{"id":"100564ae-fb79-4405-a173-4cf7403f389f","arxiv_id":"2607.19091","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Killing forms are strict local minimizers of the J1 curl-Sobolev quotient on S^n, while the same forms are unstable for J2, disproving the Frank-Loss conjecture.","lead":"On odd-dimensional spheres, this paper studies two conformally invariant curl-Sobolev ratios for differential forms and shows that the natural Killing-form family is locally optimal for the first ratio but not for the second. The second instability disproves a recent conjecture, so the true sharp constant for that inequality must be smaller than the Killing-form value.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's nonlinear stability proof is delegated to an unpublished appendix; without it the central claim is unproven.","rationale":"The reader's weakest_assumption identifies the same load-bearing gap: the nonlinear upgrade from the L² spectral gap (Theorem 4.4) to the quantitative W^{1,2n/(n+1)} stability (Theorem 1.1) is delegated to an unpublished appendix of the authors' own preprint. I checked the main spectral-gap computations and found no fatal internal error: Proposition 4.3's second-variation formula is consistent (the apparent discrepancy in the |curlφ|² coefficient is resolved by the correct second derivative of the L^p norm), the E2∩Q⊥ positivity follows from Proposition 3.4(2), and the E−1⊕E3 computation, despite minor typographical coefficient slips, still yields a positive quadratic form. These issues are not the central obstruction. The central obstruction is the missing nonlinear argument: a positive second variation on a complement of a finite-dimensional critical manifold does not automatically yield a quantitative stability estimate in a non-Hilbert critical norm, especially when the functional is invariant under an infinite-dimensional gauge group. The paper's own footnote 1 indicates the authors later proved global minimality, which would imply the quantitative stability, but that later proof is not part of this manuscript. Therefore the correct verdict remains CONDITIONAL: the main theorem is plausible and the spectral-gap foundation is sound, but the submitted proof is not self-contained and the transfer from [WZ25] is not justified in the text.","tokens_in":28282,"tokens_out":21882,"duration_ms":188807,"concrete_test":"Obtain [WZ25, arXiv:2508.09047] and check whether Appendix A's proof uses any spinor-specific fact (e.g., kernel-free Dirac operator, Clifford pointwise identities) that has no analogue for curl on middle-degree forms. Concretely, reproduce the nonlinear Taylor estimate of Appendix A on the quotient space Ω^p/dΩ^{p-1} with the norm ||curl·||_{L^{2n/(n+1)}}. If any step requires invertibility of the second derivative on a Hilbert-space complement without explicitly handling the gauge quotient by ker(d), the transfer is not identical and Theorem 1.1 needs a new, self-contained argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.1) is a quantitative nonlinear stability estimate in W^{1,2n/(n+1)}. What is actually proved in the manuscript is Theorem 4.4: the L² second variation of J1 is positive on the orthogonal complement of TξM. The proof of Theorem 1.1 then says that 'a standard quantitative argument (following [FZ22]) ... identical to [WZ25, Appendix A]' upgrades this spectral gap to the nonlinear estimate, but no such argument is included. This is load-bearing because the desired inequality has a different scaling and geometry from the Hilbert-space spectral gap: the distance is measured in the L^{2n/(n+1)} curl norm divided by the denominator, and curl has an infinite-dimensional kernel of exact forms. In [WZ25] the Dirac operator on S^n has no kernel, so the quotient by ker(d) needed here is extra structure; the transfer is not literally 'identical'. The authors' own footnote 1 states that the global-minimizer problem was solved in a subsequent preprint, reinforcing that the submitted manuscript does not contain the full nonlinear argument. If the unpublished argument does not carry over, Theorem 1.1 — the paper's main positive result — is not established. Secondary issues (the J2=J_{p,k} misidentification and the invalid Hölder bound in Proposition 5.1) affect the instability theorems, not this central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two conformally invariant Sobolev quotients for middle-degree forms on S^n, n≡3 mod 4: the curl–helicity quotient J1 and the gauge-optimized quotient J2. The main positive claim (Theorem 1.1) is a quantitative local stability estimate for J1 around the conformal family M of Killing (n−1)/2-forms and their conformal images, implying strict local minimality in W^{1,2n/(n+1)}. The paper also claims instability of this same family for J2 and for a family of generalized quotients J_{p,k}, thereby giving a strict upper bound for the sharp constant S2 and disproving the conjectured equality (1.4). Supporting material includes sharp L2 estimates for