{"id":"3c893bdd-ce51-485b-a1c7-bbb9208df1ea","arxiv_id":"2607.27044","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"There exist autonomous C^α divergence-free planar fields that fail the weak Sard property but induce no anomalous dissipation for advection-diffusion.","lead":"The paper builds a compactly supported 2D divergence-free velocity field of any Hölder regularity below 1 that fails the weak Sard property yet produces no anomalous L² dissipation for the advection-diffusion equation. This separates two notions previously conjectured to be equivalent and shows failure of weak Sard is not enough for anomalous dissipation.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates Proposition 4.3 / Corollary 4.4 as the quantitative heart of the no-dissipation half, yet that criterion is standard (triangle inequality + energy balance + explicit flow in Ψ_q-coordinates) and the paper supplies an explicit open set of parameters that make both (4.4) and (4.5) hold at once while preserving the C^{1,α} convergence needed for u ∈ C^α. The Sard-failure half is measure-theoretic and independent. Because the two sides do not pull the parameters in incompatible directions, the load-bearing assumption is satisfied rather than merely hoped for. No correction to the ACCEPT / high-confidence verdict is warranted.","tokens_in":21594,"tokens_out":902,"duration_ms":18964,"concrete_test":"Fix α = 1/2, choose concrete integers satisfying (2.16) (e.g. V = 16, 1/W = V^{ε_{2}} = 4, 1/H = V^{1+ε_{1}} = 32) and p = 3, q_{0} large; recompute the three ratios that appear in Corollary 4.4 and Proposition 3.2 (H/V, 2c log V - 1, and 2 H V^α/W^{1+α}). Verify all three are strictly less than 1 and that the resulting geometric series for ∥H-H_q∥_{C^{1,α}} still converges. If any ratio ≥ 1 the simultaneous choice fails; otherwise the parameter regime is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is internally consistent. The two halves of Theorem 1.1 rest on independent verifications that do not conflict under the stated parameter regime. Weak-Sard failure follows from the positive-measure nested affine bulk P_∞ = U^∞_x \times U^∞_y (Section 5): Ψ(P_∞) ⊂ S igcap E* and H_#(1_Ψ(P_∞) L^{2}) = L^{1}(U^∞_x) 1_{U^∞_y} dh with both factors positive, so the push-forward is non-trivial and absolutely continuous. Absence of anomalous dissipation follows from Proposition 4.3 once the same parameters satisfy both (4.4) and (4.5). The choice in Corollary 4.4 (V/H > V^{2}, c ∈ (1/(log V-log H), 1/(2 log V)), q(κ) = ⌊ c log(1/κ)⌋) makes ∥H-H_q∥_∞/κ \to 0 by the geometric sum of heights h_j ∼ (H/V)^j and makes κ ∫∫|DΦ_q|^{2} \to 0 by the bound (3.8) of order (q)^{O(1)} V^{2q}. The C^{1,α} Cauchy estimate in Proposition 3.2 simultaneously requires 2 H V^α / W^{1+α} < 1, which is compatible with (2.16) for every α ∈ (0,1) by taking rational ε_{1} < ε_{2} < (1-α)/(1+α) and large integer V. No hidden obstruction appears in the gluing of the quarter-turn maps, the measure-preservation of the limiting homeomorphism Ψ, or the energy estimates of Lemmas 4.1–4.2.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs, for every α∈(0,1), a compactly supported autonomous divergence-free velocity field u∈C^α_c(R²,R²) whose Hamiltonian fails the weak Sard property, yet for which the associated advection-diffusion equation exhibits no anomalous dissipation of L² norm (Theorem 1.1). The construction proceeds by iteratively composing measure-preserving rectangular quarter-turn maps on a tubular neighborhood of a closed curve, producing a sequence of smooth Hamiltonians H_q converging in C^{1,α} to a limit H. Failure of weak Sard is read off from a positive-measure nested affine bulk P_∞ whose image under the limiting parametrization Ψ lies in S∩E* and pushes forward to a nontrivial absolutely continuous measure on the range. Absence of anomalous dissipation is obtained from a quantitative approximation criterion (Proposition 4.3) comparing the viscous solution for H to inviscid solutions for the smooth approximants H_q, once stretching parameters are chosen so that both ||H-H_q||_∞/κ_q\to0 and κ_q∫∫|DΦ_q|²\to0.","tokens_in":22010,"tokens_out":1019,"duration_ms":48329,"significance":"The result cleanly separates two properties that recent work had suggested might be equivalent for autonomous planar fields: failure of (weak/relaxed) Sard and anomalous dissipation. It thereby disproves Conjecture 1.3 of Bagnara-Boutros-De Lellis-Mayboroda. The argument is fully constructive and self-contained, with explicit diffeomorphisms, an explicit flow in adapted coordinates, and a usable comparison criterion (Proposition 4.3) that may be of independent interest. Parameter compatibility for every α∈(0,1) is checked carefully. This is a solid, definitive counterexample paper in the active area of passive-scalar anomalous dissipation.","major_comments":[],"minor_comments":[{"comment":"Lemma 2.1 proof: typo \"diffeomorphsims\". Several other minor typos appear (e.g., spacing artifacts in the