{"id":"caee18fe-4990-4b87-a9bd-5b717642e340","arxiv_id":"2606.07823","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Derives exact mean-field spinodal surfaces for active lattice gases on six Bravais lattices via Bloch theorem reduction to a tight-binding eigenvalue problem and wavenumber perturbation.","lead":"This paper derives closed-form analytical expressions for the spinodal surface marking the onset of motility-induced phase separation in a lattice model of self-propelled particles with exclusion, using mean-field master equations linearized around the uniform state. A generalist might read it to see how discrete lattice geometry can be folded into a single coefficient that controls active phase behavior without simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the modeling choice (MF closure), but that choice is explicitly part of the claim rather than an unstated prerequisite for it. Because the paper does not assert that the MF spinodal coincides with the true microscopic instability, the derivation stands or falls on its internal algebra, which shows no evident flaw.","tokens_in":1724,"tokens_out":255,"duration_ms":14442,"concrete_test":"Re-derive the coefficient A for the square lattice from the linearized mean-field equations (without using the final formula) and confirm it matches the reported value; if it does, the perturbation step is verified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an exact closed-form spinodal within a stated mean-field closure of the master equation. The reduction via Bloch's theorem to a z-dimensional tight-binding problem, followed by a small-k perturbation, is a standard and internally consistent procedure for lattice models; lattice dependence entering only through a single scalar A is a direct consequence of the symmetry of the linearized operator at k=0. No algebraic inconsistency or hidden assumption that would invalidate the derivation is visible in the given description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript develops a mean-field theory for motility-induced phase separation in a lattice gas of self-propelled particles subject to hard-core exclusion, internal director bias, rotational diffusion, and translational diffusion. Starting from the master equation under a general mean-field closure, the homogeneous state is linearized; Bloch's theorem reduces the stability problem to a z-dimensional tight-binding eigenvalue problem whose small-k expansion yields a closed-form spinodal surface. Lattice geometry enters solely through a single exactly evaluated coefficient A for the linear, square, hexagonal, simple-cubic, bcc and fcc lattices. Translational diffusion is shown to smooth interfaces and rotational probability currents are computed for the inhomogeneous states.","tokens_in":1812,"tokens_out":340,"duration_ms":19766,"significance":"If the derivation holds, the work supplies parameter-free, analytically closed spinodal expressions across six Bravais lattices, a clean reduction of all geometric dependence to the single scalar A, and explicit non-equilibrium currents. These features constitute a reproducible, falsifiable benchmark for lattice active-matter models and enable direct comparison with simulations without fitting parameters.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the spinodal is obtained 'in closed analytical form'; a brief remark in the main text confirming that the final expression for the critical density contains no implicit numerical roots would strengthen this claim.","section":"Abstract"},{"comment":"A compact table listing the exact numerical values of A for each of the six lattices would improve readability and allow immediate cross-lattice comparison.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary accurately reflects the derivation of the exact mean-field spinodal surfaces via the Bloch reduction and the role of the single coefficient A across the six lattices.","responses":[],"tokens_in":1231,"tokens_out":68,"duration_ms":8318,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper supplies closed analytical spinodals for an active lattice gas by reducing all geometry dependence to one coefficient A evaluated on six Bravais lattices.\n\nThe derivation starts from the master equation, closes it at mean-field level, linearizes around the uniform state, invokes Bloch's theorem to reach a z-dimensional tight-binding eigenvalue problem, and performs a small-k expansion. Within those steps the algebra produces explicit formulas without fitted parameters or extra assumptions. The collapse to a single scalar A follows directly from symmetry at k=0 and is evaluated exactly for the linear, square, hexagonal, simple cubic, bcc, and fcc cases. They also track how added translational diffusion smooths interfaces and compute the rotational currents in the inhomogeneous states.\n\nThe calculation is internally consistent and stays within the stated mean-field closure. No algebraic gaps or hidden circularity appear in the procedure. The main modeling choice is the mean-field approximation itself, which is standard for these models and is not presented as exact beyond that level.\n\nThe limitation is the usual one for mean-field treatments of active exclusion: correlations can shift the actual instability, but the paper does not claim otherwise. The stress-test found no inconsistency that would invalidate the reported expressions.