eigenforms, a classification of curl eigenforms, a conformal invariant related to the first curl eigenvalue, and a lower bound for S1 in dimension 3 by comparison with the Dirac operator. The instability computations are mostly explicit, but the proof of the central quantitative stability theorem is not contained in the manuscript: it delegates the nonlinear step to an unpublished preprint.","tokens_in":28606,"tokens_out":25495,"duration_ms":236490,"significance":"If Theorem 1.1 were fully established, it would be a genuine analogue for the curl operator of the Aubin–Talenti stability theory, and the claimed local minimality of the conformal Killing family in the conformally invariant topology would be a substantial result. The paper contains several correct and useful ingredients: the second-variation formula, the L2 spectral gap in Theorem 4.4, the sharp L2 eigenform estimates in Proposition 3.4, and explicit negative directions for J2 and for the generalized quotients. The disproof of the conjecture S2 = ((n+1)^2/4)ω_n^{2/n} is also significant. However, the strongest theorem is currently conditional on an argument that is not reproduced, so the significance of the paper as a whole is not yet realized in the submitted version.","major_comments":[{"comment":"The nonlinear upgrade from the L2 spectral gap in Theorem 4.4 to the quantitative W^{1,2n/(n+1)} estimate in Theorem 1.1 is not proved in the manuscript. The text states that a 'standard quantitative argument (following [FZ22])' applies and that the implementation is 'identical to [WZ25, Appendix A]'. This is load-bearing, and it is not literally automatic: [WZ25] is an unpublished preprint treating the Dirac operator, whose kernel is trivial, whereas the curl operator has the infinite-dimensional kernel of exact forms; moreover the distance in Theorem 1.1 is measured through the L^{2n/(n+1)} curl norm normalized by the helicity denominator, not through the L2 curl norm used in Theorem 4.4. Footnote 1 further indicates that the global-minimizer and stability statement was obtained in a subsequent preprint. As written, the proof of Theorem 1.1 is incomplete; the nonlinear argument must be","section":"Section 4, proof of Theorem 1.1"}],"minor_comments":[{"comment":"The statement 'when k=(n−1)/2 and p=2n/(n+1), one has J_{2n/(n+1),(n−1)/2}=J2' is false. For these parameters q=2n/(n−1), so the numerator of J_{p,k} has exponent q/p=(n+1)/(n−1), while J2 has numerator exponent (n+1)/n; the denominators also differ, since J2 contains the gauge infimum and J_{p,k} does not. The J2 instability is proved separately in Section 5, so this does not destroy Theorem 1.2, but the claimed identification is a mathematical error and should be corrected.","section":"Section 1, definition of J_{p,k}"},{"comment":"The sentence 'By conformal invariance, Proposition 5.1 implies Theorem 1.2' overstates the implication. Proposition 5.1 directly proves the J2 part of Theorem 1.2; the J_{p,(n−1)/2} part is proved later in Section 6 using different computations. The text should be reworded to avoid the suggestion that the full theorem follows from Proposition 5.1 alone.","section":"Section 5, end"},{"comment":"The proof of Lemma 6.1 is only sketched, with several identities stated after 'direct (though somewhat lengthy) computations'. Since Theorem 1.3 rests on these identities, the authors should either provide the full verification or clearly mark the lemma as a computational assertion with details available in a supplement.","section":"Section 6.1, Lemma 6.1"},{"comment":"The paragraph before the footnote says the authors 'cannot prove or disprove' global minimality of J1, while the footnote announces that the problem was solved in a subsequent preprint and that quantitative stability follows. This is confusing in a manuscript whose Theorem 1.1 already claims quantitative stability. Please clarify the logical status of Theorem 1.1 with respect to the later preprint.","section":"Section 1, footnote 1"},{"comment":"For the record, the denominator estimate (∫|α−dφ|^{2n/(n−1)})^{(n−1)/n} ≥ ω_n^{−1/n}∫|α−dφ|^2 is correct by Hölder; the equality analysis in Proposition 5.1 is also consistent. I do not find the error in this step that a preliminary review note suggested.","section":"Section 5, Proposition 5.1"},{"comment":"There is a typo: 'for any α∈W^{1,2n/(n+1)} with with ∫<curlα,α>>0' should read 'with' once. Similar small typographical issues occur elsewhere (e.g., 'particulary').","section":"Theorem 1.1 statement"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the delegation of the proof of Theorem 1.1 to the authors' own unpublished preprint [WZ25]. In a journal submission, this is not acceptable for the central theorem. The false identification of J_{p,k} with J2 is also a concrete error that should be fixed. On the other hand, the L2 spectral-gap computation and the instability constructions are substantial and appear sound; the paper is likely recoverable after the missing nonlinear argument is supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about this one. The spectral analysis for J1 around Killing forms is genuinely good: Theorem 4.4's quadratic-form estimates on each eigenspace summand are clean, and Proposition 3.4's sharp L2 estimates are the right tool. The instability results for J2 and the generalized quotients are new and disprove the Frank–Loss / Mederski–Szulkin conjecture. Appendix A's classification of curl eigenforms is a useful standalone result. The local stability is proved in the conformally invariant W^{1,2n/(n+1)} topology for all n≡3 mod 4, which exceeds the earlier C0/C1 results in dimension 3.