{"id":"1b53fd2d-0c3b-48fc-a53b-66866f8aa244","arxiv_id":"2506.09025","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Feedback Ising models exhibit super-stable mixed phases at zero temperature when feedback strength exceeds a threshold, with a full classification of associated phase transitions.","lead":"Mean-field Ising models with a coupling that depends on the system's own magnetization can hold stable mixed phases even at zero temperature, provided the feedback is strong enough. The result gives a tractable framework for modeling how systems switch between ordered states through intermediate phases.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The stability classification hinges on an unspecified Filippov convention for Eq. (9); without it C0 is not a well-defined set of equilibria, and the claimed finite-time linear decay is a singular-limit artifact not present at any finite β.","rationale":"The paper's algebra for the linear FIM checks out: g(m)=-m-1.5γm^2, the vertex is at m=-1/(3γ), and the branch with g'(m)>0 is the candidate stable MP. Eq. (18) follows from comparing h' with g'(m)m' on the two sides of C0. The genuine gap is that Eq. (9) is a discontinuous ODE, and treating C0 as an equilibrium manifold, plus all super-stability and trapping statements, requires a Filippov or equivalent convention that is never stated. This is not merely cosmetic, because with the alternative convention sgn(0)=0 the stable MP branch disappears. However, the finite-β linearization F'(m)=-1-β(1-m^2)g'(m) shows that for g'(m)>0 the branch survives as a stable isolated equilibrium for every β>0 and converges to the zero-temperature branch as β→∞. What is fragile is the literal super-stability claim: finite-time linear decay and exact trapping are singular-limit properties, not properties of any finite-temperature regularization. The paper should either state the Filippov interpretation explicitly or qualify 'always super-stable' as a zero-temperature singular-limit statement. The reader's conditional verdict remains appropriate; our read does not change it.","tokens_in":9582,"tokens_out":19726,"duration_ms":211072,"concrete_test":"Regularize Eq. (9) to Eq. (4) with finite β and, for the linear FIM with γ=1 and h=0, solve dm/dt=-m+tanh(β(m+1.5m^2)) from m0=-0.5. For β=1,10,100, locate the attracting equilibrium near the zero-temperature stable branch m=-2/3+O(1/β) and record its stability and convergence time; also ramp h(t) at small Ω and record the h value at which the trajectory leaves the branch. If the branch points converge to C0 and remain stable as β→∞, while the escape interval tends to Eq. (18), the zero-temperature classification is confirmed; if the branch destabilizes or the escape interval diverges from Eq. (18), the central claim needs revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (9), dm/dt=-m+sgn(h-g(m)), is undefined on the curve C0 where h=g(m). The paper nevertheless treats the entire C0 as a set of equilibria and uses the trapping condition Eq. (18) to classify MPs as super-stable. If sgn(0)=0, only m=0 on C0 is an equilibrium, so the stable MP branch would not exist; if one adopts the Filippov differential inclusion, Eq. (18) is recoverable, but the paper never states this interpretation. This matters because the central existence and super-stability claims live exactly on this discontinuity. The related finite-β question is more benign: near a point m0 with g'(m0)>0, the regularized equilibrium has linearization F'(m0)=-1-β(1-m0^2)g'(m0)<0 for all β>0, so the stable branch survives regularization. What does not survive is the literal linear-in-time, finite-time decay: at finite β convergence is exponential with rate proportional to β, and exact trapping is replaced by an approximate condition. Thus the gap is a missing well-posedness statement rather than a demonstrated instability, but the paper's strongest claims about MP stability and phase-transition order depend on resolving it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a mean-field Ising model with a magnetization-dependent coupling f(m), concentrates on the linear feedback case f(m)=1+γm, and studies the zero-temperature limit in which the Glauber dynamics reduces to dm/dt=−m+sgn(h−g(m)). It identifies a curve C0 of mixed-phase (MP) equilibria, classifies their stability by the sign of g′(m), and claims that stable MPs are super-stable, meaning that perturbations decay linearly in time and vanish in finite time. It also derives a trapping condition for time-varying magnetic