{"id":"38cc3176-9c1d-402b-b3ca-53cf54cbeb5c","arxiv_id":"2412.05639","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A collection of invariant measures is the freezing collection for some potential if and only if it has constant entropy and is the equilibrium collection for some potential.","lead":"This paper proves that, in any dynamical system, a collection of invariant measures freezes at low temperature exactly when it has constant entropy and is already an equilibrium collection for some potential. It also shows freezing potentials are dense under upper semicontinuous entropy, while non-freezing potentials are generic under specification.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's proof falsely asserts F is closed; this breaks Edwards's separation step, but the gap is repairable by working with cl(F), so the central claim likely survives as a conditional accept.","rationale":"The reader identified the correct load-bearing weakness: the proof of Theorem 1 relies on the false claim that the equilibrium set F is closed, and this invalidates the Edwards separation step as written. I agree with that diagnosis. However, the concern is repairable rather than fatal. For any point ν in cl(F)\\F, the continuity of ∫φ and the fact that F is exactly the equilibrium set for φ force h(ν)<h0, where h0 is the constant entropy on F. This makes it possible to replace the non-lower-semicontinuous g with a lower-semicontinuous function g' that is 0 on the closed convex set cl(F) and ||φ|| outside. If cl(F) is a face of the Choquet simplex MT(X), Edwards's theorem applies and yields the required affine continuous separator A. The resulting ψ=φ−A then satisfies Lemma 3.1, giving freezing on F for all β≥1. The other issues noted by the reader, such as the sign error in Lemma 4.4 and the questionable rational-intersection equality in Theorem 5, are real but secondary; they do not affect the central equivalence in Theorem 1. Because the main theorem's proof can likely be corrected with a modest revision, the appropriate verdict is CONDITIONAL acceptance rather than outright rejection. The concrete test above would settle whether the repair is valid and thus whether the central claim holds.","tokens_in":10863,"tokens_out":28234,"duration_ms":273014,"concrete_test":"Verify the proposed repair: set F0=cl(F) and g'=0 on F0, ||φ|| outside. Check that F0 is a face of MT(X), e.g., using openness of the map (x,y)↦λx+(1−λ)y for λ∈(0,1), and prove that every ν∈F0\\F satisfies h(ν)<h0, using the fact that otherwise ν would be an equilibrium state for φ. Then apply Edwards's theorem to f=|∫φ dν| and g' to obtain an affine continuous A with f≤A≤g'. If ψ=φ−A satisfies the two conditions of Lemma 3.1 (F is the equilibrium set for ψ and ∫ψ is maximized on F), then Theorem 1 is correct as stated. If F0 is not a face or the entropy gap fails, the concern is fatal.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 1 (Section 3.1), after defining g with g|F=0 and g|MT\\F=||φ||, the text states 'since F is convex, F is a closed and convex set' and concludes g is concave and weak-* lower semicontinuous. This is false: convex equilibrium sets need not be closed, and the paper's own Example 2.2 exhibits exactly such an F. The lower semicontinuity of g fails at points of cl(F)\\F, so Edwards's theorem cannot be applied as written. This is a genuine gap in the written proof of the main theorem. It is, however, repairable. For any ν∈cl(F)\\F, take μ_i∈F with μ_i→ν. Since ∫φ dμ_i is constant on F (after normalizing ∫φ=0 on F), continuity gives ∫φ dν=0. If h(ν) equalled the constant h0 on F, then ν would be an equilibrium state for φ, contradicting ν∉F. Hence h(ν)<h0. Replacing g by g'(ν)=0 on cl(F) and g'(ν)=||φ|| on MT\\cl(F) yields a lower semicontinuous concave function, provided cl(F) is a face of MT(X) as it should be in a Choquet simplex. The Edwards argument then goes through with an affine continuous A satisfying |∫φ|≤A≤g', and A=0 on cl(F). Thus the stated equivalence is likely true, but the submitted proof needs a substantive correction before the theorem is established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies freezing phase transitions for continuous potentials on general dynamical systems. Its main theorem (Theorem 1) claims that a nonempty set F of invariant measures can be realized as the