{"id":"86cfb60f-645d-48f7-bc1f-3070e47acff1","arxiv_id":"2607.25782","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In the spherical SK model with non-reciprocal couplings, time-translation invariance is restored for any asymmetry, yet the steady state violates the fluctuation-dissipation theorem via an exponentially vanishing ratio.","lead":"This paper solves the long-time dynamics of a spin glass with asymmetric, one-way interactions, showing that the system becomes stationary but still violates a classic equilibrium connection between fluctuations and response. It provides exact formulas in two limits, separating FDT violations caused by aging from those caused by broken detailed balance.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"TTI stability for Γ≤0 at all finite β is asserted, not proven; the zero-temperature singular limit (Ref. [31] reports aging) makes the extrapolation from t_w=30–40 and Laplace analyticity non-trivial.","rationale":"The reader's weakest_assumption is identical to the load-bearing gap I identify: the TTI convergence and stability at all finite β is assumed rather than proven. My review adds two concrete aggravating details. First, the zero-temperature limit is a genuine singular point: the Introduction attributes to Ref. [31] an explicit two-time (aging) behavior at β^{-1}=0, while the paper's claimed Γ=0 solution extrapolates to C(τ)=1, a TTI state. This does not automatically refute the finite-β claim, but it means the stability and convergence are non-uniform in β, and the finite-t_w numerics at β≤2 cannot rule out much longer aging transients at lower temperature. Second, the 'no instability' argument from Laplace-space analyticity only shows the proposed solution has no growing poles; it does not prove the full nonlinear dynamical system is attracted to it. The paper's exact finite-β solutions are internally consistent, and the numerical match at the shown β and t_w is evidence in favor, so I do not call for rejection. However, the central claim is conditional on a stability result that is neither derived nor fully tested. The reader already marked the paper CONDITIONAL; my concern does not move that verdict, so UNCHANGED is the appropriate output.","tokens_in":11122,"tokens_out":32176,"duration_ms":322155,"concrete_test":"Perform a linear stability analysis of the TTI fixed point of (E1) for Γ=0 and Γ=-1: perturb C=C_s+ε e^{ντ} f(τ), G=G_s+ε e^{ντ} g(τ), bλ=bλ_s+εδbλ, linearize the Volterra integro-differential system, and compute the spectrum of ν. If any eigenvalue with Re ν>0 exists at finite β, the 'stable at all β' claim is false. As an independent check, run microscopic Langevin simulations at Γ=0 and β=4–8 with t_w up to 10^5 and test whether C(t_w+τ,t_w) approaches e^{-β^{-1}τ} and λ(t) approaches sqrt(β^{-2}+g^2); if aging or a different stationary regime appears, the TTI attractor is not the physical one.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that for Γ≤0 (and transiently for 0<Γ<1 above β_c^SK) the long-time dynamics actually converges to the TTI solutions found analytically. The support offered is (i) the absence of poles in the proposed Laplace-space solutions of (E2)–(E3) and (ii) DMFT numerics at t_w=30–40. Neither is a stability proof: no linearization of the full two-time DMFT equations around the TTI state is performed, and finite t_w cannot exclude aging on longer timescales. The zero-temperature limit makes this gap concrete. The Introduction states that at β^{-1}=0 the asymmetric sSK retains an explicit two-time dependence [31], yet the Γ=0 section claims that β^{-1}→0 recovers [31] from a TTI solution with C(τ)=1. If [31] really describes aging, then the convergence to the finite-β TTI state must become singular as β→∞—a behavior not captured by the numerics at β≲2, t_w=40—or the TTI solution is not the attractor at sufficiently low temperature. The branch selection at Γ=0 (removing Res(eC+(s),s=ω_+)=0 in the End Matter) is not the deepest issue: it is equivalent to requiring a physically admissible correlation function with |C(τ)|≤1. The stability of the TTI solution itself is the load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the out-of-equilibrium Langevin dynamics of the spherical Sherrington-Kirkpatrick model with asymmetric couplings, whose symmetric and antisymmetric parts are weighted by Γ ∈ [−1,1]. Using dynamical mean-field theory (the Cugliandolo–Kurchan equations), the authors derive time-translational-invariant (TTI) reductions, obtain analytic closed forms for the correlation and response functions in the symmetric (Γ=1), uncorrelated (Γ=0), and purely antisymmetric (Γ=−1) limits, and claim that for Γ<1 these TTI solutions are stable at all finite temperatures. Consequently, they argue, the fluctuation-dissipation theorem (FDT) is generically violated in a stationary, ergodic, exponentially relaxing state, with the violation arising from broken detailed balance rather than