{"id":"89d5daae-832c-4914-b1a0-529c5118cc21","arxiv_id":"2412.11094","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"For odd, homogeneous, order-delta multipliers with -1 <= delta <= 0, there exist non-trivial weak solutions failing to conserve the Hamiltonian at every regularity Lambda^{-1}theta in C^gamma with gamma < 1 + 2delta/3, matching the rigid bound.","lead":"A new convex integration proof constructs non-unique, Hamiltonian-breaking weak solutions for a broad family of 2D active scalar equations at the sharp Onsager threshold. The result unifies the known 2D Euler and SQG flexible cases and extends to even multipliers in an appendix.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.5 is load-bearing: the Newton closure (Prop. 5.6) fails if the trilinear kernel bounds in Step 3.3 have an extra frequency factor, and the proof omits the decisive L1 estimates.","rationale":"I agree with the reader's identification of the weakest assumption. The central claim is the flexible part of the generalized Onsager conjecture for general 2D odd active scalar equations; for it to hold at the stated sharp exponents, the Newton-Nash induction must close under the parameter condition β<1+2δ/3. That closure depends directly on the material-derivative estimate for R_{q,n+1}, which in turn uses Lemma 4.5, estimate (4.12). The proof of Lemma 4.5 is the technical heart of the Newton step: Steps 3.1-3.3 and Step 6 decompose the trilinear symbol into many frequency regimes, and the Hölder gain comes precisely from L1 kernel bounds obtained via Lemma 4.2. The critical Step 3.3 is summarized with 'some tedious calculations', leaving the most delicate balance of k1, k, k2, k3 unverified. I do not see a contradiction or a clear counterexample, and the overall strategy is credible: the momentum-level formulation raises the effective homogeneity by one, the algebraic lemma handles non-symmetric stresses, and the threshold matches the known rigidity result of Isett-Ma. But the omitted kernel bounds are load-bearing rather than cosmetic. A secondary issue is that Theorem 1.3 states joint C^γ regularity, whereas the proof only establishes C^0_t C^γ_x; this is a genuine overstatement but does not change the main conclusion if Lemma 4.5 is corrected. The appropriate verdict remains CONDITIONAL: the author should provide a complete proof of Lemma 4.5, especially the L1 estimates in Steps 3.2-3.3 and 6, before the paper can be accepted as fully verified.","tokens_in":79892,"tokens_out":16695,"duration_ms":142518,"concrete_test":"Fix the mSQG multiplier m(ξ)=iξ^⊥|ξ|^{δ−1} with δ=−1/2 (so m̃(ξ)=|ξ|^{δ+1}) and the frequency configuration k1=0, k2=k3=100, k=50 in the Z3 regime of Lemma 4.5. Compute the L1 norm of the inverse Fourier transform of M^4_{s,i,k,ℓ} (the term containing ∫∫ η_i ζ_s ∇^2 m̃ dτdμ) using Lemma 4.2. If the bound is O(1) in k,k1,k2,k3, (4.12) survives this regime; if it grows like 2^{k−k1} or 2^{k2−k}, then (4.12) fails and Proposition 5.6's material-derivative bound acquires an extra λ_q/λ_{q+1} factor, breaking the Newton iteration. Equivalently, ask the author to supply the missing L1 estimates for all terms M^3,...,M^6 and independently verify that no positive power of 2^{k−k1} or 2^{k2−k} survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 5.6, which advances the Newton step, estimates the material derivative D̄_t R_{q,n+1} by expanding D̄_t S[·] and invoking Lemma 4.5 for the trilinear terms E4 (and analogous E5 terms in Lemma 6.13). Thus the bound (4.12) for S^i[u,φ,ψ] is load-bearing: if any frequency regime loses a power of λ_{q+1}/λ_q, the displayed final inequality in Prop. 5.6 is off by a factor (λ_q/λ_{q+1})^c and the induction cannot close. The proof of (4.12) is not complete. In Step 3.3 (the regime k1+250≤k≤min{k2,k3}−500, |k2−k3|≤999) the author writes the symbol as M^3+⋯+M^6 and then states, without displaying the estimates, that 'using Lemmas 4.1, 4.2 and ... with some tedious calculations' one gets ||F^{-1}(M^{...