{"id":"451a2878-831c-4dd4-ae7a-f885eb41a839","arxiv_id":"2509.04617","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that a full-rank principal symbol over complex frequencies implies recovery on curves, which yields regularizing solution operators with prescribed support and dual representation formulas.","lead":"This paper builds a general framework for solving underdetermined partial differential equations by designing integral solution formulas supported on curves. It also introduces a simple algebraic test, the finite-dimensional cokernel condition, and applies it to divergence, Killing, Hessian and Einstein constraint operators.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Boundary-supported distributions falsify the claimed \\tilde W-cokernel triviality in Theorem 1.1(1) and Proposition 5.19(1); the main solution-operator constructions remain valid.","rationale":"The reader's weakest assumption concerns the quantitative admissibility of the curve family (conditions (x-1)-(x-3)). That is a genuine scope condition, but it is a hypothesis rather than a flaw: the paper's examples (straight lines, geodesic segments) satisfy it, and the general theorems are conditional on it. The present stress-test found a concrete false statement inside the paper's own results, which the reader did not flag. The 1D divergence example shows that Theorem 1.1(1) is false as stated for negative Sobolev orders, because boundary-supported distributions such as \\delta_0 lie in \\tilde H^{-1}(U) and solve P^*Z=0 in D'(U). The proof of Theorem 5.12(3) only justifies the triviality of the \\tilde W-cokernel on compactly contained subsets; the extension to all of U is unjustified and is in fact disproved by the example. The same defect propagates into Proposition 5.19(1) and the claims that the \\tilde W-cokernel is independent of the Sobolev scale. The constructive core of the paper -- the existence of conic-type and Bogovskii-type solution operators and the W-space Poincar\\'e/Friedrich inequalities -- appears sound, since those results are formulated via W^{-s,p'}(U), where boundary-supported distributions are invisible and the duality arguments are valid. Thus the appropriate verdict is CONDITIONAL: the paper should be accepted only after the cokernel statements are corrected (e.g., by replacing \\tilde W with W, or by stating triviality for compactly contained subsets) and the interpretation of the representation formulas is made precise. This is a substantive correction, not a rejection of the main method.","tokens_in":78523,"tokens_out":33622,"duration_ms":276334,"concrete_test":"Check U=(0,1), P=\\partial_x, P^*=-\\partial_x, x(y,y_1,s)=y+s(y_1-y), U_1=(1,2). Verify that \\delta_0\\in\\tilde H^{-1}(0,1) by displaying an explicit sequence in C_c^\\infty(0,1) converging to \\delta_0 in H^{-1}(R), and verify \\langle P^*\\delta_0,\\varphi\\rangle=-\\varphi'(0)=0 for all \\varphi\\in C_c^\\infty(0,1). This is a direct counterexample to Theorem 1.1(1). Then re-derive the proof of Theorem 5.12(3) to confirm that it only gives triviality of the \\tilde W-cokernel for open sets V compactly contained in U, which is the precise source of the gap.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.1(1) claims that under (RC) with nontrapping curves, ker_{\\tilde W^{-s,p'}(U)}P^* = {0} for all s\\in R, 1<p<\\infty. The proof via Theorem 5.12(3) only establishes this for open subsets V compactly contained in U, because the representation formula \\phi = S^*P^*\\phi holds in W^{-s,p'}(U), and an element of \\tilde W^{-s,p'}(U) that is zero in W^{-s,p'}(U) may be supported on \\partial U. This is not a mere technicality. Take U=(0,1)\\subset R, P=\\partial_x (divergence), P^*=-\\partial_x, straight-line curves x(y,y_1,s)=y+s(y_1-y), and U_1=(1,2), so U\\cap U_1=\\emptyset and all hypotheses of Theorem 1.1 are satisfied. For s=-1 and p'=2, the Dirac mass \\delta_0 belongs to \\tilde H^{-1}(0,1) (it is the limit in H^{-1}(R) of C_c^\\infty(0,1) bumps of width \\varepsilon), and for every \\varphi\\in C_c^\\infty(0,1) we have \\langle P^*\\delta_0,\\varphi\\rangle = -\\langle \\delta_0,\\varphi'\\rangle = -\\varphi'(0)=0. Hence P^*\\delta_0=0 in D'(0,1) but \\delta_0\\neq 0 in \\tilde H^{-1}(0,1), contradicting Theorem 1.1(1). The same phenomenon invalidates Proposition 5.19(1), where membership of \\delta_0 in \\tilde H^s depends on s, so the asserted equality of \\tilde W-cokernels across Sobolev scales fails. The constructive statements (existence of S and \\tilde S, the W^{-s,p'}(U) Poincar\\'e/Friedrich inequalities) are not affected because there the equalities are interpreted in W^{-s,p'}(U).