from aging. Numerical integration of the full two-time CK equations is presented as corroboration, and intermediate Γ values are interpolated numerically.","tokens_in":11515,"tokens_out":5447,"duration_ms":54597,"significance":"The analytic results are elegant and, conditional on the TTI ansatz, explicit: at Γ=0 the correlation and response decay as C(τ)=e^{-β^{-1}τ} and G(τ)=e^{-bλτ}, while at Γ=−1 they are Bessel-function forms. The derivation is self-contained, involves no fitted parameters, and the analytic expressions match the numerical integration shown in Figs. 1 and 2. If the central stability claim is correct, the paper provides a valuable solvable framework that separates FDT violations of dynamical origin (aging) from those of thermodynamic origin (broken detailed balance), with potential implications for neural-network, ecological, and active-matter models. The main weakness is that the stability of the TTI solutions for Γ≤0 is asserted rather than proven, and the zero-temperature limit is not reconciled with the existing literature.","major_comments":[{"comment":"The paper's central claim is that for Γ≤0 (and transiently for 0<Γ<1 above β_c^SK) the long-time dynamics converges to the TTI solutions. However, the support provided is (i) the absence of poles in the Laplace-space solutions of (E2)–(E3) and (ii) DMFT numerics at t_w=30–40. Neither constitutes a stability proof for the full two-time nonlinear CK equations. In particular, the statement on page 3 that 'the structure of the solutions ... reveals no instability' is not backed by a linearization of the two-time equations around the TTI state. Since the entire conclusion 'non-reciprocity restores TTI as soon as Γ<1' rests on this stability, please provide a rigorous argument or an explicit linear-stability analysis, or substantially soften the claim.","section":"§usSK model Γ=0; End Matter Eqs. (E1)–(E3)"},{"comment":"The Introduction states that at β^{-1}=0 the asymmetric sSK dynamics 'retain[s] an explicit two-time dependence' [31]. Yet the Γ=0 and Γ=−1 sections claim that β^{-1}→0 recovers the results of [31] from TTI solutions, with C(τ)=1 at Γ=0 and the Bessel form at Γ=−1. If [31] describes aging, these statements are inconsistent: a TTI solution with C(τ)=1 is not an aging solution. Please reconcile this and clarify whether the convergence to the finite-temperature TTI state is singular as β→∞. The numerics at β≤2, t_w=40 do not probe this limit.","section":"Introduction vs. §usSK model Γ=0 and §asSK model Γ=−1"},{"comment":"The closed form for Γ=0 is obtained by requiring that the rightmost pole ω_+ of eC+(s) be removable, i.e., Res(eC+(s), s=ω_+)=0, which fixes bλ=sqrt(β^{-2}+g^2). This is presented as an admissibility condition, but it is not derived from the causal dynamical equations. Since a different branch choice would change the quantitative results, please justify that the full two-time dynamics selects this branch — for example, by showing that it follows from the spherical constraint C(t,t)=1 and boundedness |C(τ)|≤1, or from the long-time limit of the numerical solution.","section":"End Matter, Γ=0 branch selection"},{"comment":"The analytic evidence for FDT violation in the antisymmetric case is based entirely on the Laplace-space object X_L(s)= eG(s)/β(1−s eC+(s)), which the authors acknowledge is not related to the time-domain FDR X(t,t') by an inverse Laplace transform. The physical interpretation of X_L(s) is therefore unclear, and X_L(0)<1 is not by itself a proof that the standard FDR X(τ)≠1 at long times. Since the central claim for Γ<0 relies on this, please provide the actual time-domain FDR for the Bessel solutions or show explicitly how X_L connects to the slope of the β^{-1}χ vs C plot.","section":"§asSK model Γ=−1; 'Fluctuation dissipation ratio' in End Matter"}],"minor_comments":[{"comment":"Typo: 'non-reciprociprocity' should be 'non-reciprocity'.","section":"Introduction"},{"comment":"The caption states 'TTI is observed for all t_w (here t_w=30)', but only t_w=30 is shown. Please state which t_w values were tested and specify the range over which TTI holds.","section":"Fig. 2 caption"},{"comment":"The term eC+(bλ) evaluates the Laplace transform at a point that itself depends on bλ. This is unusual and should be explicitly explained, as it is central to the Γ=0 self-consistency.","section":"Eq. (E7)"},{"comment":"Consider renaming X_L(s) to avoid confusion with the standard time-domain FDR X(t,t'), e.g., 'Laplace-space response ratio', and clearly state in the main text that it is a diagnostic, not the physical FDR.","section":"End Matter, X_L definition"}],"recommendation":"major_revision","confidential_remarks":"The analytic TTI solutions are solid and valuable, but the stability claim for Γ≤0 is the load-bearing step and is currently supported only by pole structure and short-waiting-time numerics. The zero-temperature inconsistency with Ref. [31] is a concrete red flag that