{"id":"2cd7ad89-2553-4e7c-a9e6-b5504485199c","arxiv_id":"2408.15154","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Nonlinear stability on timescale O(ε^{-2}) is proved for a stratified rest state in the inviscid 2D Boussinesq system via refined dispersive estimates and partial symmetries.","lead":"The paper proves nonlinear stability of a stably stratified rest state in the inviscid 2D Boussinesq equations for times of order 1/ε² when the initial perturbation is small, Sobolev-regular, and localized. This uses dispersive decay from anisotropic gravity waves together with a partial-symmetries technique to control nonlinear interactions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest-assumption note already flags the smallness/localization requirement needed to close the estimates. No additional load-bearing gap is detectable from the abstract-level description of the method; the verdict therefore stays UNVERDICTED.","tokens_in":1660,"tokens_out":246,"duration_ms":12940,"concrete_test":"Confirm that the bootstrap in the main theorem closes at time O(ε^{-2}) by checking whether the nonlinear interaction terms (after application of the partial symmetries) remain controlled by the linear decay without introducing an extra ε^{-δ} factor for any δ>0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is nonlinear stability on the O(ε^{-2}) timescale for small, localized Sobolev perturbations of the stratified rest state, obtained by propagating the linear t^{-1/2} decay of internal gravity waves through nonlinear terms via partial symmetries. The abstract states that the smallness assumption closes the estimates after the linear decay is established; no internal inconsistency, hidden derivative loss, or failure of the partial-symmetry reduction is visible from the given description of the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves nonlinear stability on the timescale O(ε^{-2}) for a linearly stably stratified rest state of the inviscid 2D Boussinesq system on R², for sufficiently small, Sobolev-regular, spatially localized initial perturbations of size ε. The argument proceeds by establishing a refined version of the t^{-1/2} dispersive decay for internal gravity waves (building on [EW15]) and controlling nonlinear interactions via the method of partial symmetries introduced in [GPW23]. An analogous result is stated for the dispersive SQG equation.","tokens_in":1764,"tokens_out":425,"duration_ms":11896,"significance":"If the bootstrap closes rigorously, the result supplies a concrete long-time nonlinear stability theorem in a dissipation-free stratified fluid model where linear dispersive decay is the only source of decay. The work demonstrates that the partial-symmetries framework can be adapted to propagate anisotropic dispersive estimates through quadratic nonlinearities without derivative loss, which is a technically nontrivial extension of the cited linear and methodological tools.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction should explicitly state the precise Sobolev index s and the localization weight (e.g., weighted L^2 or H^s with |x|^k decay) required for the initial data; these parameters appear only implicitly in the smallness assumption.","section":null},{"comment":"Notation for the stratification parameter and the frequency variables in the linear dispersive estimates should be unified between the Boussinesq and SQG sections to avoid reader confusion when comparing the two results.","section":null},{"comment":"Figure 1 (if present) or the schematic of the partial-symmetry vector fields would benefit from an explicit statement of the commutation relations used to close the estimates.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is an application of two external works rather than a self-contained derivation; while this is acceptable, the editor may wish to confirm that the novelty disclosure in the introduction adequately distinguishes the new technical steps from the cited references."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary and recommendation of minor revision. We are pleased that the significance of the long-time nonlinear stability result, the refined dispersive decay, and the adaptation of the partial-symmetries framework is recognized.","responses":[],"tokens_in":1169,"tokens_out":65,"duration_ms":7117,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core result is a nonlinear stability statement on a timescale longer than what linear dispersion alone gives. They take the t^{-1/2} decay for internal gravity waves from the linear theory in [EW15], refine it, and then use the partial-symmetries approach from [GPW23] to keep the nonlinear terms under control up to O(ε^{-2}). A similar claim is made for the dispersive SQG equation. That extension to the nonlinear regime on this specific timescale is the actual new piece; the rest is careful application of existing tools to this system.","headline":"The paper gets nonlinear stability on the O(ε^{-2}) timescale for small localized perturbations of the stratified rest state in the 2D inviscid Boussinesq by refining the linear t^{-1/2} decay and feeding it through partial symmetries.","tokens_in":2268,"tokens_out":214,"would_cite":false,"duration_ms":15582,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking (D=3 forcing)","paper_passage":"dispersive effect due to anisotropic internal gravity waves... amplitude decay at a rate of t^{-1/2}... method of partial symmetries... null structure... integration by parts along S... normal forms"},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel (J uniqueness)","paper_passage":"B-norm... X-norm... localization parameters p quantifying degeneracy of Λ... null