{"id":"e9bfc1c7-4929-46c4-a024-46c7c62ad4f5","arxiv_id":"2607.05130","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Driven Curie-Weiss specific heat diverges at a lowered Curie point (exponents α=1 and ≈0.86) coinciding with susceptibility, with ferro-para coexistence at large drive via Floquet stability.","lead":"The driven Curie-Weiss magnet has a completed phase diagram in temperature and drive strength: ferro and para phases can stably coexist at large amplitude and frequency, and the nonequilibrium specific heat diverges at the same critical point as the susceptibility. This shows dynamical criticality that equilibrium thermodynamics cannot produce.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged numerical and empirical soft spots.","rationale":"The paper’s strongest claim is supported by three independent diagnostics (C, χ, Floquet μ) that coincide on the same β_c and by an explicit low-T asymptotic (V.6) that proves ferro stability for A=1-ϵ_{0} with ϵ_{0}~ (log eta)/(4eta) together with a large-ω paramagnetic stability bound A>2/π. These analytic pieces do not rely on the undervived formula (III.2) or on the precise value 0.86. The reader correctly isolates the constant-rate mean-field ODE as the modelling assumption, yet that assumption is the standard Curie-Weiss closure and is used consistently; no internal contradiction appears. Hence the CONDITIONAL verdict already given by the reader is appropriate and needs no further adjustment.","tokens_in":17607,"tokens_out":608,"duration_ms":5684,"concrete_test":"Recompute the log|C| vs log|β-β_c| slopes of Fig. 3 for at least three independent (A,ω) pairs with averaging windows \tau_avg≥10^{3} periods and report 95% bootstrap intervals on the fitted α; if the intervals for eta\nearroweta_c exclude 1 and those for eta\nearroweta_c remain consistent with ≈0.86, the reported exponents stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (C and χ diverge at the same dynamical β_c, with ferro-para coexistence for large A,ω at low T) rests on the closed mean-field ODE (II.3) and the Floquet multiplier μ=exp(∫ a(t)dt) with a(t)=-1+β sech^{2}[β(m*+h(t))]. That construction is internally consistent: the same μ\to1 controls both the stability boundary (V.2–V.3) and the poles of σ_{1},ς (VI.7–VI.8) and of χ (VII.2). The only residual soft spots are already noted by the reader—the purely numerical α≈0.86 and the undervived high-T guess (III.2)—neither of which undermines the coincidence of the divergences or the existence of the coexistence region.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper completes the phase diagram of the mean-field Curie–Weiss magnet driven by a time-periodic field h(t)=A cos(ωt). Using the closed ODE dm/dt=tanh[β(m+h(t))]-m, it shows that the nonequilibrium specific heat C (defined via AC calorimetry) and the DC susceptibility χ diverge at the same dynamical critical inverse temperature β_c(A,ω)<1. For small driving the transition is second-order with fitted exponents α=1 (β↓β_c) and α≃ 0.86 (β↑β_c); for larger A a first-order regime appears together with a region of stable ferro–para coexistence. Floquet multipliers of the linearized periodic orbit control both the stability boundaries and the poles of C and χ, establishing that the criticality is dynamical.","tokens_in":17789,"tokens_out":954,"duration_ms":7451,"significance":"If the results hold, the work supplies a clean, analytically tractable mean-field example in which nonequilibrium calorimetry reveals a divergent specific heat that is absent in equilibrium, together with a genuine coexistence of stable ferromagnetic and paramagnetic periodic orbits. The Floquet analysis (Secs. V–VII) and the high-T/low-T expansions of χ (Appendix A) give a transparent link between critical slowing-down and the observed divergences. These features are of clear interest for the statistical mechanics of driven systems and for the broader program of nonequilibrium thermodynamics.","major_comments":[{"comment":"The claim α≃ 0.86 for β↑β_c (abstract and Sec. III, Fig. 3a) rests on a purely numerical linear fit of log|C| versus log(β_c-β) without reported uncertainties, fit ranges, or finite-time checks. Because the same Floquet factor μ\to1 that produces the exact α=1 pole on the ferromagnetic side also governs the approach from below, an analytic or at least systematically controlled numerical determination of the subcritical exponent is needed before the asymmetric value can be regarded as established.","section":null},{"comment":"Equation (III.2) is introduced as an “unexpectedly accurate guess” that is then used to extract the shift β_c=1+A^{2}/(2ω^{2}). While the high-T formula (III.1) is derived in Appendix A, the intermediate expression (III.2) is not. Either a controlled