generic-uniqueness mechanism may transfer to constrained problems over symbolic systems with locally constant constraints, since the feasible set there is again a slice of the simplex and the correspondence argument should carry over.","The duality formula is an infinite-dimensional linear program; one could test it numerically on finite-type examples by computing the constrained maximum and the infimum over admissible triples.","If the face-intersection lemma used in the realization proof is replaced by a direct proof, the realization theorem would extend beyond finite-type constraints to all compact metrizable systems; the paper's examples all satisfy finite type, so the gap is narrow.","In ergodic optimal transport, the generic uniqueness result suggests that for a typical continuous cost, the optimal invariant coupling between two ergodic measures is unique—a concrete, checkable consequence of the paper's framework."],"forward_implications":["Whenever M_C(X,T) is non-empty, the objectives with a unique maximizer form both a residual and a prevalent set—uniqueness is the typical outcome, not a rare one.","Non-emptiness of the constrained feasible set has a clean certificate: it fails exactly when the constraint space forces a strictly positive function; otherwise every objective has an optimizer.","For finite-type constraints—covering subsystems, fixed rotation vectors, relative measures, and ergodic optimal transport—every closed face of the feasible set is realized as the optimizer set of some continuous objective.","The dual formula gives a practical bound certificate: any admissible triple (f, g, c) provides an upper bound for the constrained maximum, and the infimum saturates the bound."],"fun_headline_variants":["Typical constrained ergodic objectives have unique optimizers","Unique solutions dominate constrained ergodic optimization","Constrained ergodic optimization: uniqueness is typical","For most continuous objectives, constrained ergodic optimizers are unique","Under zero-average constraints, typical objectives yield a single winner"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The realization theorem depends on an unproved lemma about closed faces of finite-codimensional slices of compact convex sets; if that lemma is false, Theorem 1.9 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Typical constrained ergodic objectives have unique optimizers","Unique solutions dominate constrained ergodic optimization","Constrained ergodic optimization: uniqueness is typical","For most continuous objectives, constrained ergodic optimizers are unique","Under zero-average constraints, typical objectives yield a single winner"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000572,"raw_usage":{"total_tokens":2545,"prompt_tokens":754,"completion_tokens":1791,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":1715}},"tokens_in":498,"tokens_out":1791,"duration_ms":12660,"temperature":1.0,"reasoning_tokens":1715,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T07:23:26.922762+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a continuous map T and a constraint set C with non-empty M_C(X,T) such that the objectives with more than one maximizer form a set with non-empty interior, or a set that is not negligible in the prevalent sense; such a system would directly contradict the generic/prevalent uniqueness theorem.","supporting_citations":[],"review_version":1}