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for constructing regularizing right-inverses (up to finite rank) of underdetermined differential operators and, by duality, integral representation formulas and Poincaré-/Friedrich-type inequalities for their adjoints. The key device is the 'recovery on curves' condition (RC), which yields curve-supported Green's functions, followed by smooth averaging over a family of curves to obtain pseudodifferential solution operators of optimal order. The paper also introduces the finite-dimensional cokernel condition (FC), an algebraic full-rank condition on the complex principal symbol, proves via Hilbert's Nullstellensatz that (FC) implies (RC), and verifies (FC) for divergence, Hessian, Killing, conformal Killing, and linearized Einstein constraint operators. Detailed formulas are worked out in the appendices for constant-curvature backgrounds. The constructive parts of the theory appear sound and are presented with care; however, one central cokernel claim involving the spaces \\tilde W^{-s,p'}(U) is false as stated.","tokens_in":78934,"tokens_out":10326,"duration_ms":96165,"significance":"If the cokernel statements are corrected, this is a substantial and useful paper. It unifies and generalizes Bogovskii's and Oh–Tataru's divergence formulas and Reshetnyak's representation formulas, gives a checkable algebraic criterion for a large class of geometric operators, and provides explicit integral kernels with prescribed support. The construction is self-contained and non-circular: (FC) is an algebraic hypothesis, and the solution operators are built directly from it. The proof structure is detailed, with careful tracking of derivative counts and a clean use of Hilbert's Nullstellensatz in Section 7. The main defect found below is localized to the boundary-sensitive cokernel statements for \\tilde W^{-s,p'}(U); the actual construction of S and \\tilde S and the W-based inequalities appear unaffected.","major_comments":[{"comment":"The cokernel claim ker_{\\tilde W^{-s,p'}(U)} P^* = {0} in Theorem 1.1(1) is false as stated. The proof via Theorem 5.12(3) only establishes the statement for open sets V compactly contained in the auxiliary open set \\tilde U, but \\tilde W^{-s,p'}(U) contains distributions supported on \\partial U, and the condition 'P^*Z = 0 in D'(\\tilde U)' depends on the choice of \\tilde U. Concrete counterexample: take U=(0,1)\\subset R, P u = \\partial_x u, so P^*\\varphi = -\\partial_x\\varphi, and use straight-line curves x(y,y_1,s)=y+s(y_1-y) with U_1=(1,2); all hypotheses of Theorem 1.1 are satisfied. For s=-1, p'=2, the Dirac mass \\delta_0 belongs to \\tilde H^{-1}(0,1), and choosing \\tilde U=(0,2) we have P^*\\delta_0=0 in D'((0,2)), because test functions supported in (0,2) vanish near 0. Yet \\delta_0\\neq 0 in \\tilde H^{-1}(0,1). This also disproves the claimed independence of the choice of \\tilde U in (1.4): taking \\tilde U=(-1,2) gives P^*\\delta_0\\neq 0. The statement should be repaired, for example by requiring P^*Z=0 in an appropriate \\tilde W-based sense or by replacing the \\tilde W cokernel statement with a local statement on sets compactly contained in U; the constructive Parts (2) and (3) do not appear to be affected.","section":"Theorem 1.1(1), §5.3.2, §2.3"},{"comment":"Proposition 5.19(1) inherits the same defect and is false as stated. With U=(0,1), P=\\partial_x, and the same choice \\tilde U=(0,2), the distribution \\delta_0 lies in ker_{\\tilde H^{-1}(0,1)}P^* under the definition used in (1.4), but \\ker P^* on (0,1) consists only of constants. Hence the asserted equality ker_{\\tilde W^{-s_0,p_0}(U)}P^* = ker_{\\tilde W^{-s_1,p_1}(U)}P^* = \\ker P^* fails across Sobolev scales. The proof of Proposition 5.19 uses the representation formula in W^{-s,p'}(U), which does not control boundary-supported contributions to \\tilde W^{-s,p'}(U). This point is already visible from the discussion in Section 2.3, where the authors explicitly note that \\tilde W^{s,p}(U) may contain distributions supported in \\partial U. The proposition should be reformulated so that boundary-supported distributions are either excluded by the definition of the cokernel or handled by a genuinely local statement.","section":"Proposition 5.19(1), §5.4.3"}],"minor_comments":[{"comment":"In (Φ-4), the phrase 'for the maximal degree 1 that occurs in (1.8)' appears to contain a typo; it