{"id":"cc1f40dd-72ea-4984-aab2-0eb8d56ae0b5","arxiv_id":"2412.14866","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For operators of reduced constant rank, Korn-Maxwell-Sobolev inequalities hold after subtracting a projection correction, including the limiting p=1 case.","lead":"This paper proves new Korn-Maxwell-Sobolev inequalities for constant coefficient differential operators that satisfy a reduced constant rank condition, a weaker hypothesis than the ellipticity used in prior work. As a result, previously out-of-reach combinations such as (trace, Curl) become valid after adding a projection correction on the left-hand side.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.5 is false as stated: the projection Π_B is not well-defined when B lacks global constant rank, and the proof conflates it with the projection inside ker A; a counterexample with reduced but not global constant rank refutes the general claim.","rationale":"The reader's weakest assumption is the external ellipticity complex theorem for the p=1 case. While that is a legitimate external dependence, the more serious flaw is in the main theorem for all 1<p<n: it uses Π_B, a projection onto ker B[ξ] in V, but only assumes reduced constant rank on ker A. When B does not have global constant rank, Π_B is not a bounded projection and the left-hand side need not even belong to the Sobolev space. The proof of Lemma 3.1 (in particular the step applying Lemma 2.1 with E=ker A) produces a projection inside ker A, not Π_B, and the two are conflated. The concrete example above satisfies the reduced constant rank hypothesis but yields an infinite left-hand side with a finite right-hand side, so Theorem 2.5(i) is false as stated. The paper needs an additional global constant rank assumption on B (or alternatively a different correction operator defined as the projection inside ker A), and the proof must be rewritten accordingly. Therefore the reader's ACCEPT verdict should be changed to REJECT.","tokens_in":1123,"tokens_out":1447,"duration_ms":415439,"concrete_test":"Run the counterexample: n=3, V=R^3, A(x,y,z)=z (so ker A = span{e1,e2}), W=R^2, B[ξ] = [0, |ξ|^2, ξ_2^2; 0, 0, ξ_1^2], k=2. Choose P=(0,φ,0) with φ∈C_c^∞ and φ̂ nonzero on {ξ_1=0}. Compute the e3-component of P−Π_B P in Fourier space: it equals 1_{ξ_1=0} · ξ_2^2|ξ|^2/(ξ_2^4+|ξ|^4) · φ̂(ξ). Verify that its inverse Fourier transform is supported on x_1=0 and is not in L^{p*}(R^3), so its W^{1,p*}-norm is infinite. Meanwhile A[P]=0 and BP=(Δφ,0) has finite L^p norm for any 1<p<3. If the norm is indeed infinite while the right-hand side is finite, Theorem 2.5(i) fails. This check settles the concern.","verdict_should_be":"REJECT","load_bearing_attack":"The theorem assumes only that B[ξ] restricted to ker A has constant rank, but the correction term uses Π_B, the Fourier projection onto ker B[ξ] in the whole space V. If B does not have constant rank, the symbol ξ ↦ Π_{ker B[ξ]} is discontinuous and Π_B is not a bounded Fourier multiplier, so the left-hand side is not defined as a standard tempered distribution. In Lemma 3.1 the proof invokes Lemma 2.1 with E = ker A, producing a projection inside ker A onto ker(B[ξ]|_{ker A}); that projection is not Π_B. The proof silently equates the two. A concrete counterexample shows failure: take n=3, V=R^3, ker A = span{e1,e2}, W=R^2, and B[ξ] = [0, |ξ|^2, ξ_2^2; 0, 0, ξ_1^2]. Then B[ξ]|_{ker A} has rank 1 for all ξ ≠ 0, so the hypothesis holds, but B has rank 2 on the set ξ_1≠0 and rank 1 on ξ_1=0, hence B lacks global constant rank. For P = (0, φ, 0) with φ∈C_c^∞, A[P]=0 and BP = (Δφ,0) ∈ L^p for any p>1, so the right-hand side of Theorem 2.5(i) is finite. The e3-component of P − Π_B P has Fourier multiplier 1_{ξ_1=0} · ξ_2^2|ξ|^2/(ξ_2^4+|ξ|^4) · φ̂(ξ), whose inverse Fourier transform is a distribution supported on the hyperplane x_1=0; it is not in W^{1,p*}(R^3) (its gradient has a nontrivial singular part). Thus the left-hand side is infinite while the right-hand side is finite, contradicting Theorem 2.5(i). The intended applications like A=tr, B=Curl have B globally constant rank, so the theorem could be repaired by adding that assumption, but as stated it is false.