{"id":"4a31acd8-2775-46bc-bcfd-e8a51448a1a1","arxiv_id":"2507.22882","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Equilibrium distributions of coarse observables under non-commuting charges are shown to follow a generalized Boltzmann form with weak values, and anomalous weak values reveal non-classicality at equilibrium.","lead":"This paper extends an observable-based approach to statistical mechanics to systems with non-commuting conserved charges, and shows it can predict equilibrium outcomes of coarse measurements without knowing the energy spectrum. It also finds that such systems can display non-classical behavior, witnessed by anomalous weak values, even at equilibrium.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main numerical support for Eq. (14) is in-sample: Fig. 3 uses εj and qa_j extracted from the same equilibrium data being predicted, so the 'efficient prediction' claim rests on an untested end-to-end linear ansatz (Eqs. 29–30).","rationale":"The concern is not about disagreement with consensus; it is about whether the reported numerics support the strongest claim. The reader identified the linear ansatz as fragile; I agree partly, but the sharper issue is that the main TVD plot never uses that ansatz, so the 'efficient prediction' headline is not directly tested. This is an internal-support problem, not an outside-context objection. The equilibrium equations are derived cleanly and Theorem I.1 (degenerate if and only if non-Abelian symmetry) appears sound in finite dimension; the infinite-dimensional formulation is plausible with the given spectral definitions. The anomalous weak values are an interesting independent observation, and the First Equilibrium Equation check is a legitimate consistency test. Those parts deserve credit. However, Eq. (14) as stated contains εj and qa_j that are defined through Rj and Ra_j at equilibrium; therefore any test that obtains them from the simulated ρ∞ cannot falsify the formula, because the Second Equilibrium Equation is built into the definitions and the fitted multipliers. The only falsifiable version is the one that obtains these quantities from the initial state via Eqs. (29)-(30), or another independent prescription. That version is not evaluated. The manuscript itself flags the lack of derivation of Eq. (29), so the missing test is acknowledged indirectly. A single out-of-sample end-to-end numerical check would settle whether the central claim holds as stated. Because the numerical evidence is incomplete rather than contradictory, no rejection is warranted; conditional acceptance with a required additional test matches the evidence. No code or data are supplied, which makes this missing test impossible for readers to run independently; that reinforces the need for the authors to provide it.","tokens_in":38285,"tokens_out":7360,"duration_ms":92780,"concrete_test":"Reproduce Fig. 3 end-to-end: fix the coefficients in Eqs. (29)-(30) using two or three training values of θ (for example θ=0 and θ=π/8) and use the resulting εj and qa_j, together with β_A and μ_a fitted to the energy/charge constraints, to predict pest_j for held-out θ values (π/16, 3π/16, π/4). Report TVD against pj(t) for each observable and N=12,14,16, and compare with the TVD reported in Fig. 3. Also report the difference between εj from Eq. (29) and εj=Rj(t)/pj(t) on the held-out points. If the end-to-end TVD remains below 1%, the efficient-prediction claim is supported; if it exceeds the Fig. 3 values substantially, the paper should be revised to present Eq. (14) as a consistency relation unless the linear ansatz is independently justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing weakness is that the numerical verification of the central formula does not test the claimed predictive scheme. In Sec. IV.D, to compute pest_j, the authors 'first obtain εj ≈ Rj(t)/pj(t) and qa_j ≈ Ra_j(t)/pj(t)' from the time-averaged evolution, and then fix β_A and μ_a by fitting or by constraints on the same data. Thus Fig. 3's total-variation distance below 1% shows that the equilibrium distribution is close to a Gibbs form whose 'energy levels' and chemical potentials are read off from that very distribution; it is a consistency check of Eqs. (10)/(14), not an out-of-sample prediction. The genuinely predictive route is Eqs. (29)-(30), which estimate εj and qa_j from E, ΔE, qa without spectral data. The authors explicitly state this ansatz is empirical and lacks an analytical derivation, and Fig. 2 validates it only for εj and qa_j, not for the resulting probability distributions. The end-to-end TVD obtained when εj and qa_j are replaced by the linear-ansatz estimates is never reported. Since the abstract and title promise 'efficient predictions,' the missing end-to-end test is the soft spot. If the linear-ansatz errors amplify in pest_j, the central claim reduces to a postdictive parametrization of equilibrium data.