{"id":"7def4497-1038-42f2-8b23-952ad196be52","arxiv_id":"2608.05027","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In a conserved two-species model with non-reciprocal interactions, structural and dynamic correlations are governed by different correlation lengths, with a new exponent ν_n controlling the dynamic one.","lead":"This paper uses renormalization group calculations to show that in a model of two species with one-way (non-reciprocal) interactions, the length scale controlling structure and the length scale controlling dynamics diverge with different powers near the critical point. This suggests living cells could speed up or slow down internal droplet responses by tuning metabolic activity without changing the droplets' structure.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"NR-regime exponent ν_n is inferred from the OCH crossover scaling hypothesis, not computed at the NR± fixed point; this is the least secure link in the central claim.","rationale":"The paper presents a careful two-loop RG analysis of the conserved NRCH model, and the derivation of the OCH fixed point, its instability, and the crossover length ξ× is internally consistent. Credit is due for the explicit two-loop calculation of β_κ and η, including the non-standard integrals in Appendix D, and for the clear identification of the potential condition leading to the OCH model. However, the central claim of a new dynamic exponent ν_n in the NR regime is not backed by a direct fixed-point calculation at μ=±∞. The scaling hypothesis Eq. (90) is a plausible crossover ansatz, and the leading-order values are consistent with the NR± exponents (γ'_α0 = 0 gives ν_α0 = 1/2 at leading order), but the O(ε²) correction advertised in Table I is not computed. This is precisely the load-bearing step for the claim that ξ_d and ξ_s are governed by different exponents. A two-loop evaluation of γ'_α0 at NR± would settle whether ν_n = 1/2 + O(ε²) holds or whether the exponent receives a larger shift. Since the authors acknowledge the need for such checks, a conditional verdict is appropriate, and my reading does not change that verdict.","tokens_in":38529,"tokens_out":11191,"duration_ms":127411,"concrete_test":"Independently compute the two-loop β-function for the physical non-reciprocal coupling α0 at the NR± fixed point using the dimensional-regularization framework of Section D, i.e., evaluate γ'_α0 = β_j ∂_j ln Z'_α0 at μ=±∞ to O(ε²). Then extract ν_n = 1/(2 - γ'_α0) and compare with 1/2 + O(ε²) from Eq. (94). If γ'_α0 ≠ 0 at O(ε²), or if the pole of the renormalized response function (Eq. B46) yields a length scale inconsistent with the scaling hypothesis Eq. (90), the central claim needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that ξ_d diverges as |σ|^{-ν_n} with ν_n = 1/2 + O(ε²) in the manifestly non-reciprocal regime (Table I, Eqs. (90)-(94)) is not derived from the NR± fixed points that actually control that regime. The exponent ν_n is taken as 1/λ_σ, the eigenvalue of the OCH fixed point, and the large-|s| asymptote G(s) ~ |s|^{-ν_n} is assumed via the scaling hypothesis Eq. (90). At the NR± fixed point (Eq. 125), the authors only give leading-order exponents, γ'_r = 2ε/5 and γ'_α0 = 0, and the O(ε²) correction to γ'_α0 is never computed. Thus the O(ε²) in ν_n is a placeholder, and the entire NR-regime scaling behavior rests on the unverified assumption that the pole of the response function F_d(s,y) in Eq. (B37) obeys the crossover scaling with the OCH eigenvalue. If the NR± fixed point has a different two-loop γ'_α0, the value of ν_n in the NR regime would shift, and the predicted separation of length scales ξ_d ≠ ξ_s could be altered. The paper itself concedes this: 'To verify this conjecture, numerical analysis, higher order perturbative calculations, or non-perturbative approaches, could be used' (Section V).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the non-reciprocal Cahn-Hilliard (NRCH) model near its critical point using perturbative dynamical renormalization group (RG) techniques, with both one-loop momentum-shell and two-loop Callan-Symanzik calculations. The authors identify a line of fixed points (the odd Cahn-Hilliard, OCH, fixed points) that satisfy a potential condition, and a pair of stable manifestly non-reciprocal fixed points (NR±). They argue that the structural correlation length ξ_s always scales with the Wilson-Fisher exponent ν = 1/2 + ε/10, while the dynamic correlation length ξ_d exhibits multiple scaling regimes: an effective-equilibrium regime where ξ_d ∼ ξ_s, and manifestly non-reciprocal regimes where ξ_d ∼ |σ|^{-ν_n} with ν_n = 1/2 + O(ε²), where σ = α0 − θr. The crossover between