{"id":"8ab6d562-c43d-4125-8662-0ce971768885","arxiv_id":"2412.05354","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Local KMS equilibrium states of focusing NLS and Hartree flows on T^d for d=1,2,3 coincide, on mass sublevel sets, with truncated Gibbs measures.","lead":"For focusing nonlinear Schrödinger and Hartree equations on a torus, the paper proves that the natural truncated Gibbs measures are stationary and satisfy a local KMS equilibrium condition, and that, under a regularity assumption, every local KMS state coincides with a truncated Gibbs measure on the mass ball. This is the first characterization of thermal equilibria for these focusing PDEs, where no global Gibbs measure exists.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof of Theorem 2.23 is logically coherent; Lemma 5.4 is delicate but its Case 5 gap is fillable, and the regularity restriction is explicitly scoped.","rationale":"The reader's CONDITIONAL verdict is reasonable and I do not propose changing it. The weakest assumption flagged by the reader, Lemma 5.4, is indeed the most load-bearing step for the converse, because if B_R failed to be H1-connected then Aida's irreducibility theorem could not be applied and Theorem 2.23 would not follow. However, after checking the path constructions, I found no concrete error. The straight-line reduction to w_n is justified by openness of O(u); the parallel cases 1-4 satisfy the mass bounds by monotonicity or by cancellation with a nonzero orthogonal component; and the general Case 5 reduces to a continuous selection problem between the two continuous curves F1 and F2, which is solvable by the standard homeomorphism of the epigraph/hypograph region to a rectangle. The paper leaves this selection implicit, which warrants referee attention but not rejection. The second issue, the regularity assumption ρ ∈ D1,2 ∩ L4, is explicit and restricts the theorem's scope rather than invalidating its statement; the advertised 'all local KMS states' is therefore an overstatement only if one ignores the qualifiers in Theorem 2.23 and Remark 2.24. I also checked the sign flow in Proposition 5.1, the use of Proposition 4.1, and the passage from (5.3) to the constant via Aida's theorem; these are internally consistent. I did not find any internal contradiction or a counterexample to the central claim as formally stated. Hence no adjustment to the reader's verdict is needed, though a referee should fill in the Case 5 selection argument and, ideally, test the regularity assumption's necessity.","tokens_in":81794,"tokens_out":28166,"duration_ms":296393,"concrete_test":"Verify Lemma 5.4 Case 5 by explicit construction: for admissible a = ~M(u) ∈ (-R,R), λ ∈ R, N = ||u_n||^2, W = ||w⊥||^2 with |a + λ(2+λ)N + W| < R, build f(t) = sqrt(F1(t) + δ(t)(F2(t)-F1(t))) with δ continuous, δ(0) = -F1(0)/(F2(0)-F1(0)), δ(1) = (1-F1(1))/(F2(1)-F1(1)), and check F1(t) < f(t)^2 < F2(t) for all t ∈ [0,1]. If a counterexample to this one-dimensional feasibility problem exists, H1-connectedness fails; if not, the proof of Theorem 2.23 is secure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the paper in good faith and could not locate a load-bearing flaw in the central conjecture as stated. The converse Theorem 2.23 is conditional on two explicit hypotheses: the density regularity ρ ∈ D1,2(μ0) ∩ L4(μ0) and the H1-connectedness of the mass sublevel set B_R established in Lemma 5.4. The regularity hypothesis is a genuine scope restriction relative to the phrase 'all possible local KMS equilibrium states' in the abstract and introduction, but Theorem 2.23 and Remark 2.24 state it plainly, so it is a framing caveat rather than an internal inconsistency. The geometric Lemma 5.4 is the most intricate point: for d = 2, 3, B_R is non-convex, and the proof of path connectedness splits into cases depending on λ ∈ (-∞,-2], (-2,-1), and the general case. The general case after (5.24) is not given an explicit path; the authors instead assert the existence of a continuous f satisfying F1 < f^2 < F2 from endpoint and positivity checks. This is