{"id":"fe2bcdad-b3ce-4401-a4d2-ddc33fa365ca","arxiv_id":"2505.16843","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the mean-field spherical model in random fields, the paper classifies metastates and overlap distributions: continuous symmetry breaking for d>=2, with non-self-averaging, replica-symmetry-breaking overlaps when fields are scaled by 1/sqrt(n).","lead":"This paper analyzes an exactly solvable model of many interacting spins with random local fields, showing how the large-volume behavior depends on the spin dimension. It describes how the system randomly selects among a continuum of magnetized states, and how overlaps between two copies show spin-glass-like fluctuations for volume-scaled fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.4 and 5.5 cluster-set equalities omit closure points produced by their own proofs: delta_{(r*)^2} for overlaps and nu^0_Omega for scaled-field Gibbs measures are a.s. cluster points, so the printed sets are not closed and the equalities are false as stated.","rationale":"The reader's rationale already identified the Theorem 5.4/5.5 closure problem as a genuine error and made it the basis for the CONDITIONAL verdict. I agree this is the most load-bearing concern: it is not a matter of an extra technical assumption but a false equality in the stated complete description of cluster points. The paper's own proofs construct the omitted points, so the check is decisive. I therefore do not move the verdict; the paper should remain CONDITIONAL pending the set-closure correction. The d=2 third-moment assumption in Theorem 4.10 is a real limitation of the d=2 Newman-Stein result and is honestly flagged by the authors; it is secondary to the closure error because it is stated as an explicit hypothesis rather than contradicting the displayed theorem. The main metastate, overlap, and non-self-averaging results (Theorems 1.1, 1.2, 4.7, 4.10, 5.3) are not affected by the cluster-set omission, so no REJECT-level concern is warranted.","tokens_in":55318,"tokens_out":13070,"duration_ms":116469,"concrete_test":"Analytically compute the closure of the two displayed families. In Theorem 5.4 take z_k = k e_1 and evaluate R^{ab}_1*(r* gamma_{z_k} tensor r* gamma_{z_k}) on a bounded-Lipschitz test function; the Laplace method gives the limit delta_{(r*)^2}, and the second moment of the finite-z measure differs from (r*)^4, so the atom is genuinely absent from the displayed family. In Theorem 5.5 take z_k = k e_1 and show that integral gamma_{z_k}(dOmega) nu^0_Omega converges to nu^0_{e_1}. Since the paper's own cluster-set definition (2.11) is closed, these limits must be included; if they are, the theorem statements and the corresponding table row need the closure correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central 'complete description' claims include the cluster-point characterizations for the scaled-field model, and those displayed equalities are internally false. In Theorem 5.4 the proof explicitly considers an unbounded subsubsequence S_{n_k}/sqrt(n_k); by compactness of S^{d-1} the directions bS_{n_k} have a subsubsequence converging to Omega, and the Laplace estimate of Lemma 5.3 gives R^{ab}_1*(r* gamma_{S/sqrt(n)} tensor r* gamma_{S/sqrt(n)}) -> delta_{(r*)^2}. This atom is not of the form R^{ab}_1*(r* gamma_z tensor r* gamma_z) for any finite z in R^d, since for finite z the measure has a density on [-(r*)^2,(r*)^2] for d>=2. Because the cluster set is closed by (2.11), the equality in Theorem 5.4 must be replaced by the closure, i.e. the displayed family together with delta_{(r*)^2}. The same defect appears in Theorem 5.5's first bullet: unbounded subsequences concentrate gamma_{S/sqrt(n)} to delta_Omega, so every pure state nu^0_Omega is a cluster point of mu^{h/sqrt(n)}_n, while the displayed finite-z tilt mixtures are non-product measures. The table entry 'Cluster points, lattice/cont. RF {nu^0_z, z in R^d}' inherits the omission unless 'set' is silently read as 'closure', which conflicts with the theorems' equality signs. This is a corrigible error in the statement of results, not a failure of the main overlap-convergence or metastate theorems, but it invalidates the printed descriptions of chaotic size dependence.