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes Observable Statistical Mechanics to Hamiltonians with non-commuting conserved charges. It derives first and second equilibrium equations from a maximum-entropy principle, expresses the solution as a Gibbs-like distribution (Eq. 14) whose effective 'energy levels' and 'charge levels' are the real parts of weak values of H and the charges, and shows that these weak values can be anomalous (non-classical) at equilibrium. It also proves an equivalence between Hamiltonian degeneracy and the presence of a non-Abelian symmetry group (Theorem I.1). Numerical simulations on a non-integrable SU(2)-symmetric spin chain are used to verify the equilibrium equations and to compare the resulting distribution with the non-Abelian thermal state.","tokens_in":38571,"tokens_out":7330,"duration_ms":97473,"significance":"If the framework is correct, it provides a novel structural characterization of equilibrium distributions of coarse observables under non-commuting charges, connects equilibrium physics to weak values and Kirkwood-Dirac quasiprobabilities, and offers a potential route to predictions that avoid full diagonalization of the Hamiltonian. The derivation in Appendix A4 is careful and explicit, Theorem I.1 is clean and correctly proved for both finite and infinite dimensions, and the numerical checks of the First and Second Equilibrium Equations are extensive and supportive. The main weakness is that the 'efficient predictions' advertised in the title and abstract are not tested end-to-end: the numerical validation of Eq. (14) uses quantities extracted from the very equilibrium data being predicted, and the empirical linear ansatz (Eqs. 29-30) that is supposed to make the scheme predictive is validated only for the weak-value real parts, not for the resulting probability distributions.","major_comments":[{"comment":"The numerical verification of Eq. (14) is in-sample. The text states that to compute pest_j the authors 'first obtain εj ≈ Rj(t)/pj(t) and qa_j ≈ Ra_j(t)/pj(t)' from the time-averaged evolution, and then fix β_A and μ_a by fitting or optimization on the same data. Thus Fig. 3 shows that the equilibrium distribution is consistent with a Gibbs form whose level parameters are read off from that very distribution; it does not demonstrate prediction of equilibrium without knowledge of the energy eigenvalues and eigenvectors. The genuinely predictive route, Eqs. (29)-(30), is never propagated to the probability distributions, and no total-variation distance is reported for the end-to-end scheme. I request an out-of-sample test: estimate εj and qa_j via Eqs. (29)-(30) for held-out initial states or observables, compute pest_j, and report the TVD against the time-averaged pj(t).","section":"Sec. IV.D and Fig. 3"},{"comment":"The linear ansatz for εj and qa_j is explicitly admitted to be empirical and to lack an analytical derivation. Fig. 2 validates the ansatz only for εj and qa_j, using two or three data points to fix the coefficients, and it does not quantify how errors in the ansatz propagate into the final probability distribution. Since the 'efficient prediction' claim rests on this ansatz, the paper should either supply a derivation or provide a more extensive out-of-sample validation that includes the resulting distributions, not just the intermediate weak-value estimates.","section":"Sec. IV.C, Eqs. (29)-(30), and Fig. 2"},{"comment":"The validity of Eq. (14) is conditional on small mutual information between the coarse observable and the energy and charges. The paper gives heuristic arguments for this condition, but it never directly verifies the condition in the numerical experiments. Given that this small-mutual-information assumption is load-bearing for the maximum-entropy step, I suggest computing Ieq(A,H) and Ieq(A,Qa) from the simulated data and reporting them alongside the equilibrium-equation checks. This would confirm that the studied observables actually lie in the regime where the few-constraint maximum-entropy treatment is justified.","section":"Sec. II.D and Appendix A2"}],"minor_comments":[{"comment":"The notation '!' before '=' is used nonstandardly, apparently to mean 'is set to zero' or 'should equal'. This is confusing, especially combined with the surrounding equations; please define the notation explicitly or replace it with standard equality/stationarity notation.","section":"Eqs. (9), (10), (16), (18)"},{"comment":"The procedure for determining β_A and μ_a^A is described only vaguely as 'either analytically or via numerical optimization'. Please specify for each observable which method was used, and report the fitted parameter values or the optimization objective, so that the results are reproducible.","section":"Sec. IV.D"},{"comment":"The inset legend states that the grey