regimes is controlled by the gap exponent Δ = 1 + ε/5. The paper also derives two-loop results for the β-function of the nonreciprocal stiffness κ and the anomalous dimension η.","tokens_in":38836,"tokens_out":6396,"duration_ms":83327,"significance":"If the results hold, the prediction of two distinct correlation lengths with different scaling exponents in a conserved, non-reciprocal active system is conceptually important and potentially relevant for biological phase separation controlled by metabolic activity. The paper makes a serious technical effort: it presents two complementary RG schemes, provides explicit two-loop integral evaluations in the appendices, and produces falsifiable predictions (e.g., ξ_d ≠ ξ_s in the NR regimes, with a specific crossover exponent). The RG calculation is internally consistent at leading order, and the identification of the OCH fixed line and its instability is a useful contribution. However, the central claim that the NR-regime dynamic exponent is ν_n = 1/2 + O(ε²) rests on a scaling hypothesis around the OCH fixed point, not on a calculation at the NR± fixed points that actually control the NR regime; the paper itself calls for numerical or higher-order verification. This gap materially limits the current strength of the main claim.","major_comments":[{"comment":"The exponent ν_n governing ξ_d in the manifestly non-reciprocal regime is taken from the OCH fixed point (Eq. (94), ν_n = 1/λ_σ with λ_σ = 2), but the NR regime is controlled by the NR± fixed points, not by OCH. At NR±, the paper gives only leading-order exponents (Eq. (126): γ'_r = 2ε/5, γ'_α0 = 0) and does not compute the scaling dimension of the σ direction at that fixed point. The crossover scaling form Eq. (90) and its asymptote G(s) ∼ |s|^{-ν_n} are therefore an assumption, not a derivation from the fixed point that governs the NR regime. Since the separation of length scales and the novel exponent are the central results, the authors must either compute the two-loop exponents at NR± (or otherwise derive the asymptote from the RG flow) or clearly state in the abstract and Table I that ν_n in the NR regime is a conjecture. As written, the claim exceeds what the calculation demonstrates.","section":"§IV B, Eqs. (125)–(126) and Appendix B 3"},{"comment":"The structural correlation length is asserted to scale as ξ_s ∼ ξ_r = r^{-ν} with no dependence on σ. This is justified only by a surmise that the scaling function X_s(y) is O(1) over the relevant range of y. Because the paper's key qualitative conclusion is that ξ_s and ξ_d behave differently, an unverified O(1) assumption for ξ_s is load-bearing. The authors should at least compute the linear-level correction to X_s(y) and argue why it remains O(1) under renormalization, or explicitly frame the structural-length scaling as part of the same conjecture.","section":"Appendix B 3, sentence beginning 'We therefore surmise that X_s ∼ O(1)'"},{"comment":"Table I lists ν_n = 1/2 + O(ε²) as a definite result for the αNR and r(S)NR regimes, while Section V admits that this is a conjecture requiring numerical or higher-order verification. This inconsistency between the presentation in the table and the stated status of the result should be resolved. The authors should either provide a derivation of the NR-regime exponent from the NR± fixed points, or mark the NR-regime entries as conjectural and adjust the abstract and conclusions accordingly.","section":"Table I and Section V"}],"minor_comments":[{"comment":"The notation D_d defined in Eq. (68) is easily confused with the noise amplitude D and the dimension d; consider using a different symbol or explicit wording.","section":"Section III B, Eq. (68)"},{"comment":"The caption refers to a 'yellow-and-black line' and a 'black dashed line' but the colors in the figure are described as black-and-yellow and dashed black; please make the descriptions consistent and unambiguous.","section":"Figure 2 caption"},{"comment":"The definition Δ ≡ 1 + ω_μ ν in Eq. (B24) is followed by Eq. (92) where Δ = ν/ν_n; the equivalence follows from ν_n = 1/λ_σ but is not stated explicitly, which can confuse the reader.","section":"Appendix B 3, Eq. (B24) and Eq. (92)"},{"comment":"Several references are to arXiv preprints (e.g., [20], [53], [58], [60], [128]); if the journal requires published references, please update where possible.","section":"Throughout"},{"comment":"In Eq. (121), the OCH exponents are written as γ*_r = γ*_α0 = 2ε/5, but the text later notes that α0 and r have different eigenvalues; please clarify that these are the flow functions, not the eigenvalues, and that the eigenvalues differ as given in Eqs. (123)–(124).","section":"Section