title block, \"Dissip A TION\").","section":"Lemma 2.1"},{"comment":"The admissible range for (V,W,H,ε₁,ε₂,p,q₀) is scattered across (2.16), (2.23), Proposition 3.2 and Corollary 4.4. A short dedicated remark collecting one explicit admissible tuple (or the full list of inequalities) would help the reader verify compatibility at a glance.","section":"Section 2.4 / Corollary 4.4"},{"comment":"In Lemma 3.3 the matrix M_q is defined with columns (∂_s Ψ_q, ∂_h Ψ_q), so det M_q ≡ -1, while (3.3) records det(∂_h,∂_s)=1. The signs are consistent but easy to misread; a one-line clarification would prevent confusion.","section":"Lemma 3.3"},{"comment":"Figure 1 and Figure 3 are helpful; adding a brief caption note that shaded subrectangles in Figure 3 are exactly the affine bulk pieces belonging to P_δ would make the link to Definition 2.3 immediate.","section":"Section 2"},{"comment":"The push-forward identity (5.1) is correct; it may be worth stressing explicitly that the full circle Ψ(T\times{h}) lies in the level set (not only the Cantor slice U^∞_x\times{h}), which is what guarantees the connected component is in E*.","section":"Proposition 5.1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is technically solid and appropriate in scope for a strong analysis journal. I see no load-bearing gaps. The minor-revision recommendation is only for presentation polish; I would accept after a light revision pass. The citation list includes several very recent/parallel arXiv preprints in the same circle; that is natural for the area and not a concern."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper gives an explicit autonomous divergence-free u in C^α_c(R^{2}) that fails weak Sard yet has vanishing anomalous dissipation for every L^{2} initial datum. That directly kills Conjecture 1.3 of Bagnara–Boutros–De Lellis–Mayboroda (up to the usual periodic extension). The two halves are independent and both check out.\n\nWhat is new is the iterative rectangular quarter-turn construction. Start with a measure-preserving tubular parametrization of an annulus that is affine on a rectangle, then compose maps that contract width and expand height by a large factor V on a nested positive-measure “affine bulk” P_q while staying identity near the boundaries. The limiting Hamiltonian is C^{1,α}, the critical set contains Ψ(P_∞) of positive measure inside E*, and the push-forward is absolutely continuous and non-trivial, so weak Sard fails by direct computation. At the same time the same parameters keep ||H-H_q||_∞ decaying geometrically in (H/V)^q and the flow deformation of the smooth approximants only polynomially-in-q times V^{2q}. Choosing q ~ c log(1/κ) with c in a nonempty open interval makes both hypotheses of the comparison criterion (Prop. 4.3) hold, so dissipation vanishes.\n\nThe math is self-contained and carefully parameterized: the C^{1,α} Cauchy estimate, the L^∞ comparison lemma, the explicit flow formula Φ_q(t,Ψ_q(s,h))=Ψ_q(s+t,h), and the energy estimates are all written with explicit constants and compatible regimes for every α∈(0,1). Citations sit where they should (ABC, JS24, BBDLM26, DGG26). No circularity; the two conclusions are verified separately.\n\nSoft spots are minor and mostly presentational. The construction is intricate (many nested rectangles, cut-offs, and free parameters V,W,H,δ_q), so a reader has to track the affine generations carefully, and there is no code. The setting is strictly 2-D autonomous, which is exactly the regime of the conjecture being disproved, so that is not a defect. The comparison criterion itself is natural and the parameter window is nonempty.\n\nThis is for people working on passive scalars, weak Sard, and necessary conditions for anomalous dissipation. It deserves a serious referee and should be engaged with; I would bring it to reading group and expect to cite the counter-example.","headline":"Clean counterexample: autonomous C^α fields can fail weak Sard without anomalous dissipation, refuting the BBDLM26 conjecture.","tokens_in":22641,"tokens_out":673,"would_cite":true,"duration_ms":14534,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35B65","76F25"],"pacs":[],"model":"grok-4.5","headline":"For every Hölder exponent below 1 there is a compactly supported autonomous 2D flow that fails the weak Sard property yet produces no anomalous dissipation.","keywords":["anomalous dissipation","weak Sard property","advection-diffusion equation","autonomous divergence-free fields","Hamiltonian flows","passive scalars","measure-preserving diffeomorphisms"],"falsifier":"Along the paper’s explicit sequence κ_q ~ exp(−q/c) for the constructed limiting field, evaluate or rigorously lower-bound κ ∫ ‖∇θ^κ‖² for a fixed smooth compactly supported initial datum; a strictly positive limsup would falsify the central claim.","tokens_in":22458,"feed_emoji":"🌀","tokens_out":963,"duration_ms":38874,"temperature":0.7,"pith_summary":"This paper constructs, for every smoothness exponent α between 0 and 1, a