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Korn-Maxwell-Sobolev (KMS) inequalities for operators B that are of reduced constant rank relative to a linear map A. The main result, Theorem 2.5, asserts that if the restricted symbols B[ξ]|_{ker A} have constant rank, then for 1<p<n the homogeneous Sobolev norm of P - Π_B Π_{ker A} P is controlled by ‖A[P]‖_{W^{k-1,p*}} + ‖BP‖_{L^p}, with a variant for p=1 under an additional reduced cancellation condition. The proof proceeds through Lemma 3.1, which states an equivalence between the reduced constant rank condition and an L^q estimate with a negative Sobolev norm of BP, followed by Sobolev embedding and, for p=1, Van Schaftingen's Bourgain-Brezis estimates.","tokens_in":9467,"tokens_out":12763,"duration_ms":103191,"significance":"If the main theorem were correct, it would extend the KMS inequality framework from the reduced elliptic setting to the reduced constant rank setting, covering the motivating example (A,B)=(tr,Curl). The paper is short, clearly written, and its proof strategy relies on established external results (constant rank estimates, potentials, limiting Sobolev inequalities), all of which are appropriate. However, the central claim is false as stated: the reduced constant rank hypothesis does not make the Fourier projection Π_B a bounded or even well-defined operator on the relevant function spaces, and a concrete counterexample satisfies the hypotheses while violating the conclusion. The advertised examples, such as B=Curl, do have global constant rank, so a repaired statement is plausible, but the current theorem overreaches.","major_comments":[{"comment":"The main theorem is false as stated because the hypothesis that B[ξ]|_{ker A} has constant rank does not imply that the full symbol family B[ξ] has constant rank, and the projection Π_B is only a well-defined bounded Fourier multiplier in the latter case. Concretely, take n=3, V=R^3, ker A = span{e1,e2}, and let B be the second-order operator with symbol B[ξ] = [[0, |ξ|^2, ξ_2^2],[0, 0, ξ_1^2]]. Then B[ξ]|_{ker A} has rank 1 for all ξ≠0, but B[ξ] has rank 2 when ξ_1≠0 and rank 1 when ξ_1=0. For P=(0,φ,0) with φ∈C_c^∞(R^3), we have A[P]=0 and BP=(-Δφ,0)∈L^p for every p>1, so the right-hand side of the claimed inequality is finite. On the Fourier side, the e3-component of P - Π_B P equals (φ̂(ξ)/2) 1_{ξ_1=0}, whose inverse Fourier transform is δ(x_1) times a Schwartz function of (x_2,x_3); this distribution is not in W^{1,p*}(R^3). Hence the left-hand side is infinite while the right-hand side is finite, contradicting Theorem 2.5(i).","section":"Theorem 2.5(i), Lemma 3.1"},{"comment":"The proof applies Lemma 2.1 with E=ker A and u=Π_{ker A}P. In that application, the projection produced by Lemma 2.1 is the orthogonal projection onto ker(B[ξ]|_{ker A}) inside ker A (equivalently, onto ker A∩ker B[ξ] within V), not the projection Π_B onto ker B[ξ] in V. These two projections differ whenever Π_B does not map ker A into itself. In the counterexample above, at frequencies with ξ_1=0 the projection Π_B sends vectors in ker A = span{e1,e2} to vectors with a nonzero e3 component, so it does not preserve ker A. The proof silently identifies the two projections and therefore does not establish the stated estimate for the left-hand side with Π_B. The same identification issue affects the necessity direction (a)⇒(b), where testing with P∈C_c^∞(R^n;ker A) gives an estimate involving Π_B rather than the restricted projection.","section":"Lemma 3.1"}],"minor_comments":[{"comment":"The upgrade from the L^q estimate in Lemma 3.1 to the W^{k-1,p*} estimate is only stated, not shown; it requires applying Lemma 3.1 to derivatives ∂^α P for |α|=k-1 and using that A and B commute with constant-coefficient derivatives. This should be spelled out.","section":"Proof of Theorem 2.5(i)"},{"comment":"The phrase 'has constant rank for all ξ∈R^n\\{0}' is ambiguous; it should say that there exists r such that rank B[ξ]|_{ker A}=r for every ξ≠0.","section":"Theorem 2.5"},{"comment":"In the displayed formula for Π_BΠ_{ker A}P with B=Curl, a brief derivation of the row-wise Biot-Savart formula would help the reader; as written, the passage from the abstract projection to the convolution kernel is implicit.","section":"Example 2.6"},{"comment":"Reference [24] is a Master's thesis; if journal policy allows, this is acceptable, but the authors may wish to cite a published version or a more