{"id":"d7b3b34e-8e41-4c52-9c7d-59bd8e361db0","arxiv_id":"2509.04693","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A chiral soft glassy rheology model predicts odd viscosity in steady shear and an odd viscoelastic spectrum with a resonance at twice the active rotation frequency.","lead":"This paper builds a model of a soft glassy material filled with actively rotating microscopic inclusions and derives its response to shear. It predicts that such chiral glasses develop an odd viscosity that grows as the rotation slows down, plus an odd viscoelastic spectrum with a resonance at twice the rotation frequency.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline predictions—odd viscosity growing as Ω→0 and the 2Ω resonance—both require the passive SGR spectrum η*_0(ω)~(iω)^{X-2}, which follows only from the exponential prior ρ(E)=e^{-E}; no microscopic justification is given for that prior in rotating-inclusion systems.","rationale":"The paper is a coherent theoretical construction: starting from a chiral extension of SGR, it derives the compact identities (12)-(13). The central physical claims, however, are the low-frequency growth of odd viscosity and the 2Ω resonance, and both hinge on the passive SGR spectrum diverging at zero frequency. The reader's weakest_assumption identifies exactly this dependence on ρ(E)=e^{-E}. The manuscript is explicit about assuming this prior and restricting to 1<X<2, so there is no internal contradiction; the concern is about the transferability of the headline predictions to 'chiral soft glassy materials' more broadly. The factor-of-two discrepancy between main-text Eq. (7) and SM Eq. (36) is a likely typo and does not alter the SM-derived scaling results. The resonance is acknowledged in the SM to be beyond linear theory, which reinforces rather than replaces the prior-distribution concern. Because the paper's claims are valid conditional on the stated SGR assumptions, the appropriate verdict remains CONDITIONAL; the reader's conditional verdict does not need to be changed.","tokens_in":19040,"tokens_out":7440,"duration_ms":83685,"concrete_test":"Recompute η*_odd(ω) from Eq. (13) using a non-exponential prior ρ(E) that keeps η*_0(0) finite—e.g. ρ(E)∝E^α e^{-E} with α>0, or a truncated exponential with a finite cutoff—and evaluate the steady limit η*_odd(ω→0) as Ω→0. If the result scales linearly in Ω rather than as Ω^{X-2}, the claimed growth is an artifact of the exponential prior, not a robust emergent property. Separately, a nonlinear oscillatory-shear solution near ω=2Ω would be needed to test whether the linear-response divergence is regularized.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equations (12)-(13) are exact linear-response identities of the chiral SGR model, but the physically highlighted consequences are not model-independent. The growth η*_odd(ω→0)~Ω^{X-2} and the divergence at ω=2Ω both come from the zero-frequency divergence of the passive SGR viscosity η*_0(ω) for 1<X<2. That divergence is produced by the power-law relaxation-time distribution P_ss(τ)~(X-1)τ0^{X-1}τ^{-X} (SM Eq. 58), which in turn follows solely from the exponential prior ρ(E)=e^{-E} (SM Section IV). For a generic prior with a finite mean relaxation time, η*_0(0) is finite; then Eq. (13) gives η*_odd(0)=-η''_0(2Ω)~O(Ω) for small Ω, so odd viscosity would vanish rather than grow as activity decreases. The paper gives no argument that the rotating-inclusion microstructure maps to the exponential prior; it simply inherits it from passive SGR. Thus the central nontrivial claim is conditional on a standard but unexamined modeling assumption. In addition, at ω=2Ω the linear response diverges because η*_0(0) diverges, and the authors concede in SM Section IV that 'linear theory should not be trusted to capture the resonance effects quantitatively'—so the resonance advertised in the abstract is not a fully established physical prediction without a nonlinear treatment.