set of equilibrium states for βψ for every β ≥ 1 (a freezing set) if and only if entropy is constant on F and F is already an equilibrium set for some potential. From this it derives corollaries for single ergodic measures and for upper semicontinuous entropy maps, plus density of freezing potentials (Theorem 4) and genericity of non-freezing potentials under specification (Theorem 5). Additional observations concern analyticity of the pressure function at freezing temperatures and a rapid switching example. The paper is clearly written and the main idea is appealing, but the proof of the main theorem contains a substantive separation-theoretic gap, and a later proof contains an inequality-direction error.","tokens_in":11126,"tokens_out":25972,"duration_ms":295829,"significance":"If the main theorem is established, it gives a complete characterization of freezing sets in arbitrary dynamical systems and directly answers, in broad generality, the question of whether equilibrium and freezing states impose the same restriction on a measure. The Edwards-separation strategy is natural and the corollaries, especially the dense freezing potentials and the residual non-freezing potentials under specification, would be valuable contributions. The manuscript also correctly draws on external theorems (Ruelle, Jenkinson, Bowen, Morris) without circularity. However, the current proof of Theorem 1 is not valid as written, so the advertised generality is not yet established; the gaps appear repairable, but they are load-bearing.","major_comments":[{"comment":"The proof asserts: 'since F is convex, F is a closed and convex set.' This is false in general, and the paper's own Example 2.2 gives a convex equilibrium set that is not closed. Therefore the function g defined by g|F=0 and g|MT(X)\\F=||φ|| need not be weak-* lower semicontinuous, and Edwards' separation theorem cannot be applied as stated. Moreover, concavity of g requires the face property of equilibrium sets, not mere convexity: if F were only convex, two points outside F could have their midpoint in F and concavity would fail. The proof can likely be repaired by replacing F with cl(F), proving that cl(F) is a face of the Choquet simplex, and checking that any ν∈cl(F)\\F has h(ν)<h0 and hence is not an equilibrium state for the constructed potential. This repair is substantive and must be written out before Theorem 1 can be considered proved.","section":"§3.1, proof of Theorem 1"},{"comment":"The proof states: 'by continuity of Ptop and Max(·), we know A = ⋂β>0 Aβ = ⋂β∈Q+ Aβ.' Continuity alone does not justify this equality: a continuous function can satisfy a strict inequality on a dense set yet attain equality at a limit point. Because the intersection over all β>0 is uncountable, A is not automatically a Gδ set. To make the argument work one must use convexity of β↦Pφ(β) and the finite asymptote βMax(φ)+h∞(φ) to show that if Pφ(q)>qMax(φ) for all rational q, then Pφ(β)>βMax(φ) for every real β>0. This argument is absent. In addition, the assertion that Bowen's theorem implies every Hölder potential lies in A is asserted without proof; the needed implication and the relevant reference should be stated explicitly.","section":"§3.3, proof of Theorem 5"},{"comment":"The displayed inequality in the proof has the wrong direction. Since β≥1, the factor (1−β) is nonpositive; because ∫φdµ is the maximum of ∫φdν, it follows that (1−β)∫φdµ ≤ (1−β)∫φdν, not ≥. The statement of the lemma is true and can be proved by writing h(ν)+β∫φdν = h(ν)+∫φdν+(β−1)∫φdν ≤ ∫φdµ+(β−1)∫φdµ = β∫φdµ, but the proof as printed is incorrect. Since Lemma 4.4 is used in Observation 4.5, this error must be corrected.","section":"§4.2, Lemma 4.4"}],"minor_comments":[{"comment":"The proof only computes the right derivative and concludes equality for all equilibrium states. One should also compute the left derivative (for β<β0) and use the existence of the derivative to obtain the equality ∫φdµ = ∂Pφ/∂β(β0).","section":"§4.1, Lemma 4.1"},{"comment":"There is a typo in the displayed inequality: 'Fdνu' should be 'Fdν'.","section":"§3.1, proof of Theorem 1"},{"comment":"The phrase 'there there exists' should be 'there exists'.","section":"Theorem 1 statement"},{"comment":"The text refers to 'Observation 4.4' in the proof, but the relevant statement is Lemma 4.4.","section":"§4.2, Observation 