{"id":"a891d08b-f502-4642-985c-842799e4487f","arxiv_id":"2512.08553","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Supercritical L± lines in asymmetric fluids obey the same singular diameter scaling as the subcritical coexistence curve, with antisymmetric correction terms.","lead":"This paper shows that the average of the two supercritical boundary lines L+ and L− in fluids bends away from the old straight-line law, with the same singular corrections already known for the subcritical coexistence curve. The result makes fluid asymmetry measurable in a wide supercritical window and may guide searches for critical fluctuations in QCD.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The antisymmetric-coefficient pattern in Eq. (7) rests on the unproved assertion θ+ = -θ− in SM S1C; the extremum condition ∂κT/∂θ=0 is not actually solved.","rationale":"The reader's weakest assumption identifies exactly the step on which the antisymmetric-coefficient pattern depends. The entire theoretical prediction of Eq. (8) is built on Eq. (7), and Eq. (7) is obtained from θ+ = -θ− in SM S1C. That statement is asserted rather than derived from the extremum condition. My own analysis suggests the antisymmetry is plausible to first order in the field-mixing parameters, because the leading asymmetric corrections to the susceptibility are odd under simultaneous reversal of h1 and θ, but the paper does not supply this derivation or qualify the order. Since this is a missing proof rather than a demonstrated inconsistency, it does not warrant rejection; it warrants a conditional acceptance pending a direct check. The reader's CONDITIONAL verdict is therefore unchanged.","tokens_in":17722,"tokens_out":36536,"duration_ms":339036,"concrete_test":"Numerically solve ∂κT/∂θ=0 using SM Eq. (S23) for a representative asymmetric parameter set (e.g., a3=1, b2=0.1, tanφ=0.3) over h1 ∈ [10^-6, 10^-1]. Compute θ+(H) and θ-(-H) and test whether θ+(H) = -θ-(-H) to first order in the mixing parameters. Then compute A1,P^+/A1,P^- and A1,ρ^+/A1,ρ^- from Eqs. (S29)-(S30); if either ratio deviates from 1 at the order of the ΔT^{Δ-1} correction, re-derive Eq. (8).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result, Eq. (8), follows from the coefficient ratios in Eq. (7): A1,P^+/A1,P^- = 1, A1,ρ^+/A1,ρ^- = 1, etc. These ratios are derived in SM S1C from the statement 'Based on the symmetric consideration, we should have θ+ = -θ−.' This statement is load-bearing and is not derived from the extremum condition that defines L±. The susceptibility in SM Eq. (S23) contains terms with different parity in θ once field-mixing terms are included. To first order in the mixing parameters (a3, b2, tanφ), the perturbing terms are odd and the antisymmetry can likely be shown, but the manuscript does not provide that derivation and does not state the order to which the result holds. If θ+(H) ≠ -θ-(-H) at the order of the ΔT^{Δ-1} or ΔT^{1−α} corrections, the cancellation pattern in Eq. (7) acquires extra contributions, and the specific exponents/coefficients in Eq. (8) would mix with other terms. The empirical fits in Fig. 3 have no error bars and exclude hydrogen, so they cannot discriminate a small θ-asymmetry from the predicted antisymmetric pattern.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a supercritical-subcritical correspondence in which the supercritical L± boundary lines are the mirror image of the subcritical coexistence curve. Using linear scaling theory together with complete (linear field-mixing) scaling, the authors derive scaling corrections to the L± loci and predict that the supercritical diameter ρ_d^> = (ρ+ + ρ−)/2 violates the rectilinear law and follows Eq. (8), with terms ∼ |ΔT|^{2β} and ∼ |ΔT|^{1−α}; a corresponding pressure diameter is also predicted. They test the resulting ratio relations, Eq. (9), against NIST fluid data for six fluids and against a two-state mean-field model for a liquid-liquid transition, and they extend the same framework to lines defined by higher-order cumulants of the particle-number distribution. The paper is clearly written and the overall structure—derivation, empirical test, model test, extension—is appropriate for the journal.","tokens_in":18202,"tokens_out":7945,"duration_ms":78098,"significance":"If the central claim holds, the paper extends the known singular-diameter physics from the subcritical coexistence curve to the supercritical region, where the scaling regime appears to be wider and thus more easily testable. This would be a valuable contribution to the long-standing discussion of asymmetry in fluid criticality. The derivation is based on established linear-scaling and complete-scaling ideas rather than on a purely empirical collapse, and the paper includes falsifiable predictions in Eq. (9) and auxiliary checks from a two-state model and cumulant lines. The main risk is the unproved relation θ+ = −θ− at the extremal points that define L±; the empirical support, while suggestive, is fit-based and excludes hydrogen, which visibly deviates. With a proper derivation of the extremal-angle relation and a more