IV B, Eq. (121)"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically detailed and likely of interest to the journal's readership. The main concern is that the central NR-regime exponent is derived from a crossover scaling hypothesis around the OCH fixed point rather than from the NR± fixed points that control the regime; this is a load-bearing gap that should be addressed. The extensive reliance on the authors' own prior results (refs. [90,91]) for the OCH interpretation is acceptable, but the new exponents should stand on their own. I would not reject the manuscript, but I would require either a two-loop calculation at NR± or a clear and prominent framing of the NR-regime exponents as a conjecture."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious RG paper that gives a credible two-length-scale scenario for conserved non-reciprocal dynamics. The structural length ξ_s keeps the Wilson-Fisher exponent, while the dynamical length ξ_d can diverge with a new exponent ν_n. That separation is the real news, and it is argued with real machinery: momentum-shell and Callan-Symanzik RG agree at one loop, and the two-loop β_κ and η are computed with dimensional regularization. The OCH fixed point and its unstable direction are clearly identified. Credit where due: the authors also flag the weak points themselves.\n\nThe soft spots are real but not disqualifying. First, ξ_s ~ r^{-ν} rests on a surmise that the scaling function X_s is O(1) (Appendix B 3). That is a gap in the structural part. Second, and more important, ν_n in the manifestly NR regime is taken from the OCH eigenvalue via a crossover scaling hypothesis; it is not computed directly at the NR± fixed points. The O(ε²) correction in ν_n is therefore a placeholder, and a different two-loop γ'_α0 at NR± would shift the predicted separation. The stress-test note gets this right. Third, there is no numerical or non-perturbative check, and the authors concede exactly that in Section V. Finally, the paper does not compare with the closely related prior preprint [130], which is an omission the reader should fix before publication.\n\nNone of this kills the central picture. The leading-order scenario—two lengths, new exponent—is internally consistent and the calculations are transparent enough to audit. The paper deserves a serious referee. I would send it out, with instructions to press on the NR± two-loop calculation and the comparison with [130].","headline":"Serious RG paper with a plausible two-length-scale picture; the new dynamic exponent is real at leading order but the O(ε²) claim rests on an unverified scaling-hypothesis step.","tokens_in":39372,"tokens_out":1747,"would_cite":true,"duration_ms":22295,"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":"Near the critical point of a non-reciprocally coupled conserved mixture, static and dynamic correlations are controlled by two different diverging length scales: a Wilson-Fisher length set by temperature and a new dynamic length set by…","keywords":["non-reciprocal Cahn-Hilliard model","critical dynamics","dynamic correlation length","renormalization group","odd mobility","Wilson-Fisher exponent","time-reversal symmetry breaking","active matter"],"falsifier":"Simulate the NRCH Langevin equations numerically near the critical point in $d=3$ and measure the decay of the equal-time correlation $C^{ET}(x)$ and of the total linear response $\\chi^T(x)$; if $\\xi_d$ does not scale as $r^{-\\nu_n}$ or $|\\alpha_0-\\theta r|^{-\\nu_n}$ with $\\nu_n \\approx 1/2$ while $\\xi_s \\sim r^{-\\nu}$ with $\\nu \\approx 0.6$, the central claim fails. A cheaper check is to recompute the two-loop integrals $H_1$ and $H_2$ with an independent regularization scheme and verify that the OCH fixed point remains unstable with $\\omega_\\mu = 2\\epsilon/5$.","tokens_in":38303,"feed_emoji":"🧪","tokens_out":8780,"duration_ms":89308,"temperature":0.7,"pith_summary":"This paper studies the non-reciprocal Cahn-Hilliard (NRCH) model, a minimal description of two conserved species that interact without action-reaction symmetry, close to its phase-separation critical point. It argues that static and dynamic correlations are controlled by two distinct diverging length scales: the static correlation length $\\xi_s$ follows the equilibrium Wilson-Fisher exponent $\\nu = 1/2+\\epsilon/10$, while the dynamic correlation length $\\xi_d$ diverges with a new exponent $\\nu_n = 1/2 + O(\\epsilon^2)$ in the manifestly non-reciprocal regime. The non-reciprocal coupling $\\alpha_0$, set by catalytic reaction rates, acts as a second tuning parameter alongside temperature, so a system can change its dynamic response without