}_{s,i,k,ℓ})||_{L1}≲1. This L1 kernel bound is exactly what converts symbol smoothness into the Hölder gain; it is not a routine detail because the symbols contain factors such as ξ/|ξ+η+ζ|, (η+ζ)_s/|ξ|, and double integrals of ∇^2 m̃, with three separated scales present. A missing factor 2^{k−k1} or 2^{k2−k} would make (4.12) false and invalidate the Newton step. Similar hand-waving appears in Step 3.1 ('one can easily verify') and in Lemma 4.4 ('proof ... omitted'), but Step 3.3 is decisive. The theorem also states joint C^γ(R×T^2) regularity while the proof only yields C^0_t C^γ_x, but this overstatement is secondary to the closure issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to prove the flexible part of a generalized Onsager conjecture for general 2D odd active scalar equations. For a Fourier multiplier symbol m that is odd, divergence-free, homogeneous of order δ ∈ [-1,0], and smooth away from the origin, the main theorem asserts that for every (1+δ)/2 ≤ γ < 1+2δ/3 there exist non-trivial weak solutions with compact temporal support such that Λ^{-1}θ ∈ C^γ(R_t × T^2_x), and these solutions fail to conserve the Hamiltonian. The proof works at the level of the potential field v = -∇^⊥ θ, reformulating the equation as an ASM-Reynolds system, and uses a Newton-Nash convex integration scheme following Giri-Radu. The main technical novelty is a set of bilinear and trilinear Fourier multiplier estimates, in particular Lemma 4.5, together with an algebraic lemma that provides an anti-divergence operator adapted to the general multiplier.","tokens_in":80287,"tokens_out":9607,"duration_ms":85997,"significance":"If the proof is completed, this would resolve the flexible part of the Onsager-type conjecture for the entire class of odd active scalar equations with homogeneous symbols of order δ ∈ [-1,0], unifying the previously known cases of 2D Euler (δ=-1) and SQG (δ=0), and matching the rigidity results of Isett-Ma. The paper introduces a genuinely new algebraic decomposition for the stress in the absence of symmetry of the tensor ξ ⊗ ∇ m̄(ξ), and the trilinear Fourier multiplier framework appears to be novel. These are potentially significant contributions to the convex integration literature. However, the central technical lemma is not fully proved in the submitted text, and the stated regularity of the solutions exceeds what the proof establishes.","major_comments":[{"comment":"The proof of Lemma 4.5 asserts the L1 kernel bound ||F^{-1}(M^{...}_{s,i,k,ℓ})||_{L1} ≲ 1 in Step 3.3 by 'some tedious calculations' without displaying the derivative estimates. These bounds are load-bearing: Proposition 5.6 invokes (4.12) for the E4 terms, and a loss of a single power of 2^{k-k1} or 2^{k2-k} in this regime would break the Newton-step closure. Please provide the explicit derivative bounds for the symbols M^3,...,M^6 and the resulting kernel estimates in Steps 3.1-3.3 (and the analogous claims in Steps 4-6), with all powers of the frequency scales recorded.","section":"Section 4, Lemma 4.5, Step 3.3"},{"comment":"Theorem 1.3 states Λ^{-1}θ ∈ C^γ(R_t × T^2_x), but the proof in Section 2.3 only establishes that {v_q} is Cauchy in C^0_t C^γ_x and that R_q → 0 in that space. No estimates controlling the time Hölder seminorm are given. The theorem should be restated as C^0_t C^γ_x, or the missing time-regularity argument should be provided.","section":"Theorem 1.3 and Section 2.3"},{"comment":"Lemma 4.4 is stated with 'the proof is almost the same ... and thus omitted', and Lemmas 6.8 and 6.9 similarly omit details. These lemmas feed directly into the estimates of the bilinear and trilinear multiplier bounds used in Propositions 5.6 and 6.14. Given that Lemma 4.3's proof is lengthy and the operators in Lemma 4.4 have a different structure, the omission leaves a gap in the verification of (4.6)-(4.7). Please include the proofs or a precise reduction to Lemma 4.3.","section":"Lemma 4.4 and Lemmas 6.8-6.9"}],"minor_comments":[{"comment":"The abstract states 'θ ∈ C^0_t C_x^{2δ/3-}', but Theorem 1.3 concerns Λ^{-1}θ ∈ C^γ; the relationship between these two regularities should be clarified and the apparent mismatch resolved.","section":"Abstract and Theorem 1.3"},{"comment":"Theorem D.1 is stated for even multipliers, but the proof is only a sketch that ends with 'we just conclude the main results here without providing details'. If this theorem is part of the paper's