should presumably read 'the maximal degree that occurs in (1.8)'.","section":"Definition 1.4"},{"comment":"The spelling of Reshetnyak is inconsistent: 'Retshenyak' appears in Remark 1.13 and once in the introduction, while the standard spelling is used elsewhere.","section":"Remark 1.13"},{"comment":"The claimed converse implication from (wRC) to (RC∨) is stated without proof. Since this fact is explicitly not used later, it would be clearer to defer it to an appendix or to give the short proof indicated by (5.7).","section":"Remark 5.3"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper with a real but localized flaw. The counterexample with the one-dimensional divergence operator is decisive against Theorem 1.1(1) and Proposition 5.19(1) as written, but the constructive core—existence of S and \\tilde S, the W-space estimates, and the (FC)-implies-(RC) machinery—appears sound. I recommend major revision rather than rejection, with the expectation that the authors can reformulate the \\tilde W-cokernel statements (and any applications in the companion paper [38] that rely on them) in a way that accounts for boundary-supported distributions. The companion-paper Theorem 1.15(3) should also be checked for the same issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper deserves a serious referee. The main new idea is the (FC) condition and the proof that it implies (RC) via maximal graded augmented systems using Hilbert's Nullstellensatz. That is a genuine structural result and a good unification of Bogovskii, Oh-Tataru and Reshetnyak. The kernel estimates in Sections 4 and 5 are detailed and plausible, and the worked examples in Theorem 1.14 are short and checkable.\n\nThere is, however, a real problem with some of the cokernel claims. Theorem 1.1(1) says that under (RC) with nontrapping curves, ker_{\\tilde W^{-s,p'}(U)} P^* = {0} for all s and p. The proof goes through Theorem 5.12(3), which only establishes triviality for kerns in \\tilde W^{-s,p'}(V) when V is an open set compactly contained in U. The representation formula \\phi = S^*P^*\\phi is an identity in W^{-s,p'}(U); it does not see boundary-supported distributions in \\tilde W. Here is the counterexample: U=(0,1), P=d/dx, straight line curves with U1=(1,2). The distribution \\delta_0 lies in \\tilde H^{-1}(0,1), P^*\\delta_0 = 0 in D'(0,1), and \\delta_0 is not zero in \\tilde H^{-1}(0,1). So Theorem 1.1(1) is false as stated. The same objection kills Proposition 5.19(1)'s equality of \\tilde W-cokernels across Sobolev scales.\n\nThis is not a fatal flaw for the main construction. The solution operators S and \\tilde S, and the W^{-s,p'} Poincare/Friedrich inequalities, are not affected. But the overstatement is in two headline results and should be corrected, probably by restricting the triviality/invariance claims to the W scale or to compactly contained subsets, and by adding a remark about boundary-supported distributions in \\tilde W spaces.\n\nThe citation pattern looks honest, and the companion-paper references are clearly separate. The paper is long and technical, but the structure is clear. I would send it to peer review and expect the authors to fix the \\tilde W statements. It will be a useful reference for people working on divergence-type operators, GR constraint equations, and rigidity inequalities.","headline":"A substantial, mostly correct general theory for integral solution operators, but the stated \\tilde W-cokernel triviality claims are false as written due to boundary-supported distributions.","tokens_in":79486,"tokens_out":3085,"would_cite":true,"duration_ms":29223,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A01","35A08","35S05","35Q75"],"pacs":[],"model":"deepseek-v4-flash","headline":"A full-rank complex symbol guarantees curve-supported integral solution operators for a wide class of underdetermined PDEs, and dual representation formulas for overdetermined adjoints.","keywords":["underdetermined PDEs","solution operators","recovery on curves","finite-dimensional cokernel condition","graded augmented systems","singular integral kernels","Korn inequalities","Einstein constraint equations"],"falsifier":"A concrete check is to take $P h = \\partial_j\\partial_k h^{jk}$ on $\\mathbb{R}^d$ and substitute the explicit kernel $K^{ij}_\\eta$ from Appendix A.2.3 into the identity $P K_\\eta(\\cdot,y) = \\delta_0(\\cdot-y) - b_\\eta(\\cdot,y)$; if the identity fails for any smooth compactly supported $\\eta$, the central Green's-function claim is