fields, classifies four types of phase transitions, computes Maxwell points, and concludes that stable MPs can be ground states only for γ>1. The paper is analytic, with numerical simulations used to illustrate the predicted trajectories.","tokens_in":9869,"tokens_out":8612,"duration_ms":97469,"significance":"If the main claims hold, the feedback Ising model is a useful exactly solvable extension of the Curie-Weiss model: it shows that stable mixed phases can exist at zero temperature with partly ferromagnetic coupling, something the classical mean-field Ising model does not allow. The derivations are explicit and checkable: the piecewise-linear flow in Eq. (13), the trapping condition in Eq. (18), and the Maxwell-point formulas in the End Matter are all transparent and internally consistent. The model has no fitted parameters; γ is a modeling parameter, and the dynamical-simulation parameters are fully specified in the End Matter. The main caveat is that the central stability statements are made on the discontinuity curve C0 of the sgn function, where the governing equation is not a well-defined ODE without an additional convention; this is a missing well-posedness statement rather than a demonstrated instability.","major_comments":[{"comment":"Eq. (9) is not a well-defined ordinary differential equation on the curve C0, where h=g(m) and sgn(0) is undefined. With the common convention sgn(0)=0, only m=0 on C0 is an equilibrium of Eq. (9), so the entire stable-MP branch would not exist. The paper treats the whole curve C0 as a set of equilibria and derives the trapping condition Eq. (18); this is valid only under a Filippov sliding interpretation (or an equivalent convex-selection convention for sgn(0)). Please state this convention explicitly, justify it as the β→∞ limit of the smooth Glauber dynamics, and provide the Filippov sliding vector field from which Eq. (18) follows. This point is load-bearing because the existence and super-stability of mixed phases are the central claims of the paper.","section":"Equilibria and their stability, Eq. (9)"},{"comment":"The claimed linear-in-time, finite-time decay of perturbations to a stable MP is a singular-limit artifact that does not survive at any finite β. For finite β, near a point m0 with g′(m0)>0 and h=g(m0), the linearization of Eq. (4) has derivative F′(m0)=−1−β(1−m0^2)g′(m0)<0, so perturbations decay exponentially with a rate proportional to β, and the exact trapping condition Eq. (18) becomes only approximate. The stable branch itself does survive regularization, so the central existence claim is unaffected, but the paper should either state explicitly that super-stability is a property of the exact β→∞ sliding solution, quantify the finite-β crossover, or soften the abstract and main-text wording so that the finite-time linear decay is not presented as a finite-temperature phenomenon.","section":"Equilibria and their stability, finite-β regularization"}],"minor_comments":[{"comment":"The abstract states that the paper discusses basic properties of finite-dimensional FIMs, but the body contains only two brief, speculative sentences about spatial patterns and no actual finite-dimensional analysis; please either add a short section on finite-dimensional FIMs or temper the abstract's claim.","section":"Abstract / Finite-dimensional FIMs"},{"comment":"The terms 'second-order' and 'third-order' are used for the inexact type-1 and type-4 transitions without a formal definition of the order in terms of derivatives of a thermodynamic quantity; a brief definition or an asymptotic derivation would make the classification verifiable.","section":"Phase transitions"},{"comment":"The reconstruction formula f(m)=−(2/m^2)∫_0^m g(u)du is stated without naming the regularity conditions on g that guarantee a nonsingular f; please specify the mild assumptions mentioned in the text.","section":"Inverse problem, Eq. (20)"},{"comment":"Reference [29] contains a formatting artifact in the author name ('Ho/suppress lyst'); please correct it.","section":"References"},{"comment":"The discussion of Comte's law of three stages is clearly labeled a possible application, but it is disconnected from the preceding mathematics; consider condensing it or explicitly mapping the model variables to the social stages in the text.","section":"Applications paragraph"}],"recommendation":"major_revision","confidential_remarks":"The main technical gap is the missing Filippov interpretation for Eq. (9); this is a well-posedness statement that can be added without