{"id":"b40c1514-fd22-466a-bfec-8ae717bb8b83","arxiv_id":"2606.03680","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Global regularity proven for 2D Boussinesq with fractional dissipation in subcritical regime α + β > 1 when α ≤ 2/3.","lead":"The paper establishes global regularity for the 2D fractional Boussinesq equations with dissipation exponents satisfying α + β > 1, specifically completing the case α ≤ 2/3 via nonlinear lower bounds on the fractional Laplacian and an iterative argument. A smart generalist might read it to understand progress on long-open questions about whether certain fluid models remain smooth for all time.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Iterative procedure may fail to close for small α without extra restrictions on β beyond α+β>1","rationale":"The reader's weakest_assumption already isolates the iterative closure under only α+β>1 as the critical point for the α≤2/3 case. The technical description (nonlinear lower bounds + iteration) makes this the natural place where an implicit extra condition could hide, consistent with how such arguments behave in related fractional dissipation problems.","tokens_in":1608,"tokens_out":339,"duration_ms":17466,"concrete_test":"Fix α=1/3 and β=0.8 (so α+β=1.133>1) and recompute the admissible range for the iteration parameters appearing after the nonlinear lower bound is applied; check whether the resulting a-priori bound on the highest norm remains finite or forces an extra restriction on β.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for α≤2/3 rests on deriving nonlinear lower bounds for (-Δ)^{α/2}u and feeding them into an iterative procedure that upgrades regularity. The subcritical assumption α+β>1 is invoked to absorb the nonlinear terms, but the iteration constants (arising from the lower bound and the choice of iteration thresholds) typically deteriorate as α decreases. Nothing in the abstract or the stated method rules out the possibility that, for fixed α+β-1 small and α sufficiently small, the bootstrap cannot be closed at the required regularity level without an implicit lower bound on β of the form β>1-α+δ(α) with δ(α)>0.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves global regularity of solutions to the 2D incompressible Boussinesq system with fractional dissipation (-Δ)^{α/2}u and (-Δ)^{β/2}θ in the subcritical regime α + β > 1. It treats the remaining case α ≤ 2/3 (after prior work for α > 2/3) by deriving nonlinear lower bounds on the fractional Laplacian and closing an iterative regularity bootstrap.","tokens_in":1744,"tokens_out":282,"duration_ms":16207,"significance":"If correct, the result supplies the sharpest known global regularity statement for this system under the weakest possible assumptions on the dissipation exponents, completing the subcritical theory.","major_comments":[{"comment":"The abstract and method description invoke the subcritical condition α + β > 1 to absorb nonlinear terms in the iteration, but provide no explicit control on how the iteration constants (from the nonlinear lower bound and threshold choices) behave as α → 0 with α + β - 1 fixed and small. This leaves open whether the bootstrap closes without an implicit extra restriction β > 1 - α + δ(α) for some δ(α) > 0 when α is small.","section":"Abstract / method outline"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying a point that merits clarification in the presentation of the iteration. We address the concern below and will make a targeted revision to improve transparency on constant dependence.","responses":[{"response":"The nonlinear lower bounds on the fractional Laplacian are constructed so that the resulting constants depend continuously on α and β. The iterative thresholds are then chosen proportionally to the gap α + β − 1; this choice remains admissible for any fixed pair satisfying α + β > 1, including arbitrarily small positive gaps and arbitrarily small α. No auxiliary δ(α) > 0 is imposed. To make this dependence explicit, we will insert a short paragraph after the statement of the main theorem that records the functional dependence of the iteration constants on α and β.","revision_made":"partial","referee_comment":"[Abstract / method outline] The abstract and method description invoke the subcritical condition α + β > 1 to absorb nonlinear terms in the iteration, but provide no explicit control on how the iteration constants (from the nonlinear lower bound and threshold choices) behave as α → 0 with α + β - 1 fixed and small. This leaves open whether the bootstrap closes without an implicit extra restriction β > 1 - α + δ(α) for some δ(α) > 0 when α is small."}],"tokens_in":1198,"tokens_out":300,"duration_ms":16017,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper finishes the global regularity question for the 2D incompressible Boussinesq equations with fractional dissipation in the subcritical regime α + β > 1, specifically covering the case α ≤ 2/3 that was left open by the authors' previous work on α > 2/3.\n\nThey introduce nonlinear lower bounds for the fractional Laplacian and run an iterative procedure to upgrade regularity. This is the main new technical step and lets them minimize assumptions on the parameters, giving what they call the sharpest result in this family.\n\nThe approach is direct and analytic, with no obvious circularity. If the estimates close as claimed, it completes a standard program for this model and supplies a concrete advance for specialists tracking parameter thresholds in dissipative fluid equations.