false.","tokens_in":78303,"feed_emoji":"📐","tokens_out":7560,"duration_ms":70315,"temperature":0.7,"pith_summary":"This paper builds a general machine for solving underdetermined linear PDEs of divergence type and, by duality, for representing solutions to their overdetermined adjoints. The central claim is that a purely algebraic condition on the principal symbol—full rank for every nonzero complex frequency—guarantees the existence of integral solution operators whose kernels are supported on prescribed curves and which gain the full order of the operator. If true, this turns a previously case-by-case matter into a checkable recipe that covers the divergence, Hessian, Killing, conformal Killing, linearized scalar curvature, and Einstein constraint operators. The paper proves the condition is equivalent, in constant-coefficient settings, to the finiteness of the formal cokernel, and shows how the resulting representation formulas yield Poincaré- and Korn-type inequalities.","feed_headline":"A symbol test guarantees curve-supported solutions to many PDEs","feed_subtitle":"One algebraic condition covers divergence, Killing, and Einstein constraint operators and yields Korn-type inequalities.","key_machinery":"The load-bearing object is the graded augmented system: a collection of augmented variables $\\Phi^A$ consisting of the unknown $\\varphi$ and its derivatives up to order $N_0-1$, chosen so that their derivatives along any curve obey a first-order linear ODE with the data $P^*\\varphi$ entering as a forcing term. The paper proves via Hilbert's Nullstellensatz that the algebraic condition (FC) always supplies such a system, with the coefficients of the ODE built from polynomial identities $\\xi^\\alpha I = g_\\alpha(x,\\xi)p^*(x,\\xi)$. Once the augmented system exists, Duhamel's formula along a curve expresses $\\varphi(y)$ through $P^*\\varphi$ along the curve and the endpoint value, which is exactly recovery on curves; averaging over curves then gives the integral kernels with controlled singularity.","core_discovery":"The paper's core discovery is the chain (FC) implies (graded augmented system) implies (recovery on curves) implies (integral solution and representation formulas). The finite-dimensional cokernel condition (FC) asks that the principal symbol $p^*(x,\\xi)$ of the adjoint $P^*$ be injective for every $x$ in the domain and every nonzero complex covector $\\xi$; this is stronger than ordinary ellipticity, which only requires injectivity for real $\\xi$. Theorem 1.11 asserts that (FC) implies the existence of a maximal graded augmented system, a first-order ODE system along curves that encodes all derivatives of the unknown up to a fixed order; the augmented system immediately gives recovery on curves. Smooth averaging over an admissible family of curves converts the resulting curve-supported distribution kernels into classical pseudodifferential operators of order $-m$, so the solution operator gains $m$ derivatives, and the support of the kernel remains inside the prescribed union of curves. In the constant-coefficient case the three conditions collapse: (FC), (RC), and finite dimensionality of the formal cokernel are equivalent.","pith_inferences":["A likely extension, not explored here, is to treat complexes of such operators: a Bogovskii-type chain homotopy for the de Rham complex would follow from the same curve-averaging mechanism, connecting the construction to pullback and Darboux-type theorems.","The non-effectiveness of the finite-rank correction $Q$ noted in the paper suggests that quantitative bounds for Poincaré constants will have to come from special solutions or from the completely integrable case; testing the method on non-completely integrable variable-coefficient operators would clarify how much of the gain is genuinely new.","Since (FC) is only sufficient in variable-coefficient settings, one could look for variable-coefficient operators satisfying (RC) but failing (FC), which would mark the boundary of the algebraic condition as a detection criterion."],"forward_implications":["For any operator satisfying (FC), the construction yields a right inverse up to finite rank that gains $m$ derivatives and whose kernel is supported on prescribed curves; data with suitable support can be solved with solutions supported in the corresponding unions of curves.","By duality, every such operator produces an integral representation $\\varphi = S^*P^*\\varphi$ modulo finite-rank terms, hence Poincaré- or Friedrich-type and Korn-type inequalities on domains that