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Curie--Weiss spherical model with $d$-dimensional vector spins and i.i.d. random fields, for both non-scaled and volume-scaled fields. For non-scaled fields it proves chaotic size dependence: for $d\\ge 3$ the cluster points of the finite-volume Gibbs measures are pure product states indexed by the sphere, while for $d=2$ the cluster set also contains $z$-tilted mixtures indexed by the recurrent set of the disorder random walk. It derives the Aizenman--Wehr metastate as an explicitly computed sphere average of pure states, identifies the Newman--Stein metastate limit in terms of a sphere-projected Brownian motion, and shows that the overlap distribution converges to a trivial atom. For volume-scaled fields it derives a random overlap approximation governed by $\\gamma_{S_n/\\sqrt n}$, and from it concludes non-self-averaging, replica-symmetry-breaking, and non-ultrametric overlap behavior for $d\\ge 2$, together with explicit metastate formulas. The proofs use a finite-volume mixture representation, uniform convergence of shifted microcanonical measures, Laplace concentration estimates, and random-walk limit theorems.","tokens_in":55570,"tokens_out":6854,"duration_ms":58278,"significance":"If the results stand, the paper provides a rigorous, parameter-free demonstration of random symmetry breaking and spin-glass-like overlap behavior in a solvable mean-field vector spin model. The AW metastate density is derived rather than fitted, the scaled-field overlap limit in Theorem 1.2 is a concrete falsifiable prediction, and the proof strategy via mixture representations and random-walk asymptotics is broadly reusable. The main weaknesses are not in the concentration machinery but in the printed cluster-point equalities: several displayed sets are not closed and omit atoms that the proofs themselves produce. These errors affect the advertised 'complete description' of chaotic size dependence, but they are local and repairable.","major_comments":[{"comment":"In Theorem 5.4 the displayed equality $\\mathrm{clust}(R^{ab}_n\\ast(\\mu^{h/\\sqrt n}_n\\otimes\\mu^{h/\\sqrt n}_n)) = \\{R^{ab}_1\\ast(r^*\\gamma_z\\otimes r^*\\gamma_z): z\\in\\mathbb R^d\\}$ is not correct as stated because the right-hand side is not closed, while cluster sets are closed by definition (2.11). The proof itself produces the missing atom: for an unbounded subsequence $S_{n_k}/\\sqrt{n_k}$, compactness of $S^{d-1}$ gives a direction $\\Omega\\in S^{d-1}$, and the Laplace estimate in Lemma 5.3 yields convergence to $\\delta_{(r^*)^2}$. For finite $z\\in\\mathbb R^d$ and $d\\ge 2$, the measure $R^{ab}_1\\ast(r^*\\gamma_z\\otimes r^*\\gamma_z)$ has a density on a non-trivial interval, so $\\delta_{(r^*)^2}$ is not in the displayed family. The equality should be replaced by the closure, equivalently by the displayed family together with $\\delta_{(r^*)^2}$. The same omission occurs in the first bullet of Theorem 5.5: unbounded subsequences make every pure state $\\nu^0_\\Omega$ a cluster point, while the finite-$z$ tilt mixtures are non-product measures, so the set must include $\\{\\nu^0_\\Omega: \\Omega\\in S^{d-1}\\}$. Theorem 5.2, $d=2$ case, has the analogous missing atom $\\delta_{(r^*)^2+\\|y^*\\|^2}$, and the Table 2 cluster-point row inherits the defect if the word 'set' is read literally rather than as 'closure'.","section":"Theorem 5.4; Theorem 5.5; Theorem 5.2"},{"comment":"The $d=2$ results on the Newman--Stein metastate are stated under Assumption A alone, but they rely on Lemma 4.8, whose $o(1)$ claim uses multivariate Berry--Esseen bounds with rate $O(n^{-1/2})$. That rate requires the finite absolute third moment $\\mathbb E|h_j(i)|^3<\\infty$, which is not part of Assumption A. Without this moment assumption, the almost-sure identification of the NS metastate in $d=2$ is not