structure encoded in multipliers m containing factor ζ2/|ζ|"}],"headline":"2D Boussinesq stability via dispersive waves/partial symmetries; RS derives 3D + J-cost from distinction","alignment":"orthogonal","rationale":"The paper proves nonlinear stability on O(ε^{-2}) for the 2D inviscid Boussinesq system using t^{-1/2} decay of internal gravity waves (Λ(ξ)=ξ1/|ξ|), null structures in quadratic terms, integration by parts along scaling vector field S, normal forms, and bootstrap on B/X norms with frequency localizations P_{k,p}. This is classical quasilinear dispersive PDE analysis (building on [GPW23] partial symmetries). RS framework (reality_from_one_distinction, AbsoluteFloorClosure.lean, AlexanderDuality.lean forcing D=3 via circle linking/H̃^1(S^1), Cost.lean J(x)=½(x+x^{-1})-1, 8-tick periodicity, φ-ladder constants) has no structural overlap with 2D fluid dispersion or energy estimates; paper neither invokes nor parallels J-cost, ratio symmetry, or the distinction-to-spacetime chain. No contradiction arises as domains are disjoint.","tokens_in":72652,"confidence":"high","tokens_out":396,"duration_ms":9116,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The inviscid 2D Boussinesq equation maintains nonlinear stability of a stratified rest state for times of order ε^{-2}.","keywords":["inviscid Boussinesq equation","nonlinear stability","dispersive decay","internal gravity waves","partial symmetries","stratified fluids","SQG equation"],"falsifier":"A concrete small initial perturbation whose solution grows by a fixed factor before time ε^{-2} would falsify the claimed stability.","tokens_in":2558,"feed_emoji":"","tokens_out":697,"duration_ms":16270,"temperature":0.7,"pith_summary":"The paper establishes that small Sobolev-regular and localized perturbations to a linearly stable stratified rest state in the inviscid 2D Boussinesq system on the whole plane remain bounded on a time scale O(ε^{-2}). The mechanism starts from linear dispersive decay at rate t^{-1/2} produced by anisotropic internal gravity waves and extends this control to the nonlinear problem by tracking interactions with the method of partial symmetries. The same long-time bound holds for the related dispersive surface quasi-geostrophic equation. This shows that the rest state can persist for long times in the absence of viscosity once the initial disturbance is sufficiently small and localized.","feed_headline":"Boussinesq equation keeps stratified rest state stable to time 1/ε²","feed_subtitle":"Anisotropic gravity waves disperse small perturbations enough to control nonlinear growth without viscosity.","key_machinery":"Dispersive decay from anisotropic internal gravity waves at linear rate t^{-1/2}, refined and extended to nonlinear terms by the method of partial symmetries.","core_discovery":"We establish the nonlinear stability on a timescale O(ε^{-2}) of a linearly, stably stratified rest state in the inviscid Boussinesq system on R². Here ε>0 denotes the size of an initially sufficiently small, Sobolev regular and localized perturbation. A similar statement also holds for the related dispersive SQG equation. At the core of this result is a dispersive effect due to anisotropic internal gravity waves. At the linearized level, this gives rise to amplitude decay at a rate of t^{-1/2}. We establish a refined version of this, and propagate nonlinear control via a detailed analysis of nonlinear interactions using the method of partial symmetries.","pith_inferences":["The same technique may apply to other stratified or rotating fluid models that support anisotropic waves.","Relaxing spatial localization while keeping the Sobolev smallness might be possible if the decay can be localized in frequency.","The O(ε^{-2}) threshold is set by the current nonlinear estimates; sharper time scales would require stronger decay or additional cancellation."],"forward_implications":["The rest state persists without any viscous dissipation up to the stated time scale.","Linear dispersive decay can be refined and then used to control quadratic and higher interactions.","The same long-time nonlinear stability transfers directly to the dispersive SQG equation.","Partial symmetries suffice to propagate control once the linear decay rate is available."],"fun_headline_variants":["2D Boussinesq rest state stable up to time 1/ε²","Inviscid stratified rest state stable in Boussinesq to 1/ε²","Nonlinear stability of Boussinesq rest state to ε^{-2} scale","Gravity waves enable Boussinesq stability at timescale 1/ε²"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The initial perturbation is small enough in a Sobolev norm and spatially localized so that linear dispersive decay can close the nonlinear estimates.","fun_headline_variants_meta":{"raw":{"variants":["2D Boussinesq rest state stable up to time 1/ε²","Inviscid stratified rest state stable in Boussinesq to 1/ε²","Nonlinear stability of Boussinesq rest state to ε^{-2} scale","Gravity waves enable Boussinesq stability at timescale 1/ε²"]},"model":"grok-4.3","cost_usd":0.010029,"raw_usage":{"total_tokens":4441,"prompt_tokens":644,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":100287000,"prompt_tokens_details":{"text_tokens":644,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3710,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":644,"tokens_out":87,"duration_ms":19612,"temperature":1.0,"reasoning_tokens":3710,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-23T21:29:45.445347+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete small initial perturbation whose solution grows by a fixed factor before time ε^{-2} would falsify the claimed stability.","supporting_citations":[],"review_version":1}