derivation or a clear statement that the formula is only phenomenological should be supplied, since the subsequent analytic estimate of β_c relies on it.","section":null}],"minor_comments":[{"comment":"The constant attempt frequency ν=1 is fixed without discussion of possible m-dependence (mentioned only in passing in Sec. II.A). A short remark on the robustness of the phase diagram under other Glauber-type rates would strengthen the presentation.","section":null},{"comment":"Several figures (e.g., Figs. 1, 2, 10–13) lack error bars or statements of numerical resolution; adding them would help the reader assess the quality of the reported divergences and coexistence boundaries.","section":null},{"comment":"Typographical slips: “ferromagentic” (p. 3), “sustainability” for susceptibility (Fig. 7 caption), and inconsistent spacing around β_c throughout.","section":null},{"comment":"The relation of the present Floquet analysis to earlier mean-field treatments of the dynamic phase transition (Refs. [6,7]) could be stated more explicitly in the introduction.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a natural and solid continuation of the authors’ earlier calorimetry papers. The central claims are internally consistent and the Floquet construction is clean; the two major points are local and fixable. Suitable for the journal after minor revision."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper finishes the mean-field phase diagram for the driven Curie-Weiss magnet and shows that the nonequilibrium specific heat diverges at the same dynamical critical point as the DC susceptibility. That is the real news: equilibrium mean-field C has only a jump, while here C diverges (α=1 from the ferro side, ≈0.86 from the para side) and the Curie point drops with driving. They also map a genuine ferro-para coexistence region at large A and ω, with an explicit Floquet bound A>2/π at high frequency and a low-T stability window that shrinks only as (log β)/β.\n\nWhat they do well is keep everything on the same closed ODE and the same Floquet multiplier. The multiplier controls both the stability boundary and the poles that make C and χ diverge, so the coincidence of the critical points is not an accident of numerics. The high-T and low-T expansions for χ in the appendix match the simulations cleanly for small A, and the AC-calorimetry formulae are derived without hand-waving. The phase diagrams line up with earlier work (Gallardo et al.) while adding the calorimetry and the coexistence analysis that was missing.\n\nSoft spots are real but limited. The 0.86 exponent is a pure numerical fit with no error bars, and the high-T “guess” formula (III.2) is not derived—though it works surprisingly well. No code is shipped. None of that touches the central claims: the divergences coincide, the coexistence region exists, and the criticality is dynamical. The constant-rate mean-field ODE is the usual modeling choice; everything that follows is consistent with it.\n\nThis is for people who work on dynamic phase transitions or nonequilibrium response. It is a clean, usable benchmark. I would send it to referees; the science is already solid enough that the remaining fixes are straightforward.","headline":"Solid completion of the driven Curie-Weiss phase diagram: diverging nonequilibrium specific heat, Floquet coexistence, and consistent critical points, with only minor numerical soft spots.","tokens_in":18451,"tokens_out":491,"would_cite":true,"duration_ms":4900,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.70.Ln","64.60.Ht","75.10.Hk"],"model":"grok-4.5","headline":"In a driven Curie-Weiss magnet the nonequilibrium specific heat diverges at a driving-lowered Curie point, and ferro and para phases can stably coexist.","keywords":["driven Curie-Weiss model","nonequilibrium specific heat","dynamic phase transition","Floquet stability","magnetic susceptibility","phase coexistence","critical exponents","mean-field Ising"],"falsifier":"Numerically or experimentally measure the nonequilibrium specific heat and DC susceptibility of a driven mean-field Ising magnet across the predicted β_c(A, ω); if the two quantities do not diverge at the same point, or if the measured exponents deviate from 1 and ≈ 0.86, the central claim fails.","tokens_in":18493,"feed_emoji":"🧲","tokens_out":1048,"duration_ms":8939,"temperature":0.7,"pith_summary":"This paper completes the phase diagram of the mean-field Ising magnet when it is driven by a time-periodic magnetic field. The central new claim is that the nonequilibrium specific heat diverges at the same critical inverse temperature where the DC magnetic susceptibility diverges; that critical temperature falls as the drive amplitude grows, and the critical exponents are α = 1 from the