changing the paper's conclusions, so the paper is not fatally flawed. The finite-β comment is also addressable with a short regularization discussion. The abstract's overpromise about finite-dimensional FIMs should be corrected during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely new and mostly correct paper. The feedback coupling f(m)=1+γm in the Curie-Weiss model is natural and, as far as I know, previously unanalyzed. The main results — stable mixed phases for γ>1/3, the super-stability property, the four-way classification of transitions, and the Maxwell-point calculations — are derived explicitly and easy to verify. The equivalence to a 2- and 3-spin interaction Hamiltonian is a nice observation. The paper earns its place as a solvable benchmark.\n\nWhat I like: the derivations are transparent. Equation (13) follows from the sign limit, the stability condition g'(m0)>0 comes straight from the piecewise-linear flow, and the Maxwell-point algebra checks out. They also don't oversell: they state plainly that stable MPs are ground states only for γ>1, when the coupling is partly antiferromagnetic. The Comte's law application is speculative, but it is clearly presented as a suggestion, not a result.\n\nWhere it is soft: exactly as the stress-test note says, Eq. (9) is undefined on the curve C0 where h=g(m). The paper treats the whole C0 as an equilibrium set and then uses the trapping condition to call MPs super-stable. That requires an implicit Filippov convention that is never stated. Without it, the literal equation has only m=0 on C0 as an equilibrium, and the stable MP branch would not exist. With the standard Filippov differential inclusion, the paper's statements are recoverable, but the authors need to say so. The related finite-β issue is more benign: near a stable branch, the regularized linearization is negative for any β>0, so the branch survives regularization. What does not survive is the literal finite-time linear decay — at finite β the approach is exponential. So this is a well-posedness gap, not a demonstrated instability. It should be fixed in revision, not treated as a fatal flaw.\n\nThe citation pattern looks fine. They cite the relevant p-spin, Hopfield, and feedback-Ising literature. The novelty claims are accurate as far as I can tell.\n\nWho should read this: people working on mean-field spin systems, dynamical bifurcations in social or climate models, and anyone looking for a minimal model of intermediate phases. It is not a field-reorganizing paper, but it is a solid, useful one.\n\nMy recommendation: send it to peer review. A good referee will ask for the Filippov statement, a short finite-β check, and a toned-down version of the last 'poised to become a benchmark' paragraph. With those changes it will be a nice stat-mech contribution.","headline":"A clean, checkable mean-field study of magnetization-dependent coupling; the zero-temperature sliding-dynamics gap is real but patchable.","tokens_in":10368,"tokens_out":2932,"would_cite":true,"duration_ms":27953,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B20","82B26"],"pacs":["05.50.+q","64.60.Cn"],"model":"deepseek-v4-flash","headline":"When spin-spin coupling grows with magnetization, a mean-field Ising model develops stable mixed phases at zero temperature and a four-way classification of the transitions among them.","keywords":["feedback Ising model","mixed phases","super-stability","zero-temperature Glauber dynamics","mean-field Curie-Weiss model","bifurcation diagram","phase transitions","multistability"],"falsifier":"Solve or simulate the finite-temperature Glauber equation $dm/dt = -m + \\tanh(\\beta(h - g(m)))$ for the linear FIM with $\\gamma > 1/3$ and any small positive $\\beta$; if the stable mixed branch $m_1$ disappears or perturbations decay exponentially rather than linearly, then the paper's super-stability claim is confined to exactly $T = 0$.","tokens_in":9406,"feed_emoji":"🧲","tokens_out":5466,"duration_ms":65360,"temperature":0.7,"pith_summary":"The paper asks whether an Ising ferromagnet can hold stable partially ordered mixed phases when the spin-spin coupling itself depends on the current magnetization. In the zero-temperature mean-field limit, it shows the answer is yes: once the linear feedback strength exceeds $\\gamma = 1/3$, the equilibrium curve folds and a stable mixed branch $m_1$ appears below the vertex alongside an unstable mixed branch $m_2$ above it. Stable mixed phases are super-stable: perturbations decay linearly in time and vanish in