line 'in this case coinciding with the green and orange lines' is the average energy E(θ). This makes the plot hard to read; using distinct markers or colors for overlapping curves would improve clarity.","section":"Fig. 2 inset"},{"comment":"The charges are defined with a 1/N factor in the main text (Eq. (24)) and in parts of Appendix A5, but the NATS construction in Appendix A5 uses Qa = Σ_i σ_i^a without the 1/N factor. Please flag this rescaling explicitly when moving between the two conventions, as it affects the magnitudes of qa and Δqa and could confuse readers.","section":"Sec. IV.A vs. Appendix A5"},{"comment":"The abstract claims the framework can 'accurately estimate the equilibrium distribution of coarse observables without access to the energy eigenvalues and eigenvectors'. As argued in Major Comment 1, the numerical support for this specific claim is currently missing. Please either temper the claim to what is demonstrated or add the missing end-to-end test.","section":"Abstract and Sec. V"}],"recommendation":"major_revision","confidential_remarks":"The structural result (Eq. (14) and the weak-value connection) is sound and interesting, but the paper oversells the 'efficient prediction' aspect: Fig. 3 is a consistency check, not a predictive test. The missing end-to-end validation of Eqs. (29)-(30) is the key weakness. I would support publication after the authors either supply that test or revise the claims to accurately reflect what is demonstrated. The paper is well-suited to the journal's scope, and the numerical and analytical work is of good quality."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First: this is a real step forward. The extension of Observable Statistical Mechanics to conserved non-commuting charges is new, and the weak-value/Kirkwood-Dirac reading of the equilibrium equations is a genuinely nice insight. Theorem I.1 (degenerate iff non-Abelian symmetry) is elementary but useful, and the infinite-dimensional version in the appendix is handled carefully. The observed anomalous imaginary parts of charge weak values at equilibrium, with the First Equilibrium Equation still satisfied, is a solid numerical finding. The derivation of the generalized equilibrium equations (Appendix A4) is careful, and the authors are explicit that the linear ansatz, Eqs. (29)-(30), lacks an analytical derivation.\n\nThe soft spot is the gap between the framework and the 'efficient predictions' promise. Figure 3 computes epsilon_j and q^a_j from the very equilibrium distribution it then compares against; the sub-1% TVD is a consistency check of Eq. (14), not an out-of-sample test of the predictive scheme. The genuine predictive route, Eqs. (29)-(30), is validated only for epsilon_j and q^a_j themselves (Fig. 2), and the end-to-end TVD using those estimates is never reported. Since the abstract and title stress efficient prediction, missing that end-to-end test is a real omission. Also, the max-entropy justification remains heuristic (small mutual information), and beta and mu are fitted or optimized. No code or data are provided, which makes reproduction harder.\n\nI don't think the core framework is circular; the max-entropy derivation is not based on the answer, and the consistency check is meaningful evidence for the equilibrium equations. The issue is specifically the extra claim about efficient prediction without spectral information. The authors are explicit about the empirical nature of the linear ansatz, so it is a missing test rather than a deceptive one.\n\nThis is a paper for people working on non-commuting charges and generalized thermalization, and for anyone interested in weak values in thermodynamics. It deserves a serious referee; a good referee should ask for the end-to-end test, code/data, and a sharper statement about when the small-mutual-information condition holds.\n\nSend to peer review. I expect conditional acceptance after the predictive scheme is either properly tested or the claims are softened.","headline":"Solid extension of observable statistical mechanics to non-commuting charges, but the headline 'efficient prediction' currently rests on an in-sample consistency check, not the advertised linear ansatz end-to-end.","tokens_in":39172,"tokens_out":2417,"would_cite":true,"duration_ms":26078,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For systems with non-commuting charges, the equilibrium distribution of a coarse observable is a generalized Gibbs law written in terms of weak values of energy and charge.","keywords":["non-commuting charges","observable statistical mechanics","weak values","Kirkwood-Dirac quasiprobabilities","maximum entropy principle","non-Abelian thermal state","equilibration","degenerate Hamiltonians"],"falsifier":"Compute the classical mutual information $I_{\\mathrm{eq}}(A,H)$ and $I_{\\mathrm{eq}}(A,Q^a)$ for a specific coarse