claims, it requires a complete proof or should be explicitly labeled as a conjecture.","section":"Appendix D"},{"comment":"There are numerous typos and formatting artifacts, including 'comlpex', 'filed', 'Lagraingian', and a missing summation sign in equation (4.16) where 'usBs' appears without denoting the sum over s.","section":"Throughout"},{"comment":"The transport equation for ψ_{k,n+1} is solved with initial data at t=t_k, but the existence of a unique solution on the support of ˜χ_k should be justified using the flow estimates of Lemma 5.2; this is standard, but it should be stated explicitly for completeness.","section":"Section 5.3, equation (5.16)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is ambitious and the overall architecture is coherent; the main missing piece is the detailed verification of the trilinear multiplier estimates, which is a substantial gap because Lemma 4.5 underlies the Newton-step closure in Proposition 5.6. In my assessment the gap is likely fixable by carrying out the displayed 'tedious calculations', but it is significant enough that the paper should not be accepted in its current form. The overstatement of the joint Hölder regularity in Theorem 1.3 is a separate but important correction. The appendix on even multipliers is not a proof in its current form, so the abstract's claim of analogous results should be moderated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious paper on the right problem. The main theorem – existence of non-conservative weak solutions with Λ^{-1}θ ∈ C^γ for every γ < 1 + 2δ/3, matching the Isett–Ma rigidity threshold – is a genuine advance. Prior sharp flexible constructions only covered Euler (δ=−1) and SQG (δ=0); the full range −1 ≤ δ ≤ 0 was open. If correct, this closes the flexible part of the generalized Onsager conjecture for odd active scalars. That is worth a referee's time.\n\nThe architecture is coherent: working at the level of the potential v = −∇⊥(−Δ)^{-1}θ buys one degree of homogeneity, the adapted Newton–Nash iteration from Giri–Radu is appropriate, and the algebraic lemma for decomposing stresses into ξ ⊗ ∇m̄(ξ) with a multiplier-dependent anti-divergence operator is a real idea. The trilinear Fourier multiplier estimates are the technical heart, and the paper is honest about where its novelty lies.\n\nNow the soft spots, in proportion. The proof of Lemma 4.5 is not complete. That lemma is load-bearing: Proposition 5.6's Newton closure depends on estimate (4.12), and the decisive frequency regime (Step 3.3) is dispatched with 'some tedious calculations' and 'one can verify'. The stress test is concrete: if any L1 kernel bound there carries an extra factor λ_{q+1}/λ_q, the closure fails. I am not asserting it does fail – the frequency configurations look plausible, and the statements of the estimates are consistent with the claimed result – but a referee must fill in those details before the paper can be accepted.\n\nTwo smaller issues. Theorem 1.3 states joint C^γ(R×T^2) regularity, but the proof constructs C^0_t C^γ_x; the theorem statement should be corrected. And Appendix D, on even multipliers, is explicitly a sketch; it is a secondary result and clearly marked as such, so I would not hold the paper to the same standard there.\n\nBottom line: this deserves a serious referee. The main theorem is likely true, the framework is right, and the gaps are specific and apparently fillable. I would send it to peer review and ask the referee to focus on Lemma 4.5 and the regularity statement.","headline":"Plausibly correct and important result for the flexible part of the Onsager-type conjecture for general odd 2D active scalars, but the proof's key trilinear estimate is not fully verified and the stated regularity is overstated.","tokens_in":80879,"tokens_out":1689,"would_cite":true,"duration_ms":20670,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35D30","76B03","42B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every odd active scalar equation in two dimensions, the paper constructs non-conservative Hölder solutions up to the sharp Onsager threshold.","keywords":["Onsager conjecture","active scalar equations","convex integration","Newton-Nash