\n\nThe soft spot is whether the iteration actually closes uniformly for small α. The subcritical condition α + β > 1 is used to absorb nonlinear terms, but the constants from the lower bounds and iteration thresholds often worsen as α drops. Nothing visible in the abstract rules out the need for an extra margin on β of the form β > 1 - α + δ(α) with δ(α) > 0 when α is sufficiently small. The paper would need to show that the bootstrap stays controlled without such an implicit restriction.\n\nThis is written for readers already working on regularity questions for 2D fluid models with fractional dissipation. Someone following the Boussinesq literature would get value from seeing the case closed, provided the estimates hold. It is worth sending to referees because it targets a concrete open endpoint with a reproducible analytic method, even if the iteration details require verification.","headline":"This paper closes the remaining case α ≤ 2/3 for global regularity of the 2D fractional Boussinesq system under α + β > 1 by introducing nonlinear lower bounds and an iterative upgrade procedure.","tokens_in":2185,"tokens_out":424,"would_cite":false,"duration_ms":21302,"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":"Global regularity holds for the 2D fractional Boussinesq equations whenever the dissipation orders satisfy α + β > 1, including when α ≤ 2/3.","keywords":["Boussinesq equations","fractional dissipation","global regularity","subcritical regime","fractional Laplacian"],"falsifier":"An explicit smooth initial datum whose solution loses regularity in finite time while α + β > 1 and α ≤ 2/3 would falsify the global-regularity claim.","tokens_in":2529,"feed_emoji":"","tokens_out":556,"duration_ms":24066,"temperature":0.7,"pith_summary":"The paper proves that solutions to the two-dimensional incompressible Boussinesq equations with fractional dissipation remain smooth for all time under the subcritical condition α + β > 1. This completes the picture by covering the case α ≤ 2/3 that earlier work had left open. A sympathetic reader cares because the equations model buoyancy-driven fluid flow, and global regularity means smooth initial data never produce finite-time singularities. The argument minimizes the dissipation strength required by establishing new nonlinear lower bounds on the fractional Laplacian and closing estimates through iteration.","feed_headline":"Boussinesq equations stay regular when α + β exceeds 1","feed_subtitle":"The result covers weaker velocity dissipation, closing the remaining subcritical case α ≤ 2/3.","key_machinery":"Nonlinear lower bounds for the fractional Laplacian operator, deployed inside an iterative procedure that produces uniform control on solution norms.","core_discovery":"The authors establish global regularity of the 2D incompressible Boussinesq equations with fractional dissipations (-Δ)^{α/2} u and (-Δ)^{β/2} θ whenever α + β > 1. This covers the remaining regime α ≤ 2/3. The proof proceeds by deriving nonlinear lower bounds for the fractional Laplacian operator and then applying an iterative procedure to obtain the necessary a priori bounds.","pith_inferences":["The nonlinear lower-bound technique might transfer to other active-scalar equations with fractional diffusion, such as the surface quasi-geostrophic equation.","Numerical simulations of the system should remain regular for all tested initial data inside the stated parameter range.","Further sharpening of the iteration could test whether regularity persists exactly at the critical line α + β = 1."],"forward_implications":["Smooth initial data produce globally smooth solutions throughout the subcritical regime.","The threshold separating guaranteed regularity from possible singularity formation is exactly α + β = 1.","The same dissipation strength that works for α > 2/3 also works for weaker velocity dissipation.","No additional restrictions on the individual exponents are needed beyond their sum exceeding one."],"fun_headline_variants":["2D Boussinesq regular when α + β > 1","Regularity holds for fractional Boussinesq subcritical case","Boussinesq regularity established for α ≤ 2/3","Subcritical Boussinesq equations regular with α + β > 1","Boussinesq global regularity via iterative bounds for α + β > 1"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The iterative procedure closes all estimates under no stronger restriction than α + β > 1, even when α ≤ 2/3.","fun_headline_variants_meta":{"raw":{"variants":["2D Boussinesq regular when α + β > 1","Regularity holds for fractional Boussinesq subcritical case","Boussinesq regularity established for α ≤ 2/3","Subcritical Boussinesq equations regular with α + β > 1","Boussinesq global regularity via iterative bounds for α + β > 1"]},"model":"grok-4.3","cost_usd":0.00768,"raw_usage":{"total_tokens":3476,"prompt_tokens":593,"num_sources_used":0,"completion_tokens":94,"cost_in_usd_ticks":76799500,"prompt_tokens_details":{"text_tokens":593,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2789,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":593,"tokens_out":94,"duration_ms":18772,"temperature":1.0,"reasoning_tokens":2789,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T09:21:37.788107+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit smooth initial datum whose solution loses regularity in finite time while α + β > 1 and α ≤ 2/3 would falsify the global-regularity claim.","supporting_citations":[],"review_version":1}