changing its structure. The paper's significance is that it identifies a concrete mechanism—broken time-reversal symmetry combined with a conservation law—by which living or active materials could control the speed of phase separation and pattern formation using metabolic activity.","feed_headline":"Non-reciprocal activity gives critical dynamics a new exponent","feed_subtitle":"Static and dynamic correlations diverge differently, so activity can tune response speed without changing structure.","key_machinery":"The load-bearing object is the pair of correlation lengths defined from the equal-time correlation function and from the total linear response, $\\xi_s$ and $\\xi_d$. In the linear theory they satisfy $\\xi_s = \\sqrt{K/r}$ and $\\xi_d = \\sqrt{(K+i\\beta_0)/(r+i\\alpha_0)}$, so any nonzero non-reciprocal coupling makes them differ. The argument is carried by a response-field path-integral representation of the NRCH Langevin dynamics, renormalized with a momentum-shell scheme at one loop and a Callan-Symanzik scheme at two loops, using dimensional regularization. The key identity is the potential condition $f_{\\rm eq} \\propto f_{\\rm nr}$, i.e. $\\alpha_1/u = \\beta_0/K = \\alpha_0/r = \\theta$, which defines the odd Cahn-Hilliard (OCH) model with odd mobility $\\Upsilon_{ab} = \\delta_{ab} + \\theta \\varepsilon_{ab}$; on this line the system has a Boltzmann-like steady state and a fluctuation-dissipation-like identity. The central result follows from the eigenvalues of the RG flow at the OCH fixed point: $r$ and $\\sigma = \\alpha_0 - \\theta r$ are both relevant directions, with $\\lambda_r = 2 - 2\\epsilon/5$ and $\\lambda_\\sigma = 2$, producing the scaling hypothesis $\\xi_d(\\alpha_0,r) = r^{-\\nu} G_{\\sigma r}(\\sigma/r^\\Delta)$ with gap exponent $\\Delta = \\nu/\\nu_n = 1+\\epsilon/5$.","core_discovery":"Using one-loop momentum-shell and two-loop Callan-Symanzik renormalization group calculations, the paper shows that the NRCH critical point is characterized by two correlation lengths. The equal-time correlation function gives $\\xi_s \\sim r^{-\\nu}$ with the Wilson-Fisher exponent $\\nu = 1/2 + \\epsilon/10$ and anomalous dimension $\\eta = \\epsilon^2/50$, matching equilibrium model B. The linear response function instead gives $\\xi_d$, which in the effective-equilibrium regime also scales as $r^{-\\nu}$, but in the manifestly non-reciprocal regime scales as $r^{-\\nu_n}$ or $|\\alpha_0-\\theta r|^{-\\nu_n}$ with the new exponent $\\nu_n = 1/2 + O(\\epsilon^2)$. This difference is a direct signature of broken time-reversal symmetry: the fluctuation-dissipation theorem no longer ties correlation and response together. The RG analysis identifies an odd Cahn-Hilliard (OCH) fixed point, where the dynamics obeys the potential conditions and has a Boltzmann-like steady state with odd mobility, but this fixed point is unstable; generic systems flow to stable non-reciprocal fixed points NR$\\pm$, at which $\\alpha_0$ scales as its own relevant direction. Because $\\nu_n < \\nu$, the dynamic correlation length grows more slowly than the static one, and the crossover between regimes occurs at a length $\\xi_\\times \\sim |K\\alpha_0/(\\beta_0 r)-1|^{-1/\\omega_\\mu}$ with $\\omega_\\mu = 2\\epsilon/5$.","pith_inferences":["Editorial extension: if $\\nu_n = 1/2 + O(\\epsilon^2)$ holds beyond two loops, then in $d=3$ the dynamic correlation length diverges with a smaller exponent than the static one, so near criticality the response function would appear short-ranged even where equal-time correlations look critical; a measurement separating these two decays would be a sharp test.","Editorial extension: the same two-length mechanism should appear in any conserved field theory with broken time-reversal symmetry, for example active model B+ or chiral active matter; the paper's potential-condition analysis suggests the OCH line is the only place FDT-like behavior survives.","Editorial extension: the crossover length $\\xi_\\times$ could be probed experimentally in chemically active colloid or enzyme mixtures by measuring the linear response and the static structure factor simultaneously and locating the scale where their decay lengths split.","Editorial extension: a higher-order or non-perturbative RG check of the two-loop integrals (the $\\beta_\\kappa$ function and $\\eta$) is the natural next step; if those integrals are wrong, the stability of the OCH fixed point and the value of $\\nu_n$ would change, but the qualitative two-length picture would likely survive."],"forward_implications":["In the manifestly non-reciprocal regime, the dynamic correlation length diverges with exponent $\\nu_n = 