proved. The introduction flags this as a refined assumption, but Theorems 4.10 and 4.11 should state it explicitly, or state the $d=2$ column conditionally. This is load-bearing for the $d=2$ column of Table 1.","section":"Lemma 4.8; Theorem 4.10; Theorem 4.11"}],"minor_comments":[{"comment":"The title contains a spacing typo ('MET AST A TES'), and the abstract's phrase 'continuity of random product states' should read 'continuum of random product states'.","section":"Title and abstract"},{"comment":"For i.i.d. increments uniform on $\\{-1,1\\}^2$, the recurrent set $P$ is the parity sublattice $\\{(a,b): a+b\\text{ even}\\}$, not all of $\\mathbb Z^2$; the table entry $\\{\\bar\\nu^h_z: z\\in\\mathbb Z^2\\}$ and the wording of Remark 4.5 are therefore inaccurate unless a different lattice choice is intended.","section":"Table 1 and Remark 4.5"},{"comment":"Lemma 5.3 has an extra closing parenthesis in the displayed conclusion ('))) = 0'), and the proof of Theorem 5.4 writes $\\delta_{r^{*2}}$ where $\\delta_{(r^*)^2}$ is meant.","section":"Lemma 5.3 and Theorem 5.4"},{"comment":"The sentence 'when we say that $h$ satisfies (A), we mean that Assumption A hold as if the limit ... is vanishing' is confusing, because Assumption A has already been stated for the non-scaled model; this passage should be rewritten to define the scaled-field assumptions explicitly.","section":"Section 5.2, paragraph after Lemma 4.2"},{"comment":"The notation $d_{BL1}$ is used here before it is defined in Section 2.2; a one-line reminder after Theorem 1.2 would improve readability.","section":"Theorem 1.2"}],"recommendation":"major_revision","confidential_remarks":"The skeptical concern about Theorems 5.4 and 5.5 is justified and should be conveyed to the authors: the displayed cluster sets must be replaced by their closures, and the same defect affects the $d=2$ case of Theorem 5.2. These are statement-level errors that affect Table 2 and the abstract's claim of a complete description, but they do not undermine the convergence-in-law results, the AW metastate formulas, or the overlap-limit theorems. The paper is promising and the corrections are local, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a serious mathematical paper that mostly does what it claims, but the stress-test note is right. Theorems 5.4 and 5.5 are printed as equalities to a non-closed family, and the proofs themselves produce the missing limit points. That is a genuine statement-level error, not a matter of taste, and it should be fixed before publication.\n\nWhat is actually new: rigorous, exactly solvable d>=2 vector-spin examples where the Aizenman-Wehr metastate is a continuous mixture of pure states with an explicit covariance-dependent density, where the Newman-Stein metastate is governed by projected Brownian motion, and where volume-scaled fields give non-self-averaging, replica-symmetry-breaking overlaps that are continuous for d>=2 and fail ultrametricity. The d=1 results were largely in the earlier paper by the same first author; the d>=2 analysis, the density rho_P, and the scaled-field overlap results are the new content. The proofs are mostly careful: the mixture representation, Laplace concentration, and random-walk arguments are detailed and convincing, and there are no fitted parameters. The reliance on the d=1 paper for auxiliary tilting-function lemmas is legitimate background support, not circularity.\n\nSoft spots, in proportion. First, the cluster-point statements in Theorems 5.4 and 5.5 are false as written. In Theorem 5.4 the proof takes an unbounded subsubsequence of S_n/sqrt(n); compactness of directions and Laplace concentration give delta_{(r*)^2}, which is not of the form R^ab_1*(r* gamma_z tensor r* gamma_z) for finite z. The same defect appears in Theorem 5.5: unbounded directions give the product pure states nu^0_Omega, not the finite-z tilted mixtures. Since the cluster set is closed by definition (2.11), the displayed equalities should be replaced by closures. This does not undermine the main overlap-convergence or metastate theorems, but it does invalidate the printed descriptions of chaotic size dependence and the table entries. Corrigible, but needs doing.