quantitative data analysis, the paper could be a solid contribution.","major_comments":[{"comment":"The coefficient-ratio pattern in Eq. (7), which is the load-bearing input for Eq. (8), rests on the statement in SM S1C: \"Based on the symmetric consideration, we should have θ+ = −θ−\". This is asserted, not derived. The angles θ± are defined by the susceptibility-maximum condition ∂κ_T/∂θ = 0 along constant-h1 paths (SM Eqs. S23–S24). Once field-mixing corrections are included, the expansion in Eq. (S23) contains terms of different parity in θ (for example the C-term is odd in θ), so the two extrema need not be exactly opposite at the order of the retained corrections. The authors must solve the extremum condition to the required order, or explicitly state the approximation to which θ+ = −θ− holds. If the relation fails at order Δ^{β} or Δ^{1−α}, the specific terms in Eq. (8) will mix with other corrections and the central prediction is not established.","section":"SM S1C and main-text Eq. (7)"},{"comment":"The quantitative validation is not yet convincing. The quoted values kP = 0.27, kρ = 0.191, mρ = 0.07 are single fitted amplitudes with no uncertainties, no goodness-of-fit measure, and no treatment of the strong correlations between adjacent NIST data points. The data collapse in Fig. 3 is visually good, but the exponents are fixed to 3D-Ising values, so the fits only test the amplitudes; the paper should show that the data actually discriminate the predicted exponent combinations (e.g., 1−1/Δ versus plausible alternatives). In addition, hydrogen (Appendix B) deviates from the collapsed behavior without explanation. Since the paper claims universality of the coefficients for the six fluids, the exclusion of H2 and its failure need to be addressed quantitatively, not just deferred.","section":"Main text after Eq. (9), Fig. 3, Appendix B"},{"comment":"The paper states in the text after Eq. (9) that all coefficients in Eqs. (8) and (9) are universal for the six fluids, in contrast to the system-dependent subcritical coefficients. However, the explicit SM expressions for kP, kρ and mρ depend on the linear-scaling amplitudes a, k and on the field-mixing coefficients a3, b2, tanφ. No argument is provided for why these system-dependent parameters should combine into universal values. If the intended claim is only empirical, it should be stated as such; if the theory predicts universality, the derivation must show that the dependence cancels after imposing the extremum condition. As written, the theoretical basis for the fitted universal amplitudes is missing.","section":"Main text \"universal coefficients\" vs SM Eqs. (S32)–(S33)"}],"minor_comments":[{"comment":"The sentence \"Next, we show that θ is a constant along the L± lines\" is followed only by a statement that, to lowest order, the solution is independent of h1. The derivation of the extremum itself is not shown even at leading order; this is closely related to major comment 1 and should be made explicit.","section":"SM S1B"},{"comment":"The figure caption does not indicate which symbol corresponds to which fluid; a legend or a list in the caption would improve reproducibility. Also, the solid black lines are described as fitting curves, but no fit range or number of data points is given.","section":"Fig. 3"},{"comment":"The two-state model is tested only at mean-field level, where β/Δ = 1−1/Δ = 1/3, so the two correction terms in the density ratio have the same exponent. This is a consistency check for the leading correction but does not test the nontrivial antisymmetric pattern of Eq. (7) beyond the mean-field approximation.","section":"Appendix D"},{"comment":"The higher-order-cumulant analysis uses EOSs generated from the same linear scaling theory with arbitrarily chosen parameters (a = k = 0.1, φ = 30° or 82°). It is a useful consistency demonstration, but it is not an independent experimental validation; the text should make this clear.","section":"SM S2"},{"comment":"The term \"antisymmetric\" is used for coefficients whose +/− ratio is +1 (i.e., terms that do not cancel in the diameter). This is nonstandard and may confuse readers; a brief definition at first use would help.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely and interesting question and the central structure is plausible, but the missing derivation of θ+ = −θ− is a true load-bearing gap: without it, Eq. (7) and therefore Eq. (8) are conditional. The empirical validation also needs to be made quantitative and to face the hydrogen deviation. I would be willing to review a revision that adds the missing extremum calculation and a more rigorous data analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth a serious referee. The core claim — that the supercritical L± lines in asymmetric fluids inherit the same singular diameter as the subcritical coexistence curve, with antisymmetric correction coefficients — is new, and the six-fluid data collapse in Fig. 3 is striking. The derivation from Schofield linear scaling plus complete scaling is coherent, and the explicit expressions in the SM give the reader something concrete to check. The authors also make a sensible practical point: the supercritical scaling window is wider than the subcritical one, which is why the effect is visible in NIST data where the rectilinear law still looks good below Tc.