finite time, rather than exponentially. The authors classify the phase transitions at branch endpoints into four types and show that the linear model produces a three-stage transformation $m_- \\to m_1 \\to m_+$, which they connect to Comte's law of three stages. If correct, the model gives a solvable benchmark for feedback-driven phase transformations in any system where order reinforces itself.","feed_headline":"Feedback flips Ising magnet's mixed phases from unstable to stable","feed_subtitle":"A mean-field model shows that magnetization-dependent coupling creates super-stable intermediate states and a three-stage transition.","key_machinery":"The central object is the zero-temperature Glauber dynamics $dm/dt = -m + \\mathrm{sgn}(h - g(m))$, where $g(m) \\equiv -m f(m) - \\tfrac{1}{2} m^2 f'(m)$, together with the equilibrium curve $C_0 = \\{(h,m): h = g(m)\\}$. The stability of a mixed-phase point $m_0$ on $C_0$ is decided by the slope $g'(m_0)$: a positive slope makes the point super-stable, with linear decay rates $-m_0 \\pm 1$, while a negative slope makes it super-unstable, with linear growth. For linear feedback, $g(m)$ is a parabola whose vertex at $m = -1/(3\\gamma)$ creates the stable and unstable branches when $\\gamma > 1/3$. The trapping condition of Eq. (18), bounding the field derivative between $(-m_0-1)g'(m_0)$ and $(-m_0+1)g'(m_0)$, is what guarantees that perturbations decay linearly and that the system stays on a stable mixed branch.","core_discovery":"In the mean-field feedback Ising model with Hamiltonian $\\hat{H}(m) = -hm - \\tfrac{1}{2} f(m) m^2$ and linear coupling $f(m) = 1 + \\gamma m$, the zero-temperature equilibrium condition is $h = g(m)$ with $g(m) = -m(\\tfrac{3}{2}\\gamma m + 1)$. The central claim is that for $\\gamma > 1/3$, this equilibrium curve has a vertex at $m = -1/(3\\gamma)$, giving a stable mixed-phase branch $m_1$ below the vertex and an unstable mixed-phase branch $m_2$ above it, even though the coupling remains ferromagnetic over most of the magnetization range. Stable mixed phases are always super-stable, meaning perturbations decay linearly and reach zero in finite time; the trapping condition of Eq. (18) determines when a time-dependent field keeps the system on such a branch. The paper also shows that stable mixed phases can be true ground states only when $\\gamma > 1$, and that the linear FIM exhibits all four types of phase transitions at its branch endpoints, including a three-stage sequence $m_- \\to m_1 \\to m_+$.","pith_inferences":["The paper's stability criterion is stated in terms of the sign function, so the entire super-stability result is framed at exactly zero temperature; a natural extension would be to compute finite-$\\beta$ corrections and check whether the fold and the linear decay rate survive thermal smoothing.","Because the same slope condition $g'(m)>0$ is derived for general feedback functions, any empirically reconstructed bifurcation diagram with a stable intermediate branch can be used to predict the feedback strength needed to stabilize that branch, which may be useful in ecological or socio-economic tipping-point applications.","The three-stage interpretation suggests a testable minimal model: if a community's coupling increases linearly with the fraction of positive states, the intermediate stage loses stability once that fraction reaches roughly $1/3$, a prediction that could in principle be compared with historical or observational transition data.","The equivalence between the linear FIM and a model with two- and three-spin interactions means the stable mixed phases could be realized in spin systems with engineered multispin couplings, providing a concrete experimental or numerical target."],"forward_implications":["A fully ferromagnetic coupling can support stable mixed phases at zero temperature, which the classical Curie-Weiss model cannot (there, stable mixed phases require antiferromagnetic coupling).","Stable mixed phases are super-stable, so under a slowly varying field the system follows the stable branch until the field derivative violates the trapping condition in Eq. (18), giving a concrete criterion for when a driven system is thrown off equilibrium.","The linear FIM exhibits all four transition types and the three-stage sequence $m_- \\to m_1 \\to m_+$, providing a minimal model for stage-wise transformations such as those described by Comte's law of three stages.","Because Eq. (20) reconstructs a feedback function $f(m)$ from any single-valued bifurcation diagram, the framework