observable in an SU(2)-symmetric spin chain, pick an initial state or observable where these are not much smaller than the corresponding Shannon entropies, and check whether the total variation distance between Eq. (14) and the time-averaged distribution stays below 1%; a clear violation would refute the generality of the prediction.","tokens_in":38044,"feed_emoji":"⚛️","tokens_out":8067,"duration_ms":89634,"temperature":0.7,"pith_summary":"This paper tries to show that even when a quantum system conserves charges that do not commute with one another, the equilibrium distribution of a sufficiently coarse observable can be predicted without diagonalizing the Hamiltonian. The prediction is a generalized Gibbs form, Eq. (14), in which the effective single-outcome energy and charge values are the real parts of weak values of H and of the charges conditioned on that outcome. The authors verify the formula numerically on a non-integrable SU(2)-symmetric spin chain, reaching total variation distances below 1% for one- and two-site observables and beating the usual non-Abelian thermal state for two-body observables. They also prove that Hamiltonian degeneracy is equivalent to the existence of non-commuting charges, and that the imaginary parts of the charges' weak values can be nonzero at equilibrium, which signals non-classicality that would be absent if the charges commuted.","feed_headline":"Weak values predict equilibrium under non-commuting charges","feed_subtitle":"Matches spin-chain equilibrium to under 1% and beats the non-Abelian thermal state for two-body observables.","key_machinery":"The machinery is the generalized Gibbs-type equilibrium formula of Eq. (14), built from weak values defined by $O_w(\\rho,A_j) = \\mathrm{Tr}(\\rho O A_j)/\\mathrm{Tr}(\\rho A_j)$. Its real parts, $\\varepsilon_j = \\mathrm{Re}[E_w(\\rho,A_j)]$ and $q^a_j = \\mathrm{Re}[Q^a_w(\\rho,A_j)]$, act as effective outcome-resolved energy and charge entering a maximum-entropy weight; the imaginary parts must satisfy the first Equilibrium Equation via the chemical potentials. The derivation relies on a degeneracy theorem (Appendix A3) that makes the fine-grained outcome probabilities uniform within each highly degenerate eigenspace, on a small-mutual-information argument that justifies using only first moments of $H$ and $Q^a$ as constraints, and on Theorem I.1 linking degeneracy to non-Abelian symmetry.","core_discovery":"The central claim is that observable-level equilibrium in the presence of non-commuting charges is governed by the constrained maximum of the observable's Shannon entropy, and that the resulting distribution is characterized by weak values: $p^{\\mathrm{est}}_j = d_j e^{-\\beta_A \\mathrm{Re}[E_w(\\rho,A_j)] - \\sum_a \\mu_a \\mathrm{Re}[Q^a_w(\\rho,A_j)]}/Z_A$, where $\\mathrm{Re}[E_w]$ and $\\mathrm{Re}[Q^a_w]$ are the real parts of weak values of the Hamiltonian and of each conserved charge postselected on outcome $j$. The paper derives this from the first and second Equilibrium Equations, uses a degeneracy theorem to justify reducing the fine-grained distribution to the coarse one, and numerically confirms the formula for $N = 12, 14, 16$ with errors under 1% in total variation distance. It also proves a structural equivalence: a Hamiltonian is degenerate if and only if its symmetry group is non-Abelian, which links degeneracy directly to non-commuting charges. Finally, because the charges do not commute with the equilibrium state, the imaginary parts of their weak values can be nonzero at equilibrium; the paper shows this happens in about 20% of the explored SU(2)-breaking initial-state cases, witnessing Kirkwood-Dirac non-classicality without violating the Equilibrium Equations.","pith_inferences":["One could test Eq. (14) on a model with an SU(3) or larger non-Abelian symmetry: the paper predicts anomalies in imaginary weak values should appear for appropriate symmetry-breaking initial states, while the real parts may eventually become anomalous too.","If the small-mutual-information condition is the true validity boundary, then deliberately constructing an observable that is finely correlated with an energy window or a charge sector should make Eq. (14) fail by an amount proportional to that mutual information; this would give a quantitative falsifier.","The linear ansatz connecting $\\varepsilon_j$ and $q^a_j$ to first moments of $H$ and $Q^a$ is empirical; an analytical derivation would turn the method into a parameter-free predictive scheme, but until then the method needs a few calibration points, so its practical status is interpolation rather than prediction from first principles."],"forward_implications":["For any sufficiently degenerate observable, the equilibrium distribution is computable from weak values of H and the charges, without energy eigenvectors or eigenvalues.","Because degeneracy