fillable: the open region {(t,y) : 0 ≤ t ≤ 1, F1(t) < y < F2(t)} is homeomorphic to a rectangle via Φ(t,y) = (t, (y-F1(t))/(F2(t)-F1(t))), and the endpoint conditions place (0,0) and (1,1) in the correct fibers. I therefore regard the gap as a missing justification, not a demonstrated failure. The remaining steps—Proposition 5.1, the application of Aida's Proposition 5.2, and the product arguments around (5.27)—are consistent with the stated hypotheses. No significant objection identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies local equilibrium statistical mechanics for focusing NLS and Hartree equations on the torus T^d, d=1,2,3. It introduces local Gibbs measures defined with a cutoff on the (renormalized) mass M(u) and proves three main results: Theorem 2.19 shows these measures solve the Liouville equation; Theorem 2.22 shows they satisfy a local KMS condition; and Theorem 2.23 shows that, under the explicit hypothesis μ=ρdμ0 with ρ∈D^{1,2}(μ0)∩L^4(μ0), every local KMS state agrees with the Gibbs weight e^{h_I} up to a constant on the mass ball B_R. The proof of Theorem 2.23 uses Malliavin calculus, Aida's irreducibility theorem for Dirichlet forms, and a connectedness analysis of the mass sublevel set. The paper also gives a self-contained revision of Bourgain's normalizability proof via concentration inequalities.","tokens_in":81914,"tokens_out":12127,"duration_ms":117716,"significance":"If correct, this is the first local KMS characterization for focusing dispersive PDEs and provides a rigorous link between stationary solutions of the Liouville equation and local Gibbs measures. The paper is carefully written: the finite-dimensional heuristic, the Gaussian integration by parts, and the concentration arguments in Appendix B are detailed, and the main characterization is honestly stated with its hypotheses. The primary weakness is the proof of the H^1-connectedness lemma (Lemma 5.4), specifically the general case in Case 5, where an existence statement for a continuous path is asserted rather than proved. Because this lemma is necessary for Theorem 2.23, the completeness of the proof is affected. Within the stated hypotheses, I found no circularity or demonstrated mathematical error.","major_comments":[{"comment":"The proof of the H^1-path connectedness of O(u) in Case 5 (after (5.24)) is not complete. The authors reduce the problem to finding a continuous f on [0,1] with f(0)=0, f(1)=1, and F1(t)<f(t)^2<F2(t) for all t, and they verify only the endpoint and positivity conditions in (5.25). Since Lemma 5.4 is the key geometric input for Theorem 2.23, this is a load-bearing point. Please supply a rigorous construction of f or state a general lemma showing that the conditions (5.25) imply the existence of such a continuous f. For example, one can use the homeomorphism Φ(t,y)=(t,(y-F1(t))/(F2(t)-F1(t))) of the open region between F1 and F2 to a rectangle.","section":"Section 5, Lemma 5.4"}],"minor_comments":[{"comment":"The phrase 'all possible local KMS equilibrium states' overstates the conditional, localized conclusion of Theorem 2.23; please add the assumptions ρ∈D^{1,2}(μ0)∩L^4(μ0) and 'on B_R' to the claim.","section":"Abstract and Section 1"},{"comment":"In equations (4.24) and (4.25), the limit should be δ→1 (as stated in the surrounding text), not δ→0.","section":"Section 4, proof of Theorem 2.22"},{"comment":"The last term contains 'χ′_R(M)'; it should be (χ_R^{(δ)})'(M) to match the preceding terms.","section":"Equation (4.14)"},{"comment":"The displayed formula for f(t) on (t0,1] appears to be missing a division sign; it should read f(t)=((1−t)θ(t))/((1−t0)θ(t0)) f(t0) so that f is continuous at t0 and f(1)=0.","section":"Section 5, Lemma 5.4, Case 4"},{"comment":"Several symbols appear corrupted in the text (e.g., '/BD', '/greaterorsimilar'); these are likely typesetting artifacts and should be cleaned up in the final version.