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a chiral soft glassy rheology (SGR) model in which mesoscopic elements consist of an actively rotating inclusion embedded in a glassy matrix. The model combines the original SGR yielding dynamics with an active rotation frequency Ω. Using a pre-averaging approximation that is exact to linear order in strains, the authors derive closed stress-evolution equations and obtain the linear response to steady and oscillatory shear. The central formal results are Eqs. (12)–(13), which express the shear and odd viscoelastic spectra as combinations of the passive SGR spectrum η0*(ω) shifted by ±2Ω. The paper predicts that in steady shear the odd viscosity scales as Ω^{X−2} for 1<X<2, so it grows as Ω decreases, and that in oscillatory shear there is a resonance-like feature at ω=2Ω together with glassy power-law tails η*_odd(ω)∼Ωω^{X−3}.","tokens_in":19458,"tokens_out":8556,"duration_ms":83475,"significance":"If correct, the paper establishes a new class of odd viscoelastic response in disordered chiral active matter, with explicit, falsifiable scaling predictions. The derivation in the supplementary material is careful and transparent: the pre-averaging closure is justified to linear order, the noise-free algebra is presented in detail, and there are no fitted parameters—the predictions follow from the model plus the independently established passive SGR spectrum. The paper also gives a clear physical interpretation of the 2Ω resonance in terms of reinforcement of extension/compression over half a rotation period. The main limitation is that the headline predictions are conditional on the exponential yield-energy prior that produces the divergent zero-frequency passive SGR viscosity; the manuscript does not establish that this prior is the correct microscopic description for rotating-inclusion systems. With appropriate qualification of that scope, and with corrections to a small number of internal inconsistencies, the work would be a useful contribution to the odd-viscoelasticity literature.","major_comments":[{"comment":"The printed source terms in Eqs. (7)–(8) contain an extra factor of 2: they read 2k1G v_ij and 2k2G v_ij. The SM derivation, Eqs. (36)–(37), gives k1G v_ij and k2G v_ij, and the steady-state solution Eq. (9) in the main text is consistent with the SM form, not with the printed factor 2. The factor is also absent from Eq. (10). Please correct the source terms. The final spectra (12)–(13) are derived from the SM equations, so this is not load-bearing for the scaling claims, but as printed it is a formal inconsistency in the central constitutive equations.","section":"Main text Eqs. (7)–(8) vs. SM §III"},{"comment":"The prediction η_odd(0)∼Ω^{X−2} and the 2Ω-resonance both rely on η0*(ω)∼(iω)^{X−2}, which follows only from the exponential prior ρ(E)=e^{−E} with 1<X<2. For a generic prior with finite mean relaxation time, η0*(0) is finite, and Eq. (13) gives η_odd(0) ∼ 2Ω⟨τ²⟩, i.e. the odd viscosity vanishes as Ω→0 rather than growing. The paper does not provide a microscopic argument that rotating-inclusion microstructures map to the exponential prior; it inherits this prior from passive SGR. This is a load-bearing assumption behind the headline scaling. Please either justify the exponential prior for the systems described, explicitly frame the result as a property of the exponential-prior class, or test sensitivity by computing η_odd for a representative alternative prior with finite mean relaxation time.","section":"SM §IV, Eqs. (57)–(58); main text 'Steady shear flow'"},{"comment":"The SM states that 'linear theory should not be trusted to capture the resonance effects quantitatively' because the divergence at ω=2Ω is inherited from the divergent η0(0). This caveat appears only in the supplement, while the abstract and conclusions present the 2Ω resonance as a headline result. Given that the divergence is a linear-theory artifact that nonlinear effects may regularize, the main text must carry the same caveat, and the resonance should be described as a linear-response divergence requiring nonlinear treatment, not as a quantitatively established peak.","section":"SM §IV, final paragraph; Abstract and Conclusions"}],"minor_comments":[{"comment":"The sentence 'with the last term in Eq. (8) accounting for the effect of the active rotation of the inclusion' should refer to Eq. (7), since the active-rotation term appears in the inclusion stress equation, not in the matrix equation.","section":"After Eq. (8)"},{"comment":"The signs of the off-diagonal entries in the steady-state inclusion stress matrix differ between Eq. (9) and