and non-commuting charges are equivalent, relaxation effects blamed on degeneracy cannot be separated from non-Abelian symmetry effects.","Nonzero imaginary parts of charge weak values at equilibrium certify that the equilibrium description cannot be captured by a non-contextual classical model, even while the first Equilibrium Equation holds.","For the spin model studied, the method improves on the non-Abelian thermal state precisely where coupling is not weak, which is also where the NATS derivation is not justified.","The numerical agreement improves with system size from N = 12 to N = 16, suggesting the estimate becomes increasingly accurate in the thermodynamic limit."],"supporting_citations":[{"why":"Supplies the Observable Statistical Mechanics framework and the entropy-maximization arguments that this paper generalizes to non-commuting charges.","marker":"[25]"},{"why":"Provides the degeneracy theorem that lets the paper reduce fine-grained outcome probabilities to the coarse distribution $p_j$ in equilibrium equations.","marker":"[27]"},{"why":"Sets out the physics of non-commuting charges and the non-Abelian thermal state, the baseline scenario this work extends.","marker":"[2]"},{"why":"Gives the approximate microcanonical subspace condition and the local non-Abelian thermal state used for numerical comparison.","marker":"[6]"},{"why":"Defines weak values and their retrodictive interpretation, which the paper uses to give meaning to $\\varepsilon_j$ and $q^a_j$.","marker":"[44]"},{"why":"Reviews anomalous weak values and Kirkwood-Dirac quasiprobability non-classicality, the criteria used to witness equilibrium non-classicality.","marker":"[47]"},{"why":"Establishes that anomalous weak values imply the absence of a non-contextual classical model, underpinning the paper's non-classicality claim.","marker":"[50]"}],"fun_headline_variants":["Weak values predict equilibrium despite non-commuting charges","Non-commuting charges: weak values still nail equilibrium","Equilibrium from weak values, even with non-commuting charges","Anomalous weak values at equilibrium mark non-classicality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a coarse observable shares so little mutual information with the energy and charges that maximizing entropy with only the first moments of H and Qa yields the right equilibrium distribution; if that smallness fails for some observable, Eq. (14) would need extra constraints and the prediction would break down.","fun_headline_variants_meta":{"raw":{"variants":["Weak values predict equilibrium despite non-commuting charges","Non-commuting charges: weak values still nail equilibrium","Equilibrium from weak values, even with non-commuting charges","Anomalous weak values at equilibrium mark non-classicality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000187,"raw_usage":{"total_tokens":1368,"prompt_tokens":1025,"completion_tokens":343,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":641,"completion_tokens_details":{"reasoning_tokens":274}},"tokens_in":641,"tokens_out":343,"duration_ms":3698,"temperature":1.0,"reasoning_tokens":274,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:11:55.750839+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the classical mutual information $I_{\\mathrm{eq}}(A,H)$ and $I_{\\mathrm{eq}}(A,Q^a)$ for a specific coarse observable in an SU(2)-symmetric spin chain, pick an initial state or observable where these are not much smaller than the corresponding Shannon entropies, and check whether the total variation distance between Eq. (14) and the time-averaged distribution stays below 1%; a clear violation would refute the generality of the prediction.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the degeneracy theorem that lets the paper reduce fine-grained outcome probabilities to the coarse distribution $p_j$ in equilibrium equations."},{"cited_title":"Majidy, W","cited_arxiv_id":null,"evidence_quote":"Sets out the physics of non-commuting charges and the non-Abelian thermal state, the baseline scenario this work extends."},{"cited_title":"Yunger Halpern, M","cited_arxiv_id":null,"evidence_quote":"Gives the approximate microcanonical subspace condition and the local non-Abelian thermal state used for numerical comparison."},{"cited_title":"Aharonov, D","cited_arxiv_id":null,"evidence_quote":"Defines weak values and their retrodictive interpretation, which the paper uses to give meaning to $\\varepsilon_j$ and $q^a_j$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reviews anomalous weak values and Kirkwood-Dirac quasiprobability non-classicality, the criteria used to witness equilibrium non-classicality."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that anomalous weak values imply the absence of a non-contextual classical model, underpinning the paper's non-classicality claim."}],"review_version":1}