needs to be resolved, perhaps by additional low-temperature numerics or by a corrected interpretation of the β^{-1}→0 limit. If the authors can supply a stability argument and fix the zero-temperature reconciliation, the paper would be a strong candidate for acceptance. If they cannot, the claims of 'restoring TTI' and 'stationary yet FDT-violating' should be reformulated as existence results for TTI solutions rather than statements about the long-time attractor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main new thing here is the finite-temperature two-time correlation and response functions for the Γ=0 and Γ=−1 limits of the asymmetric spherical SK model. Exponential at Γ=0, Bessel at Γ=−1, both checked against direct integration of the DMFT equations, and the parametric FDR plots give a clean picture: FDT is violated while the dynamics is stationary and ergodic. That separation of aging-based from detailed-balance-based FDT violations is the real contribution, and the analytic results are a useful benchmark. The derivation from the CK equations is self-contained; no free parameters, no fitted quantities, and the Laplace-space reduction is transparent.\n\nWhere I'd push back is the stability claim. The paper says TTI solutions are stable for Γ≤0 at any finite β, but the evidence is the absence of poles in the Laplace-space equations and numerics at t_w=30–40. That is not a stability proof. The zero-temperature limit makes the gap concrete: the paper cites [31] as showing aging at β^{-1}=0, yet claims the β^{-1}→0 limit of its Γ=0 solution recovers [31]. Since the TTI correlation is e^{-β^{-1}τ}, the decay time diverges as β→∞, so the two limits do not commute. Either convergence to TTI becomes singular at low temperature, or the TTI state is not the attractor below some finite β. The paper does not address this.\n\nThe branch-selection step at Γ=0 (removing a pole in eC_+(s)) is not the real issue—it's just admissibility, |C|≤1. The real soft spot is the assumed stability of the TTI state itself. For 0<Γ<1 above β_c, the transient TTI claim rests on earlier work [23], which is reasonable.\n\nAll that said, the paper deserves a serious referee. The analytic solutions are new and likely correct as a TTI calculation; the open question is whether the dynamics indeed converges to them. I'd send it to review with a request for either a stability analysis or a more careful statement about the domain of validity, especially at low temperature. This is the kind of paper I'd bring to a reading group on disordered out-of-equilibrium systems.","headline":"New finite-T analytic solutions for the asymmetric sSK; the FDT-violation-without-aging picture is plausible, but TTI stability is asserted not proven.","tokens_in":11968,"tokens_out":3159,"would_cite":true,"duration_ms":32706,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.20.-y","75.10.Nr"],"model":"deepseek-v4-flash","headline":"In the spherical Sherrington-Kirkpatrick model, any amount of non-reciprocity suppresses aging but still leaves the fluctuation-dissipation theorem violated in the stationary state.","keywords":["non-reciprocal interactions","spin glass","spherical Sherrington-Kirkpatrick model","fluctuation-dissipation theorem","time-translational invariance","dynamical mean-field theory","detailed balance","aging"],"falsifier":"Run long-time numerical integration of the full two-time DMFT equations with waiting times much larger than a few hundred and check whether the Γ=0 or Γ=-1 solutions lose stability at any finite β, or an explicit linear-stability analysis of the TTI ansatz in the Cugliandolo–Kurchan equations; if aging or two-time scaling reappears at large tw, the central claim fails.","tokens_in":10993,"feed_emoji":"🔄","tokens_out":4923,"duration_ms":48581,"temperature":0.7,"pith_summary":"Non-reciprocity in the couplings of the spherical Sherrington-Kirkpatrick model changes the nature of its long-time dynamics: as soon as the asymmetry parameter Γ drops below 1, time-translational invariance is restored at every finite temperature, so aging disappears. Yet the stationary state is not an equilibrium one: the fluctuation-dissipation theorem (FDT) is violated even though correlation and response relax exponentially on a single time scale. The paper proves this by solving the exact dynamical mean-field equations analytically in the symmetric (Γ=1), uncorrelated (Γ=0) and antisymmetric (Γ=-1) limits, and shows the violations come from broken detailed balance, not from weak ergodicity breaking. This cleanly separates two mechanisms of FDT violation and provides a reference framework for disordered non-reciprocal systems such as neural networks, ecological communities and active matter.","feed_headline":"Non-reciprocity ends aging, keeps FDT broken","feed_subtitle":"In a spherical spin glass, any Γ<1 restores time-translational invariance, yet the steady state still violates fluctuation-dissipation.","key_machinery":"The