standard reference for the same material.","section":"References"}],"recommendation":"reject","confidential_remarks":"The manuscript is elegantly written and the intended applications are reasonable, but the main theorem is refuted by a concrete counterexample. The authors could repair the statement by assuming B has global constant rank rather than merely reduced constant rank; in that case the proof is simpler and the examples still work. However, as submitted, the central claim is false, so I recommend rejection of the current version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the core idea here is real and worth keeping — swapping reduced ellipticity for reduced constant rank and paying for it with a projection correction is a sensible extension, and the p=1 route through Van Schaftingen's cocancelling estimate is clean. But the main theorem is not correct as written, and the stress-test note is right to call it out.\n\nThe problem is that Π_B is never defined for an operator that only has reduced constant rank. The paper defines Π_A only for constant rank operators. Under the reduced constant rank hypothesis, the full symbol B[ξ] may jump rank, and the orthogonal projection onto its kernel is then a discontinuous function of ξ, not a bounded Fourier multiplier. The proof of Lemma 3.1 applies Lemma 2.1 with domain ker A, which yields the projection inside ker A onto ker(B[ξ]|_{ker A}). That projection is not the same as Π_B applied to Π_ker A P. The proof silently identifies the two, and they typically differ whenever ker B[ξ] is not contained in ker A.\n\nThe stress-test counterexample is valid. Take n=3, ker A = span{e1,e2}, and B[ξ] = [[0, |ξ|^2, ξ_2^2],[0,0,ξ_1^2]]. Restricted to ker A, this has rank 1 for every nonzero ξ, so the reduced constant rank hypothesis holds. But B itself has rank 2 for ξ_1≠0 and rank 1 for ξ_1=0, so it is not globally constant rank, and Π_B is not a bounded multiplier. For P=(0,φ,0), A[P]=0 and BP is smooth, so the right side is finite. The left side, P − Π_B Π_ker A P, has a Fourier contribution supported on the hyperplane ξ_1=0; its inverse Fourier transform has a component constant in x_1, so it is not in W^{1,p*}(R^3). The inequality fails.\n\nThis is a load-bearing flaw, not a cosmetic one. The intended applications, particularly (tr, Curl), do have B with global constant rank, and in that class the theorem is likely true. The fix is either to add the assumption that B has global constant rank, or to change the correction term to the projection inside ker A onto ker(B[ξ]|_{ker A}) and restate the theorem accordingly. The authors also need to say explicitly where Π_B is defined when B lacks global constant rank.\n\nCredit where it is due: the paper is short, the algebra in Lemma 3.1 is clean, and the use of Raita's potential theorem for the p=1 case is a sensible route. The necessity discussion in Section 4 is honest about what remains open. The minor issues are real but small: reference [24] is never cited, and the abstract seems to blur the distinction between [9] and [11].\n\nVerdict: this deserves a serious referee, but not acceptance as is. It needs a major revision to make the theorem true, either by restricting to global constant rank or by redefining the projection. I would not cite it in its current form, but I would look again after a fix.","headline":"The reduced-constant-rank idea is genuinely new and the p=1 argument is neat, but Theorem 2.5 is false as stated because Π_B is not defined when B is only reduced constant rank; the stress-test counterexample holds.","tokens_in":10036,"tokens_out":6707,"would_cite":false,"duration_ms":48585,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A23","26D10","35Q74","35Q75","46E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For operators of reduced constant rank, the paper proves a Korn-Maxwell-Sobolev inequality whose left-hand side subtracts the projection Π_B Π_{ker A} P, and establishes the borderline L^1 case under a reduced cancellation condition.","keywords":["Korn-Maxwell-Sobolev inequalities","constant rank operators","reduced constant rank","reduced cancelling","limiting Sobolev inequalities","L1 estimates","incompatible tensor fields","projection operators"],"falsifier":"Take a candidate pair such as (A,B)=(tr,Curl) and compute the