\n\nSecond, the d=2 Newman-Stein identification requires E|h_j(i)|^3 < infinity. The authors flag it explicitly, so it is not hidden, but it is load-bearing for the d=2 column.\n\nWho this is for: people working on metastates, random symmetry breaking, and solvable disordered models. It deserves a serious referee. I would take it conditional on fixing the closure issue and stating the d=2 moment assumptions cleanly. With that fix, this is a solid paper.","headline":"A solid, genuinely new rigorous analysis of metastates and overlaps in a vector-spin model, but two printed cluster-set equalities are false as stated because the proofs themselves produce additional limit points.","tokens_in":56217,"tokens_out":3082,"would_cite":true,"duration_ms":28461,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60G55","60G60","82B20","82B21"],"pacs":[],"model":"deepseek-v4-flash","headline":"For d≥2 spherical ferromagnets in random fields, the asymptotic Gibbs states are explicit random pure states, and with volume-scaled fields the overlap distribution is continuous, non-self-averaging and non-ultrametric.","keywords":["Gibbs measures","random symmetry breaking","metastates","overlaps","non self-averaging","chaotic size dependence","spherical model","random fields"],"falsifier":"Compute, by Monte Carlo, the empirical law of \\hat S_n/\\|S_n\\| for d=2 anisotropic Gaussian fields and compare it with the claimed metastate density \\rho_P(\\$\\Omega$)\\propto\\langle\\$\\Omega$,\\$Sigma^{{-1}}$\\$\\Omega$\\$rangle^{{-1}}$; a persistent mismatch would rule out Theorem 4.7. For the scaled-field overlap, histogram $R^{{a,b}}$_n for d=2 at large n and check both that it is atomless on (r^*)^2[-1,1] and that its shape tracks \\rho_{\\|S_n\\|/\\sqrt n} as the random-walk radius fluctuates.","tokens_in":54990,"feed_emoji":"🧲","tokens_out":8304,"duration_ms":69148,"temperature":0.7,"pith_summary":"This paper aims to give a complete description of how a d-dimensional spherical ferromagnet in a random field chooses its magnetized state as the volume grows, for d≥2. The authors show that, in the ferromagnetic regime, the system exhibits random symmetry breaking: the infinite-volume Gibbs state is random, and the Aizenman–Wehr metastate is supported on a continuum of pure product states \\nu^h_\\$\\Omega$ with an explicit density proportional to \\langle\\$\\Omega$,\\$Sigma^{{-1}}$\\$\\Omega$\\$rangle^{{-d/2}}$. In dimension d=2 the set of almost-sure cluster points is larger, containing also exponential tilts over recurrent values of the field-sum random walk, while for d≥3 only pure states occur. When the random fields are scaled by 1/\\sqrt n, the model shows spin-glass features: the overlap distribution is non-self-averaging and replica-symmetry-breaking, it is continuous for d≥2 and discrete for d=1, and it oscillates chaotically with volume. A sympathetic reader would care because these results turn qualitative spin-glass phenomena into explicit formulas in a model with continuous symmetry breaking.","feed_headline":"Spherical magnet order turns random for d≥2","feed_subtitle":"A solvable model yields explicit chaotic overlap laws and metastates, linking ferromagnets to spin-glass behavior.","key_machinery":"The load-bearing construction is the representation of the finite-volume Gibbs state as an integral mixture \\mu^h_n=\\$\\alpha$^h_n[\\$nu^{{x,y,h}}$_n] over shifted microcanonical product measures, with the mixing measure \\$\\alpha$^h_n exponentially tilted by \\psi^h_n(x,y)=\\frac{\\$\\beta$}{2}\\|x\\|^2+\\$\\beta$\\langle m_n,x\\rangle+\\$\\beta$\\langle s_n,y\\rangle+\\frac{d}{2}\\ln(1-\\|x\\|^2-\\|y\\|^2). Laplace-type concentration forces \\$\\alpha$^h_n to collapse onto the maximizing sphere r^*$S^{{d-1}}$\\times\\{s\\}, reducing the state to the pure Gaussian product \\nu^h_\\$\\Omega$; the direction \\$\\Omega$ is selected by the field-sum