iteration","Hamiltonian conservation","trilinear Fourier multipliers","surface quasi-geostrophic","weak solutions"],"falsifier":"Evaluate the symbol $M_{s,i}(\\xi,\\eta,\\zeta)$ of Lemma 4.5 on frequency triads with $|\\xi| \\ll |\\eta| \\approx |\\zeta|$ and check whether the claimed bound (4.12) holds; if one triad violates it, the material-derivative estimate on the Newton stress error in Proposition 5.6 breaks, so constructing such a counterexample would disprove the convergence proof.","tokens_in":79625,"feed_emoji":"🌀","tokens_out":9529,"duration_ms":81149,"temperature":0.7,"pith_summary":"The paper aims to establish the flexible half of the generalized Onsager conjecture for all two-dimensional odd active scalar equations: $\\partial_t\\theta + u\\cdot\\nabla\\theta = 0$ with $u = T[\\theta]$ a divergence-free Fourier multiplier whose odd, $\\delta$-homogeneous symbol satisfies $-1 \\le \\delta \\le 0$. The main theorem asserts that for every $(1+\\delta)/2 \\le \\gamma < 1+2\\delta/3$ there exist non-trivial, temporally compact-supported weak solutions with $\\Lambda^{-1}\\theta \\in C^\\gamma$ that violate Hamiltonian conservation. This is exactly the regularity window left open by the rigidity side in the reference cited as [23], so the result locates the phase transition for Hamiltonian conservation at the exponent $1+2\\delta/3$. The proof unifies the sharp flexible results for the 2D Euler vorticity equation and the SQG equation, and the appendix treats even multipliers as well.","feed_headline":"Flexible Onsager theorem proven for all 2D odd active scalars","feed_subtitle":"Hölder solutions that break Hamiltonian conservation exist up to the sharp exponent for every odd multiplier.","key_machinery":"The argument is carried by a Newton-Nash convex integration scheme lifted to the potential velocity $v = (-\\Delta)^{-1}\\nabla^\\perp\\theta$, whose momentum system has velocity $u = T_1[v]$ with an even multiplier of order $1+\\delta$; this extra derivative compensates for the negative homogeneity of $m$. The scheme rests on three custom pieces: a classification of $m$ into two or three independent trace-free tensors $\\xi \\otimes \\nabla\\bar{m}(\\xi)$, which yields an $m$-dependent anti-divergence operator and the algebraic Lemma 3.2 that decomposes the Reynolds stress into elementary tensors; sharp estimates for the trilinear Fourier multiplier commutators $S^0$, $S^i$ and $S^{i,j}$ in Lemma 4.5, which control material derivatives without derivative loss; and a multi-step Newton iteration with temporally disjoint profiles whose number of steps $\\Gamma$ is chosen so that gluing errors fall below the Nash error.","core_discovery":"On the paper's own terms, the discovery is Theorem 1.3: for any odd, divergence-free, $\\delta$-homogeneous Fourier multiplier $m$ smooth away from the origin with $-1 \\le \\delta \\le 0$, and for any exponent $(1+\\delta)/2 \\le \\gamma < 1+2\\delta/3$, the active scalar equation admits non-trivial weak solutions $\\theta$ with compact temporal support and $\\Lambda^{-1}\\theta \\in C^\\gamma(\\mathbb{R} \\times \\mathbb{T}^2)$. These solutions fail to conserve the Hamiltonian, while the companion rigidity result [23] guarantees conservation above that threshold; together the two statements identify $1+2\\delta/3$ as the sharp critical exponent. In the appendix, an analogous construction for even multipliers in two and three dimensions yields non-trivial Hölder solutions with exponent below $1/3$, matching the known energy-conservation threshold.","pith_inferences":["Beyond the paper, the tensor hierarchy sketched in Section 1.3 suggests the same construction should extend to $\\delta < -1$ by iterating potential fields of higher order; a testable project is to formulate the multilinear estimates those higher systems require.","The multiplier classification in Section 3.1 reveals that the exceptional symbols blocking the algebraic lemma are precisely the higher-order polynomial-like cases and the degenerate stationary case $m \\propto i\\xi^\\perp$, which hints that any future treatment of $\\delta > 0$ must first handle these exceptions.","A numerical check of the symbol $M_{s,i}(\\xi,\\eta,\\zeta)$ on frequency