offers a data-driven route to modeling multistable systems from observed phase diagrams alone.","Maxwell constructions show that stable mixed phases become true ground states in the linear FIM only when $\\gamma > 1$, at which point the coupling is partly antiferromagnetic.","A finite-temperature simulation of the Glauber equation for the linear FIM with $\\gamma > 1/3$ would settle whether the stable mixed branch and its linear decay persist away from $\\beta = \\infty$."],"supporting_citations":[{"why":"supplies the Glauber dynamics that gives the magnetization ODE in Eq. (4), the basis of the zero-temperature analysis.","marker":"[25]"},{"why":"exemplifies state-dependent couplings in neural networks, motivating feedback in the coupling rather than in the external field.","marker":"[11]"},{"why":"introduces a market Ising model with magnetization-dependent local fields, the closest prior feedback construction from which this paper differentiates itself.","marker":"[22]"},{"why":"shows feedback inserted into mean-field Landau theory can produce limit cycles, providing a contrast that coupling feedback yields different phenomenology.","marker":"[24]"},{"why":"provides Comte's law of three stages, the historical application used to interpret the linear FIM's $m_- \\to m_1 \\to m_+$ sequence.","marker":"[26]"}],"fun_headline_variants":["Feedback creates super-stable mixed phases in Ising magnets","Ising model: feedback yields stable mixed phases","Super-stable mixed phases from feedback in Ising systems","Feedback flips mixed phases to stable in Ising magnets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire stability classification is carried out at exactly zero temperature, where the thermal transition function is replaced by a sign function, and it assumes the trajectory has a well-defined sliding motion on the discontinuous equilibrium curve; if smoothing the discontinuity at finite temperature changes which mixed phases are stable, the central claim holds only in the $\\beta \\to \\infty$ limit.","fun_headline_variants_meta":{"raw":{"variants":["Feedback creates super-stable mixed phases in Ising magnets","Ising model: feedback yields stable mixed phases","Super-stable mixed phases from feedback in Ising systems","Feedback flips mixed phases to stable in Ising magnets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000147,"raw_usage":{"total_tokens":1154,"prompt_tokens":879,"completion_tokens":275,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":210}},"tokens_in":495,"tokens_out":275,"duration_ms":3997,"temperature":1.0,"reasoning_tokens":210,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:57:02.944822+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve or simulate the finite-temperature Glauber equation $dm/dt = -m + \\tanh(\\beta(h - g(m)))$ for the linear FIM with $\\gamma > 1/3$ and any small positive $\\beta$; if the stable mixed branch $m_1$ disappears or perturbations decay exponentially rather than linearly, then the paper's super-stability claim is confined to exactly $T = 0$.","supporting_citations":[{"cited_title":"Feedback-induced self-oscillations in large interacting systems subjected to phase transitions,","cited_arxiv_id":null,"evidence_quote":"supplies the Glauber dynamics that gives the magnetization ODE in Eq. (4), the basis of the zero-temperature analysis."},{"cited_title":"Neural networks and physical systems with emergent collective computational abilities,","cited_arxiv_id":null,"evidence_quote":"exemplifies state-dependent couplings in neural networks, motivating feedback in the coupling rather than in the external field."},{"cited_title":"Fluctuations of the magnetization in the p- spin curie–weiss model,","cited_arxiv_id":null,"evidence_quote":"introduces a market Ising model with magnetization-dependent local fields, the closest prior feedback construction from which this paper differentiates itself."},{"cited_title":"Opinion for- mation model for markets with a social temperature and fear,","cited_arxiv_id":null,"evidence_quote":"shows feedback inserted into mean-field Landau theory can produce limit cycles, providing a contrast that coupling feedback yields different phenomenology."},{"cited_title":"Krapivsky, Sidney Redner, and Eli Ben-Naim, A Kinetic View of Statistical Physics (Cambridge Uni- versity Press, 2010)","cited_arxiv_id":null,"evidence_quote":"provides Comte's law of three stages, the historical application used to interpret the linear FIM's $m_- \\to m_1 \\to m_+$ sequence."}],"review_version":1}