random walk S_n, whose CLT, functional CLT, and recurrence/transience properties then feed into the metastate and overlap formulas. The pure states have marginal G_j(i)/\\sqrt\\$\\beta$+r^*\\Omega_j+h_j(i), and the tilted mixtures \\bar\\nu^h_z are sphere-averages of these with weight e^{\\$\\beta$ r^*\\langle z,\\$\\Omega$\\rangle}.","core_discovery":"The central discovery is that the asymptotic Gibbs states of the d-dimensional spherical model with i.i.d. random fields admit explicit formulas in the ordered regime \\|s\\|<1, \\$\\beta$>d/(1-\\|s\\|^2). For non-scaled fields and d≥2, the finite-volume Gibbs measure is asymptotically \\nu^h_{S_n/\\|S_n\\|}, a product of Gaussians magnetized in the direction of the field-sum S_n; the Aizenman–Wehr metastate is \\kappa^h=\\int_{$S^{{d-1}}$}d\\$\\Omega$\\,\\rho_P(\\$\\Omega$)\\,\\delta_{\\nu^h_\\$\\Omega$} with \\rho_P(\\$\\Omega$)\\propto\\langle\\$\\Omega$,\\$Sigma^{{-1}}$\\$\\Omega$\\$rangle^{{-d/2}}$, so mixtures receive zero metastate weight and only pure states are visible. In d=2 the recurrent level sets of the random walk add tilted mixtures \\bar\\nu^h_z to the cluster-point set, while in d≥3 transience kills all mixtures. For volume-scaled fields h/\\sqrt n, the overlap distribution converges in bounded-Lipschitz distance to \\rho_{\\|S_n\\|/\\sqrt n}, defined through \\gamma_z-measures on the sphere; this distribution is atomless and supported on all of (r^*)^2[-1,1] for d≥2, giving replica symmetry breaking and non-self-averaging, and it is ultrametric exactly when d=1. The Newman–Stein metastate is \\$int_0^{1}$ dt\\,\\delta_{\\nu^h_{\\hat B_t}} for non-scaled fields and \\$int_0^{1}$ dt\\,\\delta_{\\bar\\$nu^{0}$_{B_t/\\sqrt t}} for scaled fields, with Brownian motion independent of the local field configuration.","pith_inferences":["If the formulas hold, the same Brownian-projection mechanism should appear in other vector spin systems with quadratic spherical constraints, suggesting that continuous random-field symmetry breaking generically produces continuous, non-ultrametric overlap laws.","The explicit density \\rho_P(\\Omega)\\propto\\langle\\Omega,\\Sigma^{-1}\\Omega\\rangle^{-d/2} is testable by Monte Carlo sampling of \\hat S_n/\\|S_n\\| under anisotropic field distributions; deviations would signal missing finite-size corrections.","The d=2 third-moment assumption appears to be technical; one could test whether the Newman–Stein limit persists for heavy-tailed fields with finite second moments by numerical evaluation of empirical metastates.","The scaled-field construction gives a continuous-spin analogue of spin-glass behavior in which chaotic size dependence and replica symmetry breaking coexist with ferromagnetic order, not with spin-glass disorder."],"forward_implications":["For d≥2, the almost-sure limit points of finite-volume Gibbs measures are fully classified: pure product states in all dimensions, plus tilted mixtures in d=2 indexed by recurrent values of the random walk.","The Aizenman–Wehr metastate assigns zero weight to mixtures, so typical large volumes are pure even when the cluster-point set contains non-product states.","For volume-scaled random fields and d≥2, the overlap distribution is atomless on the full interval (r^*)^2[-1,1], establishing replica symmetry breaking and non-self-averaging in a solvable ferromagnetic model.","Ultrametricity of the overlap distribution holds exactly in d=1 and fails for d≥2, so continuous spin symmetry is what destroys ultrametricity in this model.","The overlap for non-scaled fields is asymptotically trivial, \\delta_{1-d/\\beta}, in every dimension, despite the metastate being spread over many pure states."],"supporting_citations":[{"why":"supplies the one-dimensional spherical-model methods, the tilting-function lemmas and the d=1 results that the d≥2 argument extends.","marker":"[23]"},{"why":"defines the scaled random-field comparison and the spin-glass features (RSB, non-self-averaging, ultrametricity) that Theorem 1.2 and Theorem 1.3 are measured