\n\nThe main problem is that Theorem 1.1 is not actually proved in this manuscript. The proof stops at the L2 second variation and hands the nonlinear upgrade to an appendix of the authors' own preprint [WZ25]. That preprint treats the Dirac operator, which has no kernel, while curl has the infinite-dimensional kernel of exact forms. The authors work modulo ker(d) earlier, but the claim that the argument is 'identical' needs justification. This is load-bearing: without it, the quantitative stability and strict local minimality in Theorem 1.1 are unproven. A referee should ask for the full argument or a precise explanation of why the kernel causes no trouble. The footnote saying the global minimizer problem was solved in a later preprint only makes the gap more visible.\n\nI disagree with the Pith Report on Proposition 5.1. The denominator estimate (∫|α−dφ|^{2n/(n−1)})^{(n−1)/n} ≥ ω_n^{−1/n}∫|α−dφ|^2 is correct; it is the standard Lp–L2 norm comparison with the right constant. The strictness argument using ⟨ξ,φ_{−1}⟩∈P2 is sound. So that section is healthier than reported.\n\nThe J_{p,k}=J2 identification in Section 1 is genuinely sloppy: J_{p,k} has denominator ∫|α|^q, whereas J2 has the infimum over exact forms. Those are different functionals. The instability conclusion for J2 can likely be recovered for co-closed variations, where the infimum is achieved at φ=0 to first order, but the manuscript does not spell this out. It needs a correction, not a rewrite.\n\nWho benefits: anyone working on sharp constants for curl-type inequalities, conformal geometry, or zero-mode problems. I would send it to peer review, with the explicit requirement that the authors include or fully justify the nonlinear argument and fix the J2 wording. If they do, this becomes a strong paper.","headline":"Solid spectral analysis and a likely-correct instability result, but Theorem 1.1's proof is delegated to an unpublished appendix; needs revision before it is complete.","tokens_in":29118,"tokens_out":8845,"would_cite":true,"duration_ms":75435,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A23","46E35","58A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"On the n-sphere with n ≡ 3 mod 4, the paper proves that Killing (n−1)/2-forms are strict local minimizers of the curl-Sobolev quotient J1, while for the quotient J2 they are unstable, making the sharp constant for J2 strictly smaller than t","keywords":["curl operator","Killing forms","conformally invariant Sobolev inequalities","sharp constants","stability","differential forms","sphere","minimizers"],"falsifier":"Use the second-variation formula in Proposition 4.3 with a direction φ ∈ E2 ∩ Q^⊥: if one can find such a φ for which the quadratic form G(φ) is negative (or equality cases in Proposition 3.4(2) that are not contained in Q), then the spectral gap in Theorem 4.4 fails and Theorem 1.1 collapses. Conversely, a direct check that G(φ) remains positive for all φ in that subspace for some n ≡ 3 mod 4 would support the claim.","tokens_in":28130,"feed_emoji":"🌀","tokens_out":3910,"duration_ms":40401,"temperature":0.7,"pith_summary":"The paper studies conformally invariant Sobolev inequalities for the middle-degree curl operator on the standard sphere. It tries to establish that the natural candidate extremals—Killing forms and their conformal images—are locally optimal for one such inequality (J1) but not for another (J2). If correct, this gives a quantitative local stability estimate for J1 and disproves a conjectured formula for the sharp constant of J2. The same instability extends to a broader family of form-valued Sobolev quotients. For dimension 3, the paper also improves the known lower bound for the J1 sharp constant by relating it to a shifted spinorial inequality.","feed_headline":"Killing forms stable for one curl inequality, not the other","feed_subtitle":"Local optimality for J1 and instability for J2 settle a conjectured sharp constant on the sphere.","key_machinery":"The central object is the middle-degree curl operator curl = ∗d acting on (n−1)/2-forms, together with the two conformally invariant quotients J1 and J2. The argument relies on the conformal invariance of these quotients, the Hodge decomposition (so that exact forms can be ignored), and a classification of curl eigenspaces on the sphere. The key quantitative input is a set of sharp L2 estimates—especially