SM Eq. (45) (e.g. the 2Ωτ entries and the third-row entries). Please check the sign convention and make the two presentations consistent.","section":"Main text Eq. (9) vs. SM Eq. (45)"},{"comment":"The statement that odd viscosity grows as Ω decreases would benefit from an immediate qualifier that this is a result for the linear response in the exponential-prior regime 1<X<2, and that the limits Ω→0 and ω→0 do not commute; this is closely related to major comment 2 and would prevent casual misreading.","section":"Abstract and main text, 'grows as Ω decreases'"}],"recommendation":"major_revision","confidential_remarks":"The core derivation is sound and the model is novel, but the paper currently oversells the Ω→0 scaling and the 2Ω resonance. The stress-test concern about the exponential prior is on point and should be addressed with either a microscopic justification or an explicit scope limitation. The factor-of-2 typo and sign inconsistencies are easy fixes. I would be comfortable with publication after a revision that appropriately qualifications the load-bearing assumptions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Banerjee & Sollich's chiral SGR paper. It's a real contribution: they extend SGR to include active rotation of inclusions, and the central formal result is clean and novel—the odd viscoelastic spectrum is a finite difference of the passive SGR spectrum at shifted frequencies, η*_odd(ω) = (i/2)[η*_0(ω+2Ω) − η*_0(ω−2Ω)]. The SM derivation is careful, and the pre-averaging approximation is exact to linear order in strains. They also get concrete scalings: steady odd viscosity ~ Ω^{X−2}, high-frequency tail ~ Ωω^{X−3}, and a resonance at ω = 2Ω. That's worth a serious referee.\n\nSoft spots:\n\n1. The headline claims are less robust than the abstract suggests. The growth of odd viscosity as Ω → 0 and the 2Ω resonance both rely on the passive SGR spectrum η*_0(ω) ~ (iω)^{X−2}, which comes from the exponential prior ρ(E) = e^{−E}. That is a standard SGR assumption, but it is an assumption, not a consequence of the rotating-inclusion physics. For a generic prior with finite mean relaxation time, η*_0(0) is finite and Eq. (13) gives η*_odd(0) ~ Ω, so the odd viscosity would vanish as Ω decreases. The paper should either justify the exponential prior in this setting or clearly state that the scaling predictions are specific to that prior. As written, the abstract markets a model-dependent result as a general finding.\n\n2. The resonance at 2Ω is a divergence inherited from η*_0(0). The authors concede in the SM that linear theory should not be trusted quantitatively there, but the abstract still highlights \"resonance effects.\" That's an oversell, even if they are up front about it later.\n\n3. Minor mechanical issue: main-text Eqs. (7) and (8) have a factor 2 in the source term that contradicts both the SM and the authors' own steady-state solution in Eq. (9). Looks like a typo, but it will confuse readers.\n\nThe citation pattern is fine—they build on SGR and cite it heavily because they are using it, not padding. No parameter fitting; the predictions follow from the model.\n\nBottom line: the paper is a solid model-based contribution, and the shifted-spectrum identity is likely to be useful. It needs revision to fix the factor-of-2 typo, temper the abstract, and pin down the status of the exponential prior. Send it to peer review; a good referee can sort that out.","headline":"Clean chiral-SGR model with a neat shifted-spectrum identity for odd viscoelasticity, but the headline scalings and the 2Ω resonance rest on the standard exponential prior in SGR—a model dependence the abstract doesn't own.","tokens_in":19905,"tokens_out":2985,"would_cite":true,"duration_ms":29030,"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":"This paper predicts that chiral glassy materials with slowly rotating inclusions show an odd viscosity that grows as the rotation frequency decreases, with a resonant odd viscoelastic response at twice that frequency.","keywords":["chiral active matter","odd viscosity","odd viscoelasticity","soft glassy rheology","SGR model","active rotors","oscillatory shear","glassy dynamics"],"falsifier":"Measure the steady-shear odd viscosity of a dense suspension of active rotors in a colloidal glass while