symbol-level multiplier bound: if there exists ξ0 where the symbol of P − Π_B Π_{ker A} P cannot be controlled by |A[ξ]| + |B[ξ]|, a smoothed plane-wave sequence will violate the claimed inequality; this finite-dimensional linear-algebra computation settles the question at a stroke.","tokens_in":8794,"feed_emoji":"📐","tokens_out":8385,"duration_ms":55746,"temperature":0.7,"pith_summary":"This paper asks which operators can appear on the right-hand side of a Korn-Maxwell-Sobolev inequality when the classical elliptic condition fails. It answers that a weaker algebraic condition — reduced constant rank relative to the part map A — is sufficient, provided the left-hand side is corrected by subtracting the projection Π_B applied to the kernel-of-A projection. The authors prove the inequality for all 1<p<n, and for p=1 under an additional reduced cancellation condition, by combining a constant-rank projection estimate with a strong limiting Sobolev estimate for cocancelling operators. This extends previous classifications, which were restricted to reduced elliptic operators, and covers the previously out-of-reach combination (A,B)=(tr,Curl) in the borderline case.","feed_headline":"A projection correction unlocks constant-rank coercivity estimates","feed_subtitle":"Beyond ellipticity: subtracting a projection restores Korn-Maxwell-Sobolev bounds, even at the L1 limit.","key_machinery":"The machinery consists of three pieces: (1) the Fourier projection operator Π_A defined by projecting the Fourier transform onto ker A[ξ], which is bounded iff A has constant rank (Lemma 2.1); (2) the algebraic reduced constant rank condition on the restricted symbols B[ξ]|_{ker A}, which is shown in Lemma 3.1 to be equivalent to the projected inequality with a negative-Sobolev norm on BP; (3) for p=1, the existence of an ellipticity complex for constant rank operators — a companion operator L with ker L[ξ] = B[ξ](ker A) — and the strong Bourgain-Brezis estimate for cocancelling operators, which bridge from $L^{1}$ to the negative-Sobolev norm.","core_discovery":"The central discovery is that the reduced constant rank condition — B[ξ] restricted to ker A having constant rank for all nonzero ξ — is sufficient for the projected Korn-Maxwell-Sobolev inequality ‖P − Π_B Π_{ker A} P‖_{$W^{{k-1,p*}}$} ≤ c (‖A[P]‖_{$W^{{k-1,p*}}$} + ‖BP‖_{L^p}) for 1<p<n, and in the limiting case p=1 if additionally the intersection of the image symbols ∩_ξ B[ξ](ker A) is trivial. The proof works by splitting P into its ker A and (ker A)⊥ components, bounding the latter pointwise by A[P], and applying the constant-rank projection estimate of Lemma 2.1 to the former. For p=1, a companion ellipticity complex is used to transfer to a cocancelling operator and apply the strong limiting Sobolev estimate.","pith_inferences":["If the open necessity question in Section 4 is resolved affirmatively, reduced constant rank would be the exact threshold for the projected inequality, making the projection correction the canonical quantifier of non-ellipticity in this setting.","The projection correction Π_B Π_{ker A} may serve as the correct right-hand-side object for error estimates in mixed finite element methods for non-elliptic incompatibilities, where the trace/deviation structure appears in plasticity models.","The reliance on an ellipticity complex for the p=1 step suggests that extending the theorem to operators with nonconstant rank, or to variable coefficients, would need a new mechanism; a constructive alternative to the potential-theoretic existence result would sharpen the argument.","The explicit (tr,Curl) inequality gives a ready test case for numerical experiments in relaxed micromorphic elasticity: one can check whether the Helmholtz-type projection of the deviatoric distortion is the exact null-space correction observed in simulations."],"forward_implications":["For any linear map A and constant rank operator B with reduced constant rank, the projected inequality holds; when B is elliptic relative to ker A the projection vanishes and the classical reduced-elliptic KMS inequalities are recovered.","The combination (A,B)=(tr,Curl) is admissible: the paper derives an explicit inequality controlling P up to the Helmholtz-type projection of the deviatoric part, with the L^1 norm of Curl P on the right.","Lemma 