ferromagnetic side and α ≈ 0.86 from the paramagnetic side. For sufficiently large drive amplitude and frequency at low temperature the two phases become simultaneously stable, a coexistence that has no equilibrium counterpart. Floquet analysis of the linearized magnetization dynamics shows that the criticality is dynamical: the Floquet multiplier reaches unity, relaxation slows, and both response functions diverge. The result supplies a concrete, analytically tractable example of how time-periodic driving reshapes thermodynamic singularities and creates new nonequilibrium phase structure.","feed_headline":"Driven magnet: heat capacity diverges, ferro and para coexist","feed_subtitle":"Nonequilibrium specific heat and susceptibility share a driving-lowered Curie point with new critical exponents.","key_machinery":"Floquet multiplier of the linearized magnetization ODE: μ = exp(∫ a(t) dt) with a(t) = -1 + β sech^{2}(β[m*(t) + A cos(ωt)]); criticality occurs when μ \to 1, which forces both the specific-heat integrals and the susceptibility to diverge.","core_discovery":"The nonequilibrium specific heat of the driven Curie-Weiss model diverges at the same critical inverse temperature β_c that marks the divergence of the DC susceptibility; the new Curie point decreases with driving amplitude, the critical exponents are α = 1 for β ↓ β_c and α ≈ 0.86 for β ↑ β_c, and a regime of stable ferromagnetic-paramagnetic coexistence appears for large enough drive amplitude and frequency at low temperature.","pith_inferences":["The same Floquet criterion may locate analogous heat-capacity divergences in other Model-A systems under periodic drive, including lattice gases with local detailed balance.","Because the specific-heat singularity is absent in equilibrium, AC-calorimetry becomes a sharper experimental probe of dynamical phase transitions than susceptibility alone.","The essential critical point at (A = 1, T = 0) suggests that low-temperature switching dynamics under strong drive may be chaotic rather than periodic, inviting further dynamical-systems analysis."],"forward_implications":["The Curie temperature of a driven mean-field magnet is a decreasing function of drive amplitude and can be read off from either heat capacity or susceptibility.","A first-order dynamical transition with a finite coexistence window of ferro and para phases appears for large A and ω at low T.","The divergence of specific heat is carried by the dissipative response term that vanishes in the static (equilibrium) limit, so the singularity is genuinely nonequilibrium.","Floquet marginal stability supplies a practical diagnostic for locating dynamical critical points in other periodically driven mean-field models."],"fun_headline_variants":["Driven magnet: ferro-para coexistence plus shared heat-susceptibility criticality","Driving lowers Curie point; nonequilibrium heat capacity diverges as α=1, ≈0.86","Stable ferro and para phases coexist under strong periodic drive in Curie-Weiss","Specific heat and DC susceptibility diverge together at drive-reduced β_c","Floquet criticality: dynamical exponents for heat capacity in driven magnet"],"cache_read_input_tokens":128,"weakest_assumption_plain":"All results rest on the assumption that the macroscopic magnetization obeys the closed ordinary differential equation dm/dt = tanh[β(m + h(t))] - m with a constant attempt rate, and that this mean-field rate choice remains valid under periodic driving.","fun_headline_variants_meta":{"raw":{"variants":["Driven magnet: ferro-para coexistence plus shared heat-susceptibility criticality","Driving lowers Curie point; nonequilibrium heat capacity diverges as α=1, ≈0.86","Stable ferro and para phases coexist under strong periodic drive in Curie-Weiss","Specific heat and DC susceptibility diverge together at drive-reduced β_c","Floquet criticality: dynamical exponents for heat capacity in driven magnet"]},"model":"grok-4.5","effort":"low","cost_usd":0.006134,"raw_usage":{"total_tokens":1561,"prompt_tokens":712,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":61340000,"prompt_tokens_details":{"text_tokens":712,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":762,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":712,"tokens_out":87,"duration_ms":6318,"temperature":1.0,"reasoning_tokens":762,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T08:20:49.454498+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Numerically or experimentally measure the nonequilibrium specific heat and DC susceptibility of a driven mean-field Ising magnet across the predicted β_c(A, ω); if the two quantities do not diverge at the same point, or if the measured exponents deviate from 1 and ≈ 0.86, the central claim fails.","supporting_citations":[],"review_version":1}