1/2+O(\\epsilon^2)$, so the response length and the static structure length no longer coincide; this is an observable time-reversal-breaking signature.","The non-reciprocal coupling $\\alpha_0$ is a relevant tuning parameter, meaning catalytic reaction rates can drive the system across its critical region just as temperature does.","Since the dynamic exponent is $z = 4 - \\eta$ with $\\eta = \\epsilon^2/50$, the response time $T_d \\sim \\xi_d^z$ is extremely sensitive to the tuning parameters, so a modest change in $\\alpha_0$ can sharply speed up or slow down the response.","In the effective-equilibrium regime the model recovers model B scaling ($\\xi_d \\sim \\xi_s \\sim r^{-\\nu}$, FDT satisfied), so the new behavior is confined to scales beyond the crossover length $\\xi_\\times$.","The OCH fixed point describes a conserved, equilibrium-like odd-mobility dynamics with a Boltzmann steady state, but it is unstable; generic conserved non-reciprocal systems flow to the NR$\\pm$ fixed points at large scales."],"supporting_citations":[{"why":"Introduced the NRCH model as a minimal description of conserved species with non-reciprocal interactions.","marker":"[42]"},{"why":"Independently introduced the NRCH model and its phase-separating and pattern-forming behavior.","marker":"[43]"},{"why":"Provides the non-conserved complex Ginzburg-Landau counterpart whose non-reciprocal coupling is irrelevant, the contrast case for the conservation-law effect.","marker":"[3]"},{"why":"Companion two-loop field-theory study of the same non-conserved model, supplying techniques and the comparison for the fixed-point structure.","marker":"[4]"},{"why":"Defines equilibrium model B and its dynamical exponent, the baseline the NRCH reduces to in the effective-equilibrium regime.","marker":"[87]"},{"why":"Demonstrated effervescence, the sign change of effective non-reciprocity, which the paper's rSNR regime extends to the critical region.","marker":"[89]"},{"why":"RG treatment of non-conserved non-reciprocal (CGLE) criticality, showing flow to the equilibrium Wilson-Fisher fixed point; provides the contrast for the conserved case.","marker":"[77]"},{"why":"Derives the fluctuation-dissipation-like identity for odd conserved dynamics that the OCH fixed point obeys.","marker":"[90]"}],"fun_headline_variants":["Two correlation lengths diverge differently near non-reciprocal criticality","New exponent for dynamic correlations in non-reciprocal critical dynamics","Odd Cahn-Hilliard fixed point yields distinct dynamic scaling exponent","Activity controls response speed without altering static structure","Broken time-reversal splits static and dynamic critical exponents"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The two-loop renormalization-group calculation—in particular the flow of the non-reciprocal stiffness $\\beta_0$ and the anomalous dimension $\\eta$—is assumed correct without independent numerical or experimental verification; an error in those integrals would change the stability of the OCH fixed point and the value of $\\nu_n$.","fun_headline_variants_meta":{"raw":{"variants":["Two correlation lengths diverge differently near non-reciprocal criticality","New exponent for dynamic correlations in non-reciprocal critical dynamics","Odd Cahn-Hilliard fixed point yields distinct dynamic scaling exponent","Activity controls response speed without altering static structure","Broken time-reversal splits static and dynamic critical exponents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000497,"raw_usage":{"total_tokens":2551,"prompt_tokens":1174,"completion_tokens":1377,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":790,"completion_tokens_details":{"reasoning_tokens":1294}},"tokens_in":790,"tokens_out":1377,"duration_ms":11064,"temperature":1.0,"reasoning_tokens":1294,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:08:29.646630+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the NRCH Langevin equations numerically near the critical point in $d=3$ and measure the decay of the equal-time correlation $C^{ET}(x)$ and of the total linear response $\\chi^T(x)$; if $\\xi_d$ does not scale as $r^{-\\nu_n}$ or $|\\alpha_0-\\theta r|^{-\\nu_n}$ with $\\nu_n \\approx 1/2$ while $\\xi_s \\sim r^{-\\nu}$ with $\\nu \\approx 0.6$, the central claim fails. A cheaper check is to recompute the two-loop integrals $H_1$ and $H_2$ with an independent regularization scheme and verify that the OCH fixed point remains unstable with $\\omega_\\mu = 2\\epsilon/5$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the fluctuation-dissipation-like identity for odd conserved dynamics that the OCH fixed point obeys."}],"review_version":1}