varying the rotation frequency Omega; if the odd viscosity decreases with decreasing Omega, or if oscillatory shear shows no peak when the driving frequency equals 2Omega, the predicted growth and resonance are falsified.","tokens_in":1539,"feed_emoji":"🌀","tokens_out":4218,"duration_ms":80015,"temperature":0.7,"pith_summary":"The paper claims that disordered chiral active matter—modeled as actively rotating inclusions embedded in a soft glassy matrix—should display an odd viscoelastic response that combines the slow, power-law relaxation of glasses with a new resonant coupling to rotation. In steady shear the model predicts an odd viscosity that grows as the active rotation frequency Omega decreases, scaling as Omega^(X-2) near the glass transition; in oscillatory shear it predicts an odd viscoelastic spectrum with a resonance at driving frequency 2Omega and a high-frequency power law. The central object is the identity eta*_odd = (i/2)[eta*_0(omega+2Omega) - eta*_0(omega-2Omega)], which expresses the odd response entirely in terms of the passive glass spectrum shifted by twice the rotation frequency. If correct, this gives a concrete rheological signature of chiral activity in amorphous materials and connects the fields of odd viscosity and glassy rheology.","feed_headline":"Odd viscosity in chiral glasses grows as rotation slows","feed_subtitle":"A new chiral SGR model predicts odd viscoelasticity with a resonance at twice the spin frequency.","key_machinery":"The engine of the argument is the chiral SGR model, a mean-field soft-glassy-rheology description in which each mesoscopic element contains an actively rotating inclusion coupled to a glassy matrix. A pre-averaging closure, exact to linear order in strain, reduces the master equation to two coupled Maxwell-like stress equations, with the inclusion equation carrying the active-rotation term -Omega(epsilon_ik sigma_kj + epsilon_jk sigma_ik). Solving these equations and integrating over the exponential yield-energy distribution yields the finite-difference identity eta*_odd = (i/2)[eta*_0(omega+2Omega) - eta*_0(omega-2Omega)], which is the single formula that generates all the scaling results,","core_discovery":"The paper introduces a chiral version of the soft glassy rheology (SGR) model—an ensemble of mesoscopic elements, each an actively rotating inclusion in a glassy matrix—and shows that in linear response the inclusion stress acquires an odd component. For steady shear the odd viscosity scales as Omega^(X-2), so it grows as the active rotation frequency Omega decreases, the opposite of what a naive fast-rotation picture would suggest. In oscillatory shear, the odd viscoelastic spectrum is exactly a finite difference of the passive glassy spectrum at frequencies shifted by ±2Omega; this produces a resonance at omega=2Omega and a high-frequency power law Omega omega^(X-3). Away from zero frequen","pith_inferences":["The identity suggests a rheological probe: locating the 2Omega peak in an oscillatory-shear measurement could directly read off the active rotation frequency of inclusions.","The exponential yield-energy distribution is the input that produces the divergent low-frequency spectrum; alternative distributions or states below the glass transition would weaken or remove the predicted growth of odd viscosity, a checkable prediction for particle simulations of dense spinners.","Because the linear theory predicts large stresses near resonance, nonlinear effects such as enhanced yielding or shear-thickening should be most pronounced there, a testable extension to large-amplitude oscillatory shear.","The relaxation-time-cutoff mechanism is generic, so analogous odd viscoelastic signatures may appear in any glassy matrix containing active rotors, including biological tissues and emulsions."],"forward_implications":["Steady shear of a chiral glass should show an odd viscosity that increases as the inclusions rotate more slowly, scaling as Omega^(X-2) near the glass transition.","Oscillatory shear should show a resonance-like peak in the odd response when the driving frequency is twice the rotation frequency, reflecting reinforcement of extension and compression by the rotation.","Away from the resonance, the odd response contains both viscous and elastic parts, meaning the same material