central object is the pair of Schwinger–Dyson (Cugliandolo–Kurchan) dynamical mean-field equations for the two-time correlation C(t,t') and response G(t,t') of soft spins with a spherical constraint, parametrized by Γ. In the TTI limit these reduce to closed Laplace-space equations for the response and correlation transforms; a key identity (E5) shows that FDT holds in TTI regimes exactly when Γ=1. The Lagrange multiplier λ(t) that enforces the spherical constraint plays the role of an effective damping, and its asymptotic value bλ is fixed by analyticity and pole-removability of the Laplace-space solutions, which also selects the stable physical branch.","core_discovery":"The paper studies the dynamics of the spherical Sherrington-Kirkpatrick model with random asymmetric couplings, parametrized by Γ∈[-1,1]. It derives the full dynamical mean-field (Cugliandolo–Kurchan) equations and shows, in the time-translational invariant (TTI) regime, that the FDT is satisfied if and only if Γ=1. At Γ=0 the correlation and response are exact exponentials, C(τ)=e^{-β^{-1}τ} and G(τ)=e^{-bλτ} with bλ=√(β^{-2}+g^2), and the fluctuation-dissipation ratio decays as e^{(β^{-1}-bλ)τ}, vanishing at long times. At Γ=-1 both functions are Bessel-damped, e^{-β^{-1}τ}J1(2gτ)/(gτ), producing oscillations and a static susceptibility that stays analytic at all temperatures. In both limi","pith_inferences":["If this picture holds, FDT ratios measured in non-reciprocal active or neural systems may often be misread as aging when they are actually steady-state signatures of broken detailed balance; the model suggests checking whether relaxation is single-exponential as a first discriminator.","The exact Γ=0/Γ=-1 formulas imply that the time-dependence of the FDR—or the oscillation frequency at Γ<0—could serve as a quantitative probe of the degree of non-reciprocity in an otherwise disordered system.","The claim that TTI stability holds at all finite β for Γ≤0 could be tested by simulating non-fully-connected or diluted versions of the model: one would predict that FDT violation persists even when translational invariance is restored by asymmetric couplings."],"forward_implications":["At Γ=0 the FDT violation is captured by a single exponential timescale: correlation decays as e^{-β^{-1}τ}, response as e^{-bλτ} with bλ>β^{-1}, and the FDR vanishes as e^{(β^{-1}-bλ)τ}.","At Γ=-1 the dynamics are oscillatory with period ∝1/g and an algebraic envelope τ^{-3/2}; the static susceptibility remains finite and analytic at all temperatures, signalling the absence of an aging transition.","For any Γ<1 the spin-glass transition at β_c=1/g disappears in the sense that TTI solutions remain stable at all finite temperatures (for Γ≤0), or become TTI transiently for 0<Γ<1 after a frozen transient.","These exact solutions give a quantitative benchmark to distinguish FDT violations due to aging from those due to broken detailed balance in disordered non-reciprocal systems.","At high temperature (β→0), the FDT is recovered in all cases, so non-reciprocity only matters when interactions are strong enough to push the system away from equilibrium."],"fun_headline_variants":["Asymmetry kills aging but keeps FDT violation","Non-reciprocity: no aging, yet FDT still broken","FDT broken without aging in non-reciprocal spin glass","Antisymmetric couplings bring oscillations and FDT breakdown","Broken detailed balance, not aging, drives FDT violations"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The long-time dynamics is assumed to converge to time-translational invariant solutions for any Γ<1 at all finite temperatures; this is supported by the analytic structure of the Laplace-space equations and by numerics only up to waiting times of order 30–40, not by an explicit linear-stability proof.","fun_headline_variants_meta":{"raw":{"variants":["Asymmetry kills aging but keeps FDT violation","Non-reciprocity: no aging, yet FDT still broken","FDT broken without aging in non-reciprocal spin glass","Antisymmetric couplings bring oscillations and FDT breakdown","Broken detailed balance, not aging, drives FDT violations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000263,"raw_usage":{"total_tokens":1437,"prompt_tokens":744,"completion_tokens":693,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":623}},"tokens_in":488,"tokens_out":693,"duration_ms":7505,"temperature":1.0,"reasoning_tokens":623,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T01:29:25.615458+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run long-time numerical integration of the full two-time DMFT equations with waiting times much larger than a few hundred and check whether the Γ=0 or Γ=-1 solutions lose stability at any finite β, or an explicit linear-stability analysis of the TTI ansatz in the Cugliandolo–Kurchan equations; if aging or two-time scaling reappears at large tw, the central claim fails.","supporting_citations":[],"review_version":1}