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper does not seem to contain a circular argument; the converse direction is genuinely proven against the stated hypotheses. The main issue is the missing proof detail in Lemma 5.4, Case 5, which is fillable. If the authors supply that, I would support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper proves something new: for focusing NLS/Hartree on T^d, d=1,2,3, the one-way statement from [11] (truncated Gibbs implies local KMS) is complemented by the converse, Theorem 2.23, giving local Gibbs measures as the only local KMS states on the mass ball. Second, the proof is unusually honest about its hypotheses, but the geometric lemma carrying the converse is delicate enough that I would want a referee to verify it before trusting the characterization at full strength.\n\nWhat is actually good: the structure is sound. Proposition 3.3 is a clean finite-dimensional warm-up; the Gaussian integration by parts in Theorem 2.19 is written out with the required integrability checks; Appendix B gives a self-contained concentration-inequality proof of Bourgain's normalizability bound (2.35), including positivity of z. That is real, reproducible work. The reliance on [7,8,11] is programmatic, not circular: the converse is proved against external anchors (Aida's irreducibility, Kusuoka, Bourgain). I read the full text and found no internal contradiction.\n\nSoft spots, in order. The abstract says 'characterize all possible local KMS equilibrium states', but Theorem 2.23 requires rho in D^{1,2}(mu0) cap L^4(mu0) and only pins rho on B_R up to a constant fixed by mass outside B_R (Remark 2.24). That is a real scope restriction, though explicitly stated in the body. Second, Lemma 5.4 (H^1-connectedness of the renormalized mass sublevel set) is load-bearing; in the general case after (5.24) the authors assert existence of a continuous f without giving the path. The stress-test's fill-in via the open rectangle argument is plausible and I think it is fixable, but a referee should ask for the explicit path or a rigorous argument. Third, Lemma 3.14 uses (2.32) with -h_I; that is terse but the cutoff carries the argument, so minor.\n\nWho this is for: people working on invariant measures, KMS states, and statistical equilibria for dispersive PDEs. The arguments are long but directed. I would send it to a serious referee; the main claims are likely correct, and the paper gives enough detail that referee time is well spent. If the referee certifies Lemma 5.4 and the sign cases in Appendix B, I would accept.","headline":"A dense but genuinely new converse result: local KMS states for focusing NLS/Hartree are characterized as local Gibbs measures, modulo explicit regularity and connectedness hypotheses that need referee scrutiny.","tokens_in":82726,"tokens_out":1944,"would_cite":true,"duration_ms":22033,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","37D35","60H07","28C20","35L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For focusing NLS and Hartree equations on the torus in dimensions 1–3, every sufficiently regular local KMS equilibrium state coincides with a local Gibbs measure on the mass ball.","keywords":["KMS states","local Gibbs measures","focusing nonlinear Schrödinger equation","Hartree equation","Gaussian measures","Malliavin calculus","Dirichlet forms","concentration inequalities"],"falsifier":"Find a probability measure $\\mu=\\rho\\,d\\mu_0$ with $\\rho\\in D^{1,2}(\\mu_0)\\cap L^4(\\mu_0)$ satisfying the local KMS condition whose density on $B_R$ is not proportional to $e^{h_I}$. Concretely, the proof would break if the path construction in Lemma 5.4 failed for some $R$: for a configuration $u\\in B_R$ and a direction $w\\in B_R(u)$ with $w=\\lambda u_n + w^\\perp$, the explicit path for $\\lambda\\in(-\\infty,-2)$ or $(-2,-1)$ must stay inside $B_R$; checking those paths on $\\mathbb{T}^3$ with a numerical or analytic computation would settle the claim.","tokens_in":81334,"feed_emoji":"🌀","tokens_out":6679,"duration_ms":209730,"temperature":0.7,"pith_summary":"The paper studies statistical equilibria of focusing nonlinear Schrödinger and Hartree equations on the torus with $d=1,2,3$. Because the focusing nonlinearity makes the global Gibbs