triads with $|\\xi| \\ll |\\eta| \\approx |\\zeta|$ would test whether the cancellations asserted in Lemma 4.5 hold as stated."],"forward_implications":["Hamiltonian conservation for general 2D odd active scalars now has a sharp threshold: weak solutions conserve the Hamiltonian above $C^{1+2\\delta/3}$ and can violate it strictly below.","The same iteration applies uniformly to the mSQG family, recovering the sharp flexible results for 2D Euler in vorticity form and for SQG as endpoints $\\delta = -1$ and $\\delta = 0$.","The non-conservative solutions are compactly supported in time, so the theorem also implies failure of uniqueness and of compactness of solution sets at those regularities.","For even multipliers, the appendix gives non-trivial $C^\\alpha$ solutions with $\\alpha < 1/3$, matching the energy-conservation threshold known from the positive side."],"supporting_citations":[{"why":"Supplies the Newton-Nash iteration and the temporal-profile construction on which the scheme is built.","marker":"[17]"},{"why":"Proves the rigid half, Hamiltonian conservation above the threshold, that makes the flexible construction sharp.","marker":"[23]"},{"why":"Establishes the sharp flexible result for SQG that this paper generalizes to arbitrary odd multipliers.","marker":"[10]"},{"why":"Provides the earlier direct SQG construction and the algebraic decomposition into tensors $\\xi \\otimes \\nabla\\bar{m}(\\xi)$ that the paper adapts.","marker":"[22]"},{"why":"Introduced the potential-velocity reformulation for SQG and gave the first partial flexible result for that equation.","marker":"[4]"},{"why":"Gives the previous partial range for modified SQG equations, the baseline that Theorem 1.3 improves to sharpness.","marker":"[26]"}],"fun_headline_variants":["Sharp Onsager exponent for all 2D odd active scalars","Hamiltonian-breaking solutions for every odd active scalar","Non-conservative weak solutions exist for all 2D odd scalars","All 2D odd active scalars break Hamiltonian conservation","Onsager flexibility resolved for all 2D odd active scalars"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The iteration closes only if the sharp trilinear Fourier multiplier estimate in Lemma 4.5, for the commutators $S^0[u,\\varphi,\\psi]$, $S^i[u,\\varphi,\\psi]$ and $S^{i,j}[u,\\varphi,\\psi]$, is valid in every frequency arrangement for the full range $-1 \\le \\delta \\le 0$, and the proof of that lemma is where several steps are only summarized.","fun_headline_variants_meta":{"raw":{"variants":["Sharp Onsager exponent for all 2D odd active scalars","Hamiltonian-breaking solutions for every odd active scalar","Non-conservative weak solutions exist for all 2D odd scalars","All 2D odd active scalars break Hamiltonian conservation","Onsager flexibility resolved for all 2D odd active scalars"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001107,"raw_usage":{"total_tokens":4671,"prompt_tokens":1058,"completion_tokens":3613,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":3524}},"tokens_in":674,"tokens_out":3613,"duration_ms":21682,"temperature":1.0,"reasoning_tokens":3524,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:19:36.032531+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the symbol $M_{s,i}(\\xi,\\eta,\\zeta)$ of Lemma 4.5 on frequency triads with $|\\xi| \\ll |\\eta| \\approx |\\zeta|$ and check whether the claimed bound (4.12) holds; if one triad violates it, the material-derivative estimate on the Newton stress error in Proposition 5.6 breaks, so constructing such a counterexample would disprove the convergence proof.","supporting_citations":[{"cited_title":"Isett and A","cited_arxiv_id":null,"evidence_quote":"Provides the earlier direct SQG construction and the algebraic decomposition into tensors $\\xi \\otimes \\nabla\\bar{m}(\\xi)$ that the paper adapts."},{"cited_title":"Buckmaster, S","cited_arxiv_id":null,"evidence_quote":"Introduced the potential-velocity reformulation for SQG and gave the first partial flexible result for that equation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the previous partial range for modified SQG equations, the baseline that Theorem 1.3 improves to sharpness."}],"review_version":1}