are star-shaped with respect to the curve family or for which the curves exit the domain.","The seven geometric examples listed in Theorem 1.14—divergence, trace-free double divergence, Killing, conformal Killing, linearized scalar curvature, and Einstein constraint operators—all satisfy (FC); therefore the method applies to lower-order variable-coefficient perturbations of their principal parts.","When the principal symbol has constant coefficients, finite dimensionality of the formal cokernel is enough to guarantee the whole chain, so the method captures exactly the operators one might hope to treat by integral formulas without parametrix constructions."],"supporting_citations":[{"why":"Supplies the classical divergence solution kernel (1.2) that the paper's construction generalizes and reproduces in the divergence case.","marker":"[5]"},{"why":"Supplies the conic solution operator for the divergence equation, whose support-in-cones property the paper abstracts into the recovery-on-curves framework.","marker":"[51]"},{"why":"Provides Reshetnyak's completely integrable augmented systems and integral representation formulas for Killing and conformal Killing operators, which the paper generalizes by dropping complete integrability.","marker":"[52]"},{"why":"Provides Hilbert's Nullstellensatz, used in the proof of Theorem 1.11 to show (FC) implies a maximal graded augmented system.","marker":"[4]"},{"why":"Supplies the pseudodifferential calculus used to turn the singular integral kernel bounds into Sobolev-space boundedness of the averaged solution operator.","marker":"[56]"}],"fun_headline_variants":["Full-rank symbol test unifies PDE solution formulas","Recovery on curves: a general route to integral solutions","Finite cokernel condition yields curve-supported Green functions","One algebraic check covers divergence, Killing, and more"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction's payoff—a bounded, optimally regularizing solution operator with prescribed support—hinges on the existence of an admissible family of curves connecting each point to a prescribed endpoint set, with quantitative control on how the curves spread as they move; without such a family, the recovery identities do not give usable operators.","fun_headline_variants_meta":{"raw":{"variants":["Full-rank symbol test unifies PDE solution formulas","Recovery on curves: a general route to integral solutions","Finite cokernel condition yields curve-supported Green functions","One algebraic check covers divergence, Killing, and more"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000394,"raw_usage":{"total_tokens":2174,"prompt_tokens":1158,"completion_tokens":1016,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":774,"completion_tokens_details":{"reasoning_tokens":951}},"tokens_in":774,"tokens_out":1016,"duration_ms":9164,"temperature":1.0,"reasoning_tokens":951,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:28:36.069135+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to take $P h = \\partial_j\\partial_k h^{jk}$ on $\\mathbb{R}^d$ and substitute the explicit kernel $K^{ij}_\\eta$ from Appendix A.2.3 into the identity $P K_\\eta(\\cdot,y) = \\delta_0(\\cdot-y) - b_\\eta(\\cdot,y)$; if the identity fails for any smooth compactly supported $\\eta$, the central Green's-function claim is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical divergence solution kernel (1.2) that the paper's construction generalizes and reproduces in the divergence case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the conic solution operator for the divergence equation, whose support-in-cones property the paper abstracts into the recovery-on-curves framework."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Reshetnyak's completely integrable augmented systems and integral representation formulas for Killing and conformal Killing operators, which the paper generalizes by dropping complete integrability."},{"cited_title":"Co., Reading, MA, 1969","cited_arxiv_id":null,"evidence_quote":"Provides Hilbert's Nullstellensatz, used in the proof of Theorem 1.11 to show (FC) implies a maximal graded augmented system."},{"cited_title":"Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals , Princeton Mathematical Series, Princeton Mathematical Press, Princeton, NJ, 1993","cited_arxiv_id":null,"evidence_quote":"Supplies the pseudodifferential calculus used to turn the singular integral kernel bounds into Sobolev-space boundedness of the averaged solution operator."}],"review_version":2}