measure non-normalizable, the authors work with localized Gibbs measures obtained by cutting off the (renormalized) mass. They prove two directions: every local Gibbs measure is a local KMS state and a stationary solution of the Liouville equation, and conversely every local KMS state whose density is sufficiently regular is a local Gibbs measure on the mass ball $B_R$, up to a normalization constant and up to arbitrary mass outside $B_R$. This gives the first characterization of local thermal equilibria for focusing dispersive PDEs on the torus, and it recovers almost sure global well-posedness for these equations. The interest is that the KMS condition, a standard criterion for thermal equilibrium, singles out exactly the same measures that the Hamiltonian structure suggests.","feed_headline":"Focusing NLS equilibria are exactly its local Gibbs measures","feed_subtitle":"On the torus in dimensions 1–3, the local equilibrium condition pins the density to the Gibbs factor inside the mass ball.","key_machinery":"The argument runs through Malliavin calculus on the Gaussian measure $\\mu_0$ with covariance $A^{-1-s}$, where $A=-\\Delta+1$. The central objects are the renormalized mass $M(u)$ (Wick-ordered $\\|u\\|_{L^2}^2$ in $d=2,3$), the local Gibbs measure with sharp cutoff $\\chi^{(1)}_R(M)$, and the local KMS condition, which integrates the Poisson bracket identity against test functions supported on the mass ball. A Gaussian integration by parts formula shifts derivatives from test functions onto the Hamiltonian, while Aida's irreducibility theorem for Dirichlet forms on infinite-dimensional domains turns the resulting differential equation $\\nabla(e^{-h_I}\\rho)=0$ into constancy of $e^{-h_I}\\rho$. The delicate geometric step is proving that the sublevel set $B_R$ is $H^1$-connected, meaning connected along Cameron-Martin translation directions, even though it is not convex for $d=2,3$; the paper constructs explicit paths handling the parallel-component scaling cases $\\lambda\\in(-\\infty,-2)$ and $(-2,-1)$.","core_discovery":"The central claim is an equivalence theorem. Theorem 2.22 shows that the truncated Gibbs measure $\\mu^{(1)}$ defined by density proportional to $e^{h_I}\\chi^{(1)}_R(M)$ with respect to the Gaussian measure $\\mu_0$ satisfies the local KMS condition of Definition 2.17. Theorem 2.23 proves the converse: if $d\\mu=\\rho\\,d\\mu_0$ with $\\rho\\in D^{1,2}(\\mu_0)\\cap L^4(\\mu_0)$ and $\\mu$ is a local KMS state, then $\\rho(u)=c_0 e^{h_I(u)}$ for $\\mu_0$-almost every $u$ in the mass ball $B_R=\\{u: |M(u)|<R\\}$. Thus, inside the mass ball, local KMS equilibrium states coincide with local Gibbs measures; outside the ball the measure is arbitrary, which is why the constant $c_0$ is fixed by the total mass. Theorem 2.19 adds that these local Gibbs measures are stationary solutions of the Liouville equation, yielding almost sure global existence as a corollary.","pith_inferences":["If the equivalence extends to the full state space, then uniqueness of local equilibria holds only modulo the mass outside the ball; this suggests that ergodicity, if it holds, must be understood relative to the conserved mass, and that mixing may fail because different exterior masses coexist.","The $H^1$-connectedness of the nonconvex mass sublevel sets is a standalone geometric fact that may transfer to other constructions of truncated Gibbs measures, for example in stochastic quantization or in mean-field limits of Bose gases.","A natural testable extension is to replace the sharp mass cutoff by smooth cutoffs or by other conserved quantities; the same differential-equation-plus-irreducibility strategy would predict the same Gibbs form on each connected component of the resulting sublevel set.","The method invites a follow-up: proving ergodicity of the local Gibbs measures for the focusing flows, the second step the authors announce, using the now-complete description of the equilibrium states."],"forward_implications":["Every local Gibbs measure is stationary for the Liouville equation and satisfies the local KMS