3.1 gives an equivalence: the reduced constant rank condition is necessary and sufficient for the L^q inequality with negative-Sobolev BP norm, so any future improved inequality must be tight against this condition.","The p=1 borderline works under reduced cancellation, matching the structure known from the elliptic case and identifying the extra algebraic condition that must be checked in applications.","The constant rank framework subsumes the classical elliptic classification as a special case, showing that constant rank is a natural level of generality for projection-based coercivity estimates."],"supporting_citations":[{"why":"Supplies the constant-rank projection estimate used in Lemma 2.1 for the sufficiency direction.","marker":"[7]"},{"why":"Proves the necessity of the constant rank condition for L^p estimates, completing Lemma 2.1.","marker":"[14]"},{"why":"Provides the original coerciveness inequality for nonelliptic systems under constant rank, a basis for the sufficiency part.","marker":"[23]"},{"why":"Gives the theorem that constant rank operators admit an ellipticity complex, used in the p=1 step.","marker":"[22]"},{"why":"Supplies the strong limiting Sobolev estimate for cocancelling operators used in the p=1 case.","marker":"[25]"},{"why":"Establishes the reduced elliptic case, which the present theorem generalizes and recovers in the non-borderline range.","marker":"[9]"},{"why":"Gives the limiting p=1 classification for reduced elliptic operators, recovered here as a special case.","marker":"[11]"}],"fun_headline_variants":["Projection correction widens Korn-Maxwell-Sobolev","Constant rank operators tamed by projection fix","Beyond ellipticity: projection unlocks new bounds","Korn-Maxwell-Sobolev reaches reduced constant rank","L1 limit solved for constant rank operators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The borderline case p=1 rests on a cited theorem that every constant-rank operator admits an ellipticity complex (a companion operator L whose symbol kernels match B[ξ](ker A)); if that theorem were false or inapplicable for some pair A,B, the step bounding BP in the negative Sobolev space would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Projection correction widens Korn-Maxwell-Sobolev","Constant rank operators tamed by projection fix","Beyond ellipticity: projection unlocks new bounds","Korn-Maxwell-Sobolev reaches reduced constant rank","L1 limit solved for constant rank operators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000173,"raw_usage":{"total_tokens":1308,"prompt_tokens":1007,"completion_tokens":301,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":623,"completion_tokens_details":{"reasoning_tokens":225}},"tokens_in":623,"tokens_out":301,"duration_ms":2642,"temperature":1.0,"reasoning_tokens":225,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:50:51.150817+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a candidate pair such as (A,B)=(tr,Curl) and compute the symbol-level multiplier bound: if there exists ξ0 where the symbol of P − Π_B Π_{ker A} P cannot be controlled by |A[ξ]| + |B[ξ]|, a smoothed plane-wave sequence will violate the claimed inequality; this finite-dimensional linear-algebra computation settles the question at a stroke.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the constant-rank projection estimate used in Lemma 2.1 for the sufficiency direction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the necessity of the constant rank condition for L^p estimates, completing Lemma 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the original coerciveness inequality for nonelliptic systems under constant rank, a basis for the sufficiency part."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the theorem that constant rank operators admit an ellipticity complex, used in the p=1 step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the strong limiting Sobolev estimate for cocancelling operators used in the p=1 case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the reduced elliptic case, which the present theorem generalizes and recovers in the non-borderline range."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the limiting p=1 classification for reduced elliptic operators, recovered here as a special case."}],"review_version":1}