\n\nThis is for soft-matter theorists who work with lattice models of active particles and want analytical benchmarks rather than simulation output. A reader needing explicit spinodal surfaces for these geometries will find usable formulas. It deserves peer review because the result is new within the mean-field setting, the steps are reproducible, and the claims are scoped correctly.","headline":"This paper supplies closed analytical spinodals for an active lattice gas by reducing all geometry dependence to one coefficient A evaluated on six Bravais lattices.","tokens_in":2300,"tokens_out":390,"would_cite":false,"duration_ms":17014,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Mean-field master equation yields closed analytical spinodal surfaces for active particles on six Bravais lattices.","keywords":["motility-induced phase separation","mean-field theory","lattice gas","active particles","spinodal surface","Bravais lattices","tight-binding model","broken detailed balance"],"falsifier":"Numerical comparison of the analytically predicted spinodal densities against direct stochastic simulations of the underlying master equation on the same six lattices.","tokens_in":2620,"feed_emoji":"","tokens_out":695,"duration_ms":11912,"temperature":0.7,"pith_summary":"The paper establishes an exact analytical expression for the spinodal surface that marks the linear instability of the uniform state in a lattice gas of self-propelled particles subject to hard-core exclusion, directed hopping, rotational diffusion, and translational diffusion. It proceeds by linearizing the mean-field master equation, invoking Bloch's theorem to reduce the problem to a z-dimensional tight-binding eigenvalue problem, and performing a small-wavenumber perturbation expansion that isolates all lattice dependence into a single coefficient A. A reader would care because the resulting formulas give the critical density as an explicit function of activity, diffusion rates, and lattice type without requiring stochastic simulations, while also showing that translational diffusion smooths phase interfaces and that rotational currents persist in the inhomogeneous states.","feed_headline":"Exact spinodal surfaces derived for active particles on six lattices","feed_subtitle":"Mean-field master equation reduces stability analysis to a tight-binding problem whose expansion isolates geometry in one coefficient A","key_machinery":"The z-dimensional tight-binding eigenvalue problem obtained after linearizing the mean-field master equation and applying Bloch's theorem, whose small-wavenumber expansion isolates lattice geometry into the single coefficient A.","core_discovery":"Linearization of the mean-field master equation around the homogeneous stationary state, followed by Bloch's theorem, reduces the stability analysis to a z-dimensional tight-binding eigenvalue problem. A perturbation expansion in wavenumber near zero then produces the spinodal surface in closed form for the linear, square, hexagonal, simple cubic, body-centered cubic, and face-centered cubic lattices, with all geometric influence captured by one coefficient A that is evaluated exactly in each case. Translational diffusion is shown to smooth the interface between dense and dilute phases, and the rotational probability currents associated with the inhomogeneous states are computed explicitly.","pith_inferences":["The same reduction to a tight-binding problem could be applied to lattices with longer-range hops or to continuous-space limits.","The explicit form of A for each lattice supplies a quantitative ranking of how strongly geometry stabilizes or destabilizes the uniform phase.","The rotational currents computed for the phase-separated states offer a measurable signature that distinguishes active from passive phase separation in experiments."],"forward_implications":["The spinodal density is obtained as an explicit algebraic function of the model parameters for each lattice.","All effects of lattice geometry are confined to the single exactly computed coefficient A.","Translational diffusion reduces the sharpness of the dense-dilute interface.","Inhomogeneous stationary states carry nonzero rotational probability currents that signal broken detailed balance."],"fun_headline_variants":["Mean-field master equation yields spinodals for active particles on six lattices","Single coefficient A encodes lattice geometry in active spinodal surfaces","Bloch theorem reduces stability to tight-binding problem for active lattices","Exact closed-form spinodals for self-propelled particles on Bravais lattices"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The general mean-field approximation to the master equation is sufficient to locate the linear instability of the homogeneous state.","fun_headline_variants_meta":{"raw":{"variants":["Mean-field master equation yields spinodals for active particles on six lattices","Single coefficient A encodes lattice geometry in active spinodal surfaces","Bloch theorem reduces stability to tight-binding problem for active lattices","Exact closed-form spinodals for self-propelled particles on Bravais lattices"]},"model":"grok-4.3","cost_usd":0.009237,"raw_usage":{"total_tokens":4144,"prompt_tokens":683,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":92374500,"prompt_tokens_details":{"text_tokens":683,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3387,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":683,"tokens_out":74,"duration_ms":16812,"temperature":1.0,"reasoning_tokens":3387,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T20:18:21.265339+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical comparison of the analytically predicted spinodal densities against direct stochastic simulations of the underlying master equation on the same six lattices.","supporting_citations":[],"review_version":1}