\n\nThe soft spots are real but addressable. The stress-test note is correct: the coefficient-ratio pattern in Eq. (7) rests on the statement in SM S1C that θ+ = −θ−, which is not derived from the extremum condition ∂κ_T/∂θ = 0. The susceptibility in Eq. (S23) contains terms with different parity in θ once field-mixing terms are included, and the paper does not show at what order in the mixing parameters the antisymmetry holds. If θ+ and −θ− differ at the order of the corrections, Eq. (8) would mix with other terms. This is a genuine gap, but it looks fillable — a careful perturbation in the mixing parameters should settle it, and the empirical collapse might survive.\n\nThe empirical validation has the usual weaknesses: the amplitudes k_P, k_ρ, m_ρ are fitted without uncertainties, hydrogen deviates and is excluded, and the two-state model check uses the same theoretical framework, so it is consistency rather than independent confirmation. The universality claim for the coefficients in Eq. (9) is therefore plausible but not established. That said, the authors are honest about the hydrogen deviation and the fit-based nature of the analysis.\n\nFor a reader in critical phenomena or supercritical fluid theory, this is a stimulating paper with a testable prediction. It deserves peer review, not a desk rejection, and the referee should ask for a proper derivation of the θ± symmetry and error-aware fits.","headline":"A genuinely new supercritical analog of the singular coexistence-curve diameter, with a coherent complete-scaling derivation but a load-bearing symmetry assumption that is asserted rather than proved.","tokens_in":18576,"tokens_out":1109,"would_cite":true,"duration_ms":12859,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Asymmetric fluids bend the supercritical density mean into a singular scaling law, not a straight line.","keywords":["supercritical-subcritical correspondence","L± lines","rectilinear diameter law","asymmetric scaling corrections","complete scaling","linear scaling theory","critical point universality","higher-order cumulants"],"falsifier":"A direct check is to compute the susceptibility-maximum angles θ± from the full ∂κ_T/∂θ = 0 condition including all mixing terms, without assuming θ+ = −θ−. If θ+ + θ− ≠ 0 at finite ΔT, the ratio pattern in Eq. (7) fails and the supercritical diameter would acquire additional terms with different exponents. Experimentally, a falsifying observation would be a supercritical fluid whose measured ρ_d^> − ρ_c is dominated by an exponent clearly different from both 2β and 1−α, or whose ΔT−/ΔT+ − 1 versus |δP| curve is not described by Eq. (9) with 3D Ising exponents.","tokens_in":17674,"feed_emoji":"⚛️","tokens_out":5216,"duration_ms":50271,"temperature":0.7,"pith_summary":"This paper argues that the singular corrections long studied for the coexistence-curve diameter below a critical point also appear above it. The authors define a supercritical diameter as the mean density of the two supercritical boundary lines L+ and L−, which they propose as the supercritical mirror of the subcritical coexistence curve. Using a linear scaling theory with complete three-field mixing, they derive that this mean deviates from the rectilinear law and scales as ρ_c + d_ρ |ΔT|^{2β} + e_ρ |ΔT|^{1−α}, because certain correction amplitudes are equal rather than opposite on the two branches, so they survive when the branches are averaged. They verify the predicted ratios against NIST liquid-gas data for six fluids and a two-state model of a liquid-liquid transition, and show the same asymmetric scaling appears on the κ3=0 'symmetry line' of the order-parameter distribution. If correct, asymmetric effects are not confined to the subcritical coexistence region and should be easier to observe above Tc.","feed_headline":"Supercritical density mean obeys singular scaling, not a line","feed_subtitle":"The average of the two L± boundary lines carries the same asymmetric corrections as the coexistence-curve diameter.","key_machinery":"The machinery is the linear scaling (parametric) representation of the critical equation of state, in which the ordering and thermal fields are written as h1 = a r^{β+γ} θ(1−θ²) and h2 = r(1−b²θ²), with θ a polar angle and r a distance from the critical point. The L± lines are located by ∂κ_T/∂θ = 0 along paths parallel to the critical isochore, and to leading order their position corresponds to a constant value of θ. Complete scaling is then imposed: the physical fields Δμ, ΔT, ΔP are linear mixtures of h1, h2, h3 with coefficients a3, b2, and tanφ. Expanding the compressibility and density to subleading order in r produces the amplitude-ratio pattern of Eq. (7). The load-bearing identity i","core_discovery":"The central claim is a supercritical-subcritical correspondence: the supercritical L± lines—defined as the loci of compressibility maxima