against.","marker":"[8]"},{"why":"introduces the Aizenman–Wehr metastate and the framework of random symmetry breaking used throughout.","marker":"[1]"},{"why":"introduces the Newman–Stein metastate as the large-N limit of empirical measures along volume sequences.","marker":"[29]"},{"why":"provides the functional limit theorem for random walks used to pass from empirical measures of \\hat S_n to the Brownian-time integrals in Theorem 4.10.","marker":"[26]"},{"why":"gives the multivariate Berry–Esseen rate used in Lemma 4.8 to control the empirical fraction of short random-walk increments in d=2.","marker":"[18]"},{"why":"supplies the random-walk recurrence, transience and possible-values facts underlying the d=2 versus d≥3 cluster-point dichotomy.","marker":"[14]"},{"why":"provides the method for identifying cluster points of a recurrent normalized random walk, used in Lemma 4.2.","marker":"[21]"},{"why":"supplies equal-area sphere partitions with small diameter and zero-boundary measure used in approximating Newman–Stein metastates.","marker":"[25]"}],"fun_headline_variants":["Explicit chaos in d≥2 spherical magnet under random fields","Random fields flip spherical magnet into spin-glass behavior","d≥2 spherical magnet: chaotic overlaps, no ultrametricity","Random-field spherical magnet: explicit metastates and spin-glass signs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The d=2 Newman–Stein metastate result assumes the random fields have a finite third moment, because the proof needs a rate-of-convergence estimate that is only available under that assumption; without it, the d=2 column of the results is not established.","fun_headline_variants_meta":{"raw":{"variants":["Explicit chaos in d≥2 spherical magnet under random fields","Random fields flip spherical magnet into spin-glass behavior","d≥2 spherical magnet: chaotic overlaps, no ultrametricity","Random-field spherical magnet: explicit metastates and spin-glass signs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000596,"raw_usage":{"total_tokens":2905,"prompt_tokens":1179,"completion_tokens":1726,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":795,"completion_tokens_details":{"reasoning_tokens":1656}},"tokens_in":795,"tokens_out":1726,"duration_ms":10900,"temperature":1.0,"reasoning_tokens":1656,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:54:57.884160+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, by Monte Carlo, the empirical law of \\hat S_n/\\|S_n\\| for d=2 anisotropic Gaussian fields and compare it with the claimed metastate density \\rho_P(\\$\\Omega$)\\propto\\langle\\$\\Omega$,\\$Sigma^{{-1}}$\\$\\Omega$\\$rangle^{{-1}}$; a persistent mismatch would rule out Theorem 4.7. For the scaled-field overlap, histogram $R^{{a,b}}$_n for d=2 at large n and check both that it is atomless on (r^*)^2[-1,1] and that its shape tracks \\rho_{\\|S_n\\|/\\sqrt n} as the random-walk radius fluctuates.","supporting_citations":[{"cited_title":"Infinite volume gibbs states and metastates of the random field mean-field spherical model","cited_arxiv_id":null,"evidence_quote":"supplies the one-dimensional spherical-model methods, the tilting-function lemmas and the d=1 results that the d≥2 argument extends."},{"cited_title":"Features of a spin glass in the random field ising model","cited_arxiv_id":null,"evidence_quote":"defines the scaled random-field comparison and the spin-glass features (RSB, non-self-averaging, ultrametricity) that Theorem 1.2 and Theorem 1.3 are measured against."},{"cited_title":"Functional limit theorems for random walks","cited_arxiv_id":"1810.06275","evidence_quote":"provides the functional limit theorem for random walks used to pass from empirical measures of \\hat S_n to the Brownian-time integrals in Theorem 4.10."},{"cited_title":"A partition of the unit sphere into regions of equal area and small diameter","cited_arxiv_id":null,"evidence_quote":"supplies equal-area sphere partitions with small diameter and zero-boundary measure used in approximating Newman–Stein metastates."}],"review_version":1}