can show odd viscosity and odd elasticity simultaneously.","Active rotation cuts off the slow relaxation modes that dominate the passive glass, reducing the shear viscosity while generating an odd viscosity.","The finite-difference identity means the chiral response can be predicted directly from the passive glass spectrum, so odd viscoelasticity is inherited from ordinary glassy rheology."],"supporting_citations":[{"why":"Supplies the original SGR model—activated yielding with exponential yield energies—that the chiral model extends.","marker":"[51]"},{"why":"Provides the passive constitutive equation and the low-frequency spectrum eta*_0 ~ (i omega)^(X-2) used to derive the scaling predictions.","marker":"[52]"},{"why":"Gives the tensorial SGR formulation whose stress equations the chiral model adapts.","marker":"[53]"},{"why":"Provides the pre-averaging approximation that closes the stress dynamics exactly to linear order in strain.","marker":"[63]"},{"why":"Defines odd viscosity as the antisymmetric part of the viscosity tensor, the quantity predicted here.","marker":"[26]"},{"why":"Establishes odd viscosity in chiral active fluids, the setting this work extends to glassy disorder.","marker":"[27]"},{"why":"Reports experimental odd-viscosity flow in spinning colloids, motivating the active-rotor scenario.","marker":"[48]"},{"why":"Analyzes odd elasticity in disordered chiral active materials, a counterpart to the glassy odd viscoelasticity studied here.","marker":"[50]"}],"fun_headline_variants":["Odd viscosity in chiral glasses spikes at slow rotation","Chiral glass odd viscosity grows as spin slows down","Slow rotation raises odd viscosity in chiral glasses","In chiral glasses, odd viscosity rises when rotation drops","Odd viscoelasticity emerges in chiral glasses at slow rotation"],"cache_read_input_tokens":21632,"weakest_assumption_plain":"The predictions assume that the glassy matrix has an exponential distribution of yield energies, making the passive viscosity diverge at low frequency; if the matrix is not close enough to a glass transition, the odd viscosity would not grow as rotation slows and the resonance would disappear.","fun_headline_variants_meta":{"raw":{"variants":["Odd viscosity in chiral glasses spikes at slow rotation","Chiral glass odd viscosity grows as spin slows down","Slow rotation raises odd viscosity in chiral glasses","In chiral glasses, odd viscosity rises when rotation drops","Odd viscoelasticity emerges in chiral glasses at slow rotation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001086,"raw_usage":{"total_tokens":4355,"prompt_tokens":700,"completion_tokens":3655,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":444,"completion_tokens_details":{"reasoning_tokens":3580}},"tokens_in":444,"tokens_out":3655,"duration_ms":27936,"temperature":1.0,"reasoning_tokens":3580,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:56:06.713280+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the steady-shear odd viscosity of a dense suspension of active rotors in a colloidal glass while varying the rotation frequency Omega; if the odd viscosity decreases with decreasing Omega, or if oscillatory shear shows no peak when the driving frequency equals 2Omega, the predicted growth and resonance are falsified.","supporting_citations":[{"cited_title":"Sollich, F","cited_arxiv_id":null,"evidence_quote":"Supplies the original SGR model—activated yielding with exponential yield energies—that the chiral model extends."},{"cited_title":"Sollich, Rheological constitutive equation for a model of soft glassy materials, Physical Review E58, 738 (1998)","cited_arxiv_id":null,"evidence_quote":"Provides the passive constitutive equation and the low-frequency spectrum eta*_0 ~ (i omega)^(X-2) used to derive the scaling predictions."},{"cited_title":"Cates and P","cited_arxiv_id":null,"evidence_quote":"Gives the tensorial SGR formulation whose stress equations the chiral model adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the pre-averaging approximation that closes the stress dynamics exactly to linear order in strain."},{"cited_title":"Banerjee, A","cited_arxiv_id":null,"evidence_quote":"Establishes odd viscosity in chiral active fluids, the setting this work extends to glassy disorder."}],"review_version":1}