along paths parallel to the critical isochore—act as the supercritical counterpart of the subcritical coexistence curve, and their mean density obeys the same singular scaling law as the subcritical diameter. Concretely, when the physical fields Δμ, ΔT, ΔP are linearly mixed into the scaling fields h1, h2, h3, the amplitudes of the leading Wegner correction terms on the two branches satisfy the ratio pattern of Eq. (7): leading amplitudes are opposite in sign, while certain correction amplitudes are equal (ratio +1) across the two branches. Because equal c","pith_inferences":["If the coefficient-ratio pattern holds generally, the supercritical diameter could serve as a practical assay for complete-scaling amplitudes in fluids where the subcritical coexistence-curve diameter is too narrow to measure; the paper notes this advantage but does not develop it into a standardized protocol.","A testable extension is to compute or measure the individual asymmetry parameters a3, b2, and tanφ for each fluid and check whether their combinations reproduce the fitted k_P, k_ρ, and m_ρ; if the apparent collapse among the six fluids is not reflected in those parameters, the universality may instead reflect a broader corresponding-states relationship.","Simulations of asymmetric lattice gases or simple asymmetric potentials, where the mixing coefficients are known exactly, could decouple the validity of the θ+ = −θ− assumption from the fitting of real-fluid data; the paper does not perform such a check.","The κ3=0 result implies that experimental searches for a 'symmetry line' in supercritical matter—such as heavy-ion collision cumulant measurements—should expect the line to bend with asymmetric scaling exponents rather than with the symmetric Ising prediction."],"forward_implications":["The violation of the rectilinear diameter law is not subcritical-only: the mean density of the L± lines carries the same |ΔT|^{2β} and |ΔT|^{1−α} singular corrections, so asymmetric effects can be probed without crossing the coexistence curve.","Because the supercritical scaling regime appears larger than the subcritical one, high-quality equation-of-state data can be used to test complete-scaling coefficients; the paper demonstrates this on NIST data for O2, N2, SF6, C2H6, Ar, and CO2.","The fitted coefficients k_P, k_ρ, and m_ρ in Eq. (9) are found to be nearly universal across those six fluids, suggesting that the asymmetry parameters a3, b2, and tanφ follow a common pattern in the supercritical regime.","The same framework predicts that the κ3=0 'symmetry line' is not actually symmetric: it scales as P − P_c ∼ |ΔT|^{2Δ−1} and ρ − ρ_c ∼ |ΔT|^{1−α}, a result directly relevant to interpreting higher-order cumulant measurements near a conjectured critical point.","In the two-field-mixing limit only one antisymmetric correction survives, so the observed exponents can be used to diagnose which physical fields are actually mixed in a given system."],"fun_headline_variants":["Antisymmetric corrections govern supercritical L± mean scaling","Supercritical-subcritical correspondence yields singular diameter law","Mean of supercritical L± lines mirrors coexistence curve's diameter","Supercritical L± average breaks rectilinear law","Antisymmetric terms drive singular scaling in supercritical L± mean"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The derivation assumes that the two L± branches sit at exactly opposite parametric angles, θ+ = −θ−, stated in the supplementary material as 'based on the symmetric consideration'; if field-mixing terms shift the susceptibility-maximum angles asymmetrically, the coefficient ratios in Eq. (7) change and the specific exponents in Eq. (8) would mix with other correction terms.","fun_headline_variants_meta":{"raw":{"variants":["Antisymmetric corrections govern supercritical L± mean scaling","Supercritical-subcritical correspondence yields singular diameter law","Mean of supercritical L± lines mirrors coexistence curve's diameter","Supercritical L± average breaks rectilinear law","Antisymmetric terms drive singular scaling in supercritical L± mean"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001659,"raw_usage":{"total_tokens":6413,"prompt_tokens":724,"completion_tokens":5689,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":5609}},"tokens_in":468,"tokens_out":5689,"duration_ms":35962,"temperature":1.0,"reasoning_tokens":5609,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T17:38:25.353294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to compute the susceptibility-maximum angles θ± from the full ∂κ_T/∂θ = 0 condition including all mixing terms, without assuming θ+ = −θ−. If θ+ + θ− ≠ 0 at finite ΔT, the ratio pattern in Eq. (7) fails and the supercritical diameter would acquire additional terms with different exponents. Experimentally, a falsifying observation would be a supercritical fluid whose measured ρ_d^> − ρ_c is dominated by an exponent clearly different from both 2β and 1−α, or whose ΔT−/ΔT+ − 1 versus |δP| curve is not described by Eq. (9) with 3D Ising exponents.","supporting_citations":[],"review_version":1}