compactly supported divergence-free velocity field in the plane that is only C^α. The field fails the weak Sard property—a measure-theoretic condition on the critical set of its Hamiltonian that is already known to be necessary for anomalous energy loss—yet the associated advection-diffusion equation still sends the dissipated energy to zero as diffusivity vanishes. The construction repeatedly folds an annular region by measure-preserving maps that stretch and contract nested rectangles, yielding a limiting Hamiltonian whose critical set is large enough to break weak Sard, while intermediate smooth fields approximate the limit fast enough that diffusion cannot lock onto a positive energy fraction. The result disproves a conjecture that failure of a related Sard-type condition should be equivalent to anomalous dissipation. A reader who follows passive-scalar and 2D Hamiltonian transport cares because the example cleanly separates two phenomena that had been expected to travel together.","feed_headline":"Rough 2D flows can break weak Sard yet not dissipate","feed_subtitle":"Counterexample separates two phenomena once expected to be equivalent for passive scalars","key_machinery":"A quantitative approximation criterion comparing viscous solutions for the limiting Hamiltonian H against inviscid transport for smooth approximants H_q: if ‖H−H_q‖_∞/κ_q → 0 and κ_q times the space-time integral of |DΦ_q|² vanishes, anomalous dissipation is ruled out. The H_q are built by composing measure-preserving rectangular quarter-turn maps on nested affine bulks inside a tubular neighborhood of a closed curve, so the flow in those coordinates is an explicit translation.","core_discovery":"For every α in (0,1) there exists a compactly supported autonomous divergence-free vector field u in C^α_c(R²,R²) whose Hamiltonian fails the weak Sard property, yet for every square-integrable initial datum the unique parabolic solution of the advection-diffusion equation satisfies lim_{κ↓0} κ ∫₀¹ ‖∇θ^κ_t‖²_{L²} dt = 0.","pith_inferences":["The same nested-rectangle family appears tunable: slowing the approximation of H_q relative to κ should push the construction across a threshold into anomalous dissipation, giving a single-parameter bridge between the two regimes.","Periodic extension immediately places the counterexample on the torus, so the separation persists in the geometry most often used for idealised turbulence models.","Any successful criterion for anomalous dissipation in autonomous 2D flows will likely need a uniform positive lower bound on Lagrangian variance in the vanishing-noise limit, beyond mere failure of weak Sard."],"forward_implications":["Failure of the weak Sard property is not sufficient for anomalous dissipation of passive scalars driven by autonomous 2D flows.","The conjecture equating failure of the relaxed Sard property with anomalous dissipation is false after periodic extension of the example.","Autonomous planar examples with non-unique inviscid transport can still be dissipation-regular under vanishing diffusion.","Future sufficient criteria for anomalous dissipation in this class must track quantitative stretching rates, not only the measure of the critical set."],"fun_headline_variants":["C^α 2D flows break weak Sard without anomalous dissipation","Autonomous fields fail weak Sard yet spare L2 norm","No weak Sard, no anomalous dissipation for rough 2D flows","Counterexample separates weak Sard failure from energy loss","Compact C^α fields break weak Sard but not L2 conservation"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The no-dissipation half of the theorem rests on choosing the stretching ratios and cutoff scales so that one sequence of smooth Hamiltonians both approximates the rough limit faster than diffusivity and keeps flow deformation from blowing up too fast.","fun_headline_variants_meta":{"raw":{"variants":["C^α 2D flows break weak Sard without anomalous dissipation","Autonomous fields fail weak Sard yet spare L2 norm","No weak Sard, no anomalous dissipation for rough 2D flows","Counterexample separates weak Sard failure from energy loss","Compact C^α fields break weak Sard but not L2 conservation"]},"model":"grok-4.5","effort":"low","cost_usd":0.003675,"raw_usage":{"total_tokens":1115,"prompt_tokens":645,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":36748000,"prompt_tokens_details":{"text_tokens":645,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":402,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":645,"tokens_out":68,"duration_ms":7750,"temperature":1.0,"reasoning_tokens":402,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T12:51:17.198693+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Along the paper’s explicit sequence κ_q ~ exp(−q/c) for the constructed limiting field, evaluate or rigorously lower-bound κ ∫ ‖∇θ^κ‖² for a fixed smooth compactly supported initial datum; a strictly positive limsup would falsify the central claim.","supporting_citations":[],"review_version":1}