condition, so the KMS criterion is satisfied by the natural truncated equilibria.","Any sufficiently regular local KMS state has the exponential-of-interaction Gibbs density on the mass ball, so the local thermal equilibrium class is exactly the local Gibbs class in that region.","Focusing NLS and Hartree initial data are almost surely globally well-posed with respect to the local Gibbs measure, recovered here from the Liouville-equation method rather than by direct flow construction.","The characterization holds on the torus in dimensions one, two, and three for both the local and Hartree nonlinearities covered by Assumption 2.10."],"supporting_citations":[{"why":"Supplies Aida's irreducibility theorem for Dirichlet forms on infinite-dimensional domains, which converts the vanishing-gradient equation into constancy on connected components.","marker":"[2]"},{"why":"Provides the Liouville-equation framework from which Corollary 2.20 derives almost sure global existence.","marker":"[7]"},{"why":"Sets up Malliavin calculus, Gross-Sobolev spaces, and the KMS framework, and supplies the Gaussian integration by parts formula used throughout.","marker":"[11]"},{"why":"Gives the d=1 construction and normalizability of truncated Gibbs measures for focusing NLS.","marker":"[17]"},{"why":"Proves normalizability of the local Gibbs measure for the focusing Hartree equation in d=2,3, revisited here via concentration inequalities.","marker":"[19]"}],"fun_headline_variants":["Local KMS states are exactly Gibbs in focusing NLS","KMS-Gibbs equivalence for focusing NLS on tori","Focusing NLS: local KMS states are Gibbs measures","Local equilibrium KMS states pin Gibbs for focusing NLS","Equivalence of local KMS and Gibbs for focusing NLS"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the region of field configurations with renormalized mass below R is connected along the Cameron-Martin directions, so that the differential equation $\\nabla(e^{-h_I}\\rho)=0$ forces one constant density, and that the density is smooth enough to lie in the Malliavin Sobolev space $D^{1,2}\\cap L^4$.","fun_headline_variants_meta":{"raw":{"variants":["Local KMS states are exactly Gibbs in focusing NLS","KMS-Gibbs equivalence for focusing NLS on tori","Focusing NLS: local KMS states are Gibbs measures","Local equilibrium KMS states pin Gibbs for focusing NLS","Equivalence of local KMS and Gibbs for focusing NLS"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000793,"raw_usage":{"total_tokens":3527,"prompt_tokens":1013,"completion_tokens":2514,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":2429}},"tokens_in":629,"tokens_out":2514,"duration_ms":17149,"temperature":1.0,"reasoning_tokens":2429,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:52:03.584797+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a probability measure $\\mu=\\rho\\,d\\mu_0$ with $\\rho\\in D^{1,2}(\\mu_0)\\cap L^4(\\mu_0)$ satisfying the local KMS condition whose density on $B_R$ is not proportional to $e^{h_I}$. Concretely, the proof would break if the path construction in Lemma 5.4 failed for some $R$: for a configuration $u\\in B_R$ and a direction $w\\in B_R(u)$ with $w=\\lambda u_n + w^\\perp$, the explicit path for $\\lambda\\in(-\\infty,-2)$ or $(-2,-1)$ must stay inside $B_R$; checking those paths on $\\mathbb{T}^3$ with a numerical or analytic computation would settle the claim.","supporting_citations":[{"cited_title":"Ammari, S","cited_arxiv_id":null,"evidence_quote":"Provides the Liouville-equation framework from which Corollary 2.20 derives almost sure global existence."},{"cited_title":"Ammari, V","cited_arxiv_id":null,"evidence_quote":"Sets up Malliavin calculus, Gross-Sobolev spaces, and the KMS framework, and supplies the Gaussian integration by parts formula used throughout."},{"cited_title":"Bourgain, Invariant measures for the Gross-Pitaevskii equation , J","cited_arxiv_id":null,"evidence_quote":"Proves normalizability of the local Gibbs measure for the focusing Hartree equation in d=2,3, revisited here via concentration inequalities."}],"review_version":1}