Proposition 3.4—that control the component of an eigenform along a fixed Killing form; these estimates yield a spectral gap for the second variation of J1 in directions transverse to the conformal family M. This spectral gap is then upgraded to a nonlinear stability estimate.","core_discovery":"For n ≡ 3 mod 4, every Killing (n−1)/2-form and its conformal images form a family M of critical points for both J1 and J2, analogous to the classical extremal family of the scalar Sobolev inequality. The paper proves that near M, J1 satisfies a quantitative local stability estimate: if a form is close to M in the curl-norm, then the excess of J1 over its value at Killing forms is bounded below by a positive constant times the distance to M, squared. In particular, every element of M is a strict local minimizer of J1 in the conformally invariant space W^{1,2n/(n+1)}. For J2, the paper shows the opposite: each such critical point is unstable, and there exist variations that strictly decrease","pith_inferences":["The different behavior of J1 and J2 highlights the role of the gauge optimization in the denominator: for J1, the denominator is the helicity-type integral ⟨curl α, α⟩, while for J2 it is an infimum over exact perturbations; the latter creates negative directions that destroy local minimality.","The expected Morse index of each Killing form for J2 (the paper suggests n+1) could be computed explicitly; if confirmed, it would quantify the instability and could guide the search for true minimizers.","Since the paper leaves open whether the infimum S2 is attained, a natural next question is whether minimizers of J2 exist and are non-invariant under the symmetry group that fixes the Killing forms.","The Appendix C lower bound for n = 3, obtained via a shifted Dirac operator, could be sharpened further if the spinorial comparison in Lemma C.2 can be made into an equality case; that would pinpoint the corrections to the conjectured value of S1."],"forward_implications":["If Theorem 1.1 is correct, the Killing family M is locally the correct extremal family for J1, and a quantitative stability inequality holds in W^{1,2n/(n+1)}.","By conformal invariance, the local stability and instability results transfer from the sphere to R^n.","The instability for J2 shows that the sharp constant S2 is strictly less than (n+1)^2/4 ω_n^{2/n}, disproving the natural conjecture that the Killing forms attain it.","The instability extends to the generalized quotients J_{p,k} for each Killing (n−1)/2-form and for all 1 < p < n, as well as to the 1-form quotient with p = n/2.","A footnote reports that the authors have since confirmed global minimality of M for J1 in a later preprint; if that holds, the sharp constant S1 would equal (n+1)/2 ω_n^{1/n} and the quantitative stability would be global."],"fun_headline_variants":["Curl-Sobolev sharp constant: J1 stable, J2 unstable around Killing forms","Killing forms minimize J1 but not J2 in curl-Sobolev inequalities","Stable vs unstable: Killing forms split curl-Sobolev quotients","For n ≡ 3 mod 4, Killing forms are local minima for J1, not J2","Settling a sharp constant: Killing forms stable for J1, unstable for J2"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of Theorem 1.1 ends by invoking, without reproducing, a 'standard quantitative argument' from the authors' companion work to pass from a spectral gap of the second variation to the nonlinear stability estimate; if that argument does not carry over verbatim—especially because curl has an infinite-dimensional kernel of exact forms—the quantitative stability and strict local minimality claimed in Theorem 1.1 are not established by the submitted manuscript.","fun_headline_variants_meta":{"raw":{"variants":["Curl-Sobolev sharp constant: J1 stable, J2 unstable around Killing forms","Killing forms minimize J1 but not J2 in curl-Sobolev inequalities","Stable vs unstable: Killing forms split curl-Sobolev quotients","For n ≡ 3 mod 4, Killing forms are local minima for J1, not J2","Settling a sharp constant: Killing forms stable for J1, unstable for J2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000909,"raw_usage":{"total_tokens":3835,"prompt_tokens":926,"completion_tokens":2909,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":670,"completion_tokens_details":{"reasoning_tokens":2793}},"tokens_in":670,"tokens_out":2909,"duration_ms":18647,"temperature":1.0,"reasoning_tokens":2793,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T13:29:20.345865+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Use the second-variation formula in Proposition 4.3 with a direction φ ∈ E2 ∩ Q^⊥: if one can find such a φ for which the quadratic form G(φ) is negative (or equality cases in Proposition 3.4(2) that are not contained in Q), then the spectral gap in Theorem 4.4 fails and Theorem 1.1 collapses. Conversely, a direct check that G(φ) remains positive for all φ in that subspace for some n ≡ 3 mod 4 would support the claim.","supporting_citations":[],"review_version":1}