4.5"},{"comment":"The two examples are asserted without proof. Example 2.2 is used later to illustrate the non-closedness issue, so providing at least a brief justification or a reference for the assertions would improve the paper.","section":"§2.2, Examples 2.1 and 2.2"},{"comment":"When defining B = {φ : h∞(φ)=0}, the text says h∞ is obtained by any cluster point of equilibrium states (μβ). This presumes existence of equilibrium states for βφ; under specification existence is standard, but it would be helpful to state this explicitly.","section":"§3.3, proof of Theorem 5"}],"recommendation":"major_revision","confidential_remarks":"The main theorem's proof gap is significant but appears repairable along the lines indicated in the major comments; I do not see grounds for rejection. The paper is within the scope of the journal. The author should also be asked to supply the missing justifications in Theorem 5 and to correct Lemma 4.4."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The central claim is attractive and probably true: a collection of measures can freeze exactly when it is an equilibrium set with constant entropy. Corollary 2 and the density/genericity theorems (Theorems 4 and 5) are genuinely new and useful. But the proof of Theorem 1 is not gap-free, and I don't think the repair is as easy as the stress-test note suggests.\n\nThe proof asserts 'since F is convex, F is a closed and convex set.' That's false, and the paper's own Example 2.2 is a counterexample. The closedness is needed for the lower semicontinuity of the barrier function g in Edwards's theorem. The stress-test says one can work with cl(F) instead, but the proposed g' (0 on cl(F), ||phi|| elsewhere) is not concave in general: a convex combination of two points outside cl(F) can land in cl(F) if cl(F) is not a face, and no argument is given that cl(F) is a face in the non-u.s.c. case. So the separation step, as written and as repaired in the note, doesn't go through.\n\nThat said, the theorem may well be salvageable with a different separation argument, because F is a face for the affine function h+∫phi and its complement is convex. A referee with expertise in Choquet theory could sort this out. The rest of the paper is in better shape. Lemma 4.4 has an inequality direction error, but the result is true with a one-line fix. Theorem 4's proof is fine. Theorem 5's argument is fairly standard, though the claim that Hölder potentials lie in A needs to be checked carefully.\n\nSo my overall take: novel results, serious but repairable-seeming gaps in the main proof, and some minor errors elsewhere. I wouldn't take it as it stands, but I would send it to a good referee, especially one who can evaluate the Edwards step. The author clearly knows the literature and the question is legitimate.","headline":"The central equivalence is a good idea, but the proof of Theorem 1 has a real gap in the Edwards separation step, and the proposed repair doesn't fix it as written.","tokens_in":11676,"tokens_out":21086,"would_cite":false,"duration_ms":192269,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D35","37A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Freezing a system adds no new restriction beyond the equilibrium condition.","keywords":["freezing phase transitions","equilibrium states","zero-temperature limits","entropy map","thermodynamic formalism","invariant measures","specification property","pressure function"],"falsifier":"Take the non-closed equilibrium set $F$ from Example 2.2 (the convex hull of the two maximal-entropy measures on disjoint full shifts, with the fixed point $\\delta_\\infty$ in its closure) and determine whether any continuous $\\psi$ has $F$ as the equilibrium set of $\\beta\\psi$ for every $\\beta \\ge 1$; the absence of such a $\\psi$ would refute Theorem 1 as stated, while its existence would show the closedness gap is repairable.","tokens_in":10601,"feed_emoji":"❄️","tokens_out":15088,"duration_ms":139490,"temperature":0.7,"pith_summary":"The paper aims to prove that freezing a system—arranging that below some temperature the equilibrium states of $\\beta\\varphi$ no longer change—adds no new constraint beyond the ordinary equilibrium condition. Its main theorem says that a non-empty collection $F$ of invariant measures can be frozen by some continuous potential, in the sense that $F$ is the equilibrium set of $\\beta\\psi$ for every $\\beta \\ge 1$, exactly when $F$ is already the equilibrium set of some potential and entropy is constant on $F$. If correct, this makes freezing states and equilibrium states the same class of measures, so any ergodic measure that can appear as an equilibrium state can also be made the unique frozen state at low temperature. It also yields a concrete picture of when freezing potentials are common or rare, and shows that a true freeze is always accompanied by a non-analytic pressure function at the transition temperature.","feed_headline":"Freezing a system adds no new restriction beyond equilibrium","feed_subtitle":"Any equilibrium family with constant entropy can be frozen at low temperature by some potential.","key_machinery":"The key identity is Lemma 3.1: a potential $\\varphi$ freezes on $F$ exactly when $F$ is the equilibrium set for $\\varphi$ and all measures in $F$ attain the same maximal integral $\\int \\varphi\\,d\\mu$. The sufficiency part of Theorem 1 then runs through a separation theorem for the compact convex simplex of invariant measures (Theorem 4 from [6]): after adding a constant so that $\\int \\varphi\\,d\\mu = 0$ on $F$, the proof interpolates a weak-$*$ continuous affine function $A$ between the convex function $f(\\nu) = |\\int \\varphi\\,d\\nu|$ and the concave lower semicontinuous function $g$ that is zero on $F$ and $\\|\\varphi\\|$ off $F$, and sets $\\psi = \\varphi - A$. This makes $\\psi$ maximized precisely on $F$ while preserving the equilibrium condition, so Lemma 3.1 applies at every $\\beta \\ge 1$.","core_discovery":"The central discovery is Theorem 1: for any dynamical system $(X,T)$ with finite topological entropy and any non-empty $F \\subset M_T(X)$, there exists $\\psi \\in C(X)$ with $F$ equal to the equilibrium set of $\\beta\\psi$ for all $\\beta \\geq 1$ if and only if $h$ is constant on $F$ and $F$ is the equilibrium set of some continuous potential. The mechanism is Lemma 3.1, which shows that freezing on $F$ is equivalent to $F$ being an equilibrium set for the same potential while every measure in $F$ maximizes the integral of that potential. In the upper semicontinuous-entropy case the theorem yields freezing on the closed convex hull of any closed, constant-entropy collection of ergodic measures; in particular every ergodic equilibrium state can be frozen. The paper further proves that freezing potentials are dense in $C(X)$ when entropy is upper semicontinuous, that under specification for $\\mathbb{Z}$-actions the non-freezing potentials form a dense $G_\\delta$, and that freezing at a minimal inverse temperature forces $P_\\varphi$ to be non-analytic there.","pith_inferences":["Because the theorem equates freezing states with constant-entropy equilibrium states, a complete classification of which measures are equilibrium states would automatically settle which measures can be frozen; the two questions are one.","A natural test of the theorem's boundary is the non-closed equilibrium set from Example 2.2: determining whether a freezing potential exists for it would decide whether the equivalence needs an explicit closedness hypothesis.","The contrast between density of freezing potentials and genericity of non-freezing potentials suggests that 'most potentials freeze' and 'most potentials do not freeze' are both true in different topological senses; the physically relevant notion of typicality would need a measure on $C(X)$, not just a topology.","The rapid-switching construction (maximal-entropy equilibrium for $\\beta<1$, zero-entropy equilibrium for $\\beta>1$) is a toy model for first-order-like transitions; one could try to build analogous potentials with nonzero entropy on both sides of the switch."],"forward_implications":["For a single ergodic measure, being the freezing state of some potential is equivalent to being an equilibrium state of some potential; the low-temperature regime imposes no extra restriction.","When the entropy map is upper semicontinuous, any closed collection of ergodic measures with constant entropy can be frozen on its closed convex hull, covering lattice models from statistical physics.","Freezing potentials are dense in the uniform topology whenever entropy is upper semicontinuous, so every potential can be perturbed slightly to one that freezes.","For $\\mathbb{Z}$-actions with specification, the typical potential does not freeze: the non-freezing potentials contain a dense $G_\\delta$ even though freezing potentials are dense.","A potential that first freezes at inverse temperature $\\beta_0$ exhibits a phase transition: the pressure function is not analytic at $\\beta_0$."],"supporting_citations":[{"why":"Supplies the separation theorem for compact convex simplices of invariant measures used to construct the freezing potential.","marker":"[6]"},{"why":"Provides the realization theorem that closed collections of ergodic measures are equilibrium sets, used directly in Corollary 3.","marker":"[9]"},{"why":"Shows under upper semicontinuous entropy that ergodic measures can be realized as equilibrium states, supplying the existence input behind Corollary 2.","marker":"[15]"},{"why":"Contributes the result that a dense class of regular potentials has unique equilibrium states, used to show non-freezing potentials are dense under specification.","marker":"[1]"},{"why":"Supplies the dense $G_\\delta$ result for zero residual entropy in ergodic optimization, providing the set $B$ in Theorem 5.","marker":"[14]"}],"fun_headline_variants":["Freezing states: no extra burden over equilibrium","Low temperature freezing equals equilibrium reach","Equilibrium states can always freeze","Freezing adds no new restrictions to equilibrium","Any equilibrium set can be frozen with a potential"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the 'if' direction in Theorem 1 assumes that a convex set of equilibrium states is automatically closed, because the separation theorem it uses requires a lower semicontinuous bounding function; the paper's own Example 2.2 gives a convex equilibrium set that is not closed when the entropy map is not upper semicontinuous, so the general theorem rests on this unproved closedness.","fun_headline_variants_meta":{"raw":{"variants":["Freezing states: no extra burden over equilibrium","Low temperature freezing equals equilibrium reach","Equilibrium states can always freeze","Freezing adds no new restrictions to equilibrium","Any equilibrium set can be frozen with a potential"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000381,"raw_usage":{"total_tokens":2055,"prompt_tokens":1014,"completion_tokens":1041,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":977}},"tokens_in":630,"tokens_out":1041,"duration_ms":10544,"temperature":1.0,"reasoning_tokens":977,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:32:12.467631+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the non-closed equilibrium set $F$ from Example 2.2 (the convex hull of the two maximal-entropy measures on disjoint full shifts, with the fixed point $\\delta_\\infty$ in its closure) and determine whether any continuous $\\psi$ has $F$ as the equilibrium set of $\\beta\\psi$ for every $\\beta \\ge 1$; the absence of such a $\\psi$ would refute Theorem 1 as stated, while its existence would show the closedness gap is repairable.","supporting_citations":[{"cited_title":"On separation and approximation of real func tions defined on a Choquet simplex","cited_arxiv_id":null,"evidence_quote":"Supplies the separation theorem for compact convex simplices of invariant measures used to construct the freezing potential."},{"cited_title":"Every ergodic measure is uniquely max imizing","cited_arxiv_id":null,"evidence_quote":"Provides the realization theorem that closed collections of ergodic measures are equilibrium sets, used directly in Corollary 3."},{"cited_title":"Thermodynamic formalism: the mathematical structure of equilibrium statistical mechanics","cited_arxiv_id":null,"evidence_quote":"Shows under upper semicontinuous entropy that ergodic measures can be realized as equilibrium states, supplying the existence input behind Corollary 2."},{"cited_title":"Some systems with unique equilibrium state s","cited_arxiv_id":null,"evidence_quote":"Contributes the result that a dense class of regular potentials has unique equilibrium states, used to show non-freezing potentials are dense under specification."},{"cited_title":"Ergodic optimization for generic continu ous functions","cited_arxiv_id":null,"evidence_quote":"Supplies the dense $G_\\delta$ result for zero residual entropy in ergodic optimization, providing the set $B$ in Theorem 5."}],"review_version":1}