{"id":"832b0174-c24e-4b10-8472-41d03a043c6c","arxiv_id":"2608.09034","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims optimal Bianchi-Egnell constants for a nonlocal Sobolev inequality on the Heisenberg group, but the key strict spectral bound is unproved.","lead":"This paper tries to find the sharp stability constant for a nonlocal Sobolev inequality on the Heisenberg group, the distance-to-bubbles version of a classical problem. The main result, that this constant is attained and strictly exceeds the local one, rests on a strict inequality whose proof the authors themselves say is incomplete.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Spectral strictness (3.4) is unproved: the paper admits its cubic argument fails, and the proposed harmonic has zero cubic integral by U(1)-invariance; Theorems 3.2 (mu<=4) and 3.4 rest on it.","rationale":"The central claim is that H_NS is attained and H_NS > H_BE. The standard two-barrier strategy requires strict thresholds, and the two-peaks threshold (3.3) is derived by a separate expansion and appears plausible. The spectral threshold (3.4), however, is the barrier that prevents minimizing sequences from approaching the bubble manifold. The proof in Section 5.1 contains an explicit admission that the cubic strictness argument does not establish (3.4), and the proposed explicit spherical harmonic has zero cubic integral by U(1)-invariance. The manuscript's own computation would imply a nonzero integral for a holomorphic-degree-6 monomial, which is impossible under the rotation-invariant measure on S^{2n+1}. Consequently, condition (5.2) is not met, and the proof of (3.4) fails. Theorem 3.2 for mu<=4 and Theorem 3.4 depend directly on (3.4); the strict comparison H_NS > H_BE in Remark 3.1 relies on the full attainment result. The two-peaks bound may support the mu>4 condition (3.5), but the theorem as stated covers mu<=4 and the abstract claims optimal stability and attainment under the stated conditions. The failure is internal to the manuscript rather than a disagreement with external consensus, and no machine-checked proof or reproducible computation is offered to repair it. These considerations support the reader's REJECT verdict.","tokens_in":37734,"tokens_out":4640,"duration_ms":41408,"concrete_test":"Compute the cubic integral in (5.2) for the proposed omega analytically via U(1) phase covariance: it vanishes identically for every omega in H^{n+1}_{2,0} and n>=2, while the displayed omega is undefined for n=1. A numerical cross-check on S^5, e.g. quasi-Monte Carlo integration of zeta1^2 zeta2^2 zeta3^2 over the unit sphere, should return zero up to roundoff. If the integral is zero, the epsilon^3 strictness argument collapses and the proof of (3.4) is invalid as written.","verdict_should_be":"REJECT","load_bearing_attack":"Section 5.1 proves the strict spectral upper bound (3.4) by reducing to the sign condition (5.2). The proof requires an H^{n+1}_{2,0} spherical harmonic whose cubic contribution has a definite sign. The text explicitly states: 'the cubic strictness argument in its present form does not establish (3.4); a corrected perturbation scheme or a fourth-order expansion is needed.' The proposed repair omega = zeta1*zeta2 + zeta1*zeta3 + zeta2*zeta3 does not work. For every omega in H^{n+1}_{2,0}, the U(1)-rotation zeta -> e^{i theta} zeta gives omega(e^{i theta} zeta) = e^{2 i theta} omega(zeta), so omega^3 transforms by e^{6 i theta}; the standard rotation-invariant measure on S^{2n+1} then forces the integral of omega^3 to vanish. The displayed positive value 6 * integral of zeta1^2 zeta2^2 zeta3^2 is itself zero by the same phase-covariance argument, since that monomial has holomorphic degree 6. Thus condition (5.2) fails for every (2,0) spherical harmonic, not just for the displayed one. Because (3.4) is the compactness barrier used in the mu<=4 branch of Theorem 3.2 and is the input for Theorem 3.4, the attainment result and the strict comparison H_NS > H_BE are unsupported on the claimed parameter range. The two-peaks bound (3.3) appears to stand, but it only covers the mu>4 condition (3.5), leaving the mu<=4 case of Theorem 3.2 without a valid spectral barrier.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies quantitative stability for the critical nonlocal Sobolev inequality on the Heisenberg group, where the deficit of the Hardy-Littlewood-Sobolev-type Sobolev quotient is compared with the squared distance to the manifold of Jerison-Lee bubbles. The main results are: two strict upper bounds for the optimal stability constant H_NS (a spectral bound and a two-peaks bound, Theorem 3.1); attainment of H_NS under either mu <= 4 or the condition (3.5) (Theorem 3.2); the strict comparison H_NS > H_BE with the local Folland-Stein-Sobolev stability constant (Remark 3.1); the sharp universal upper bound 1 with equality characterization (Theorem 3.3); and a strict single-bubble upper bound for a critical-point residual quotient (Theorem 3.4). The proof strategy combines diagonalization of two nonlocal quadratic forms on complex spherical harmonics, a local perturbation around a bubble, a two-bubble interaction expansion, and a Heisenberg-group profile-decomposition compactness argument.","tokens_in":38073,"tokens_out":4597,"duration_ms":43231,"significance":"If the main results were established, the paper would be a substantial contribution: it identifies nonlocal-Heisenberg-specific compactness thresholds that differ from both the Euclidean nonlocal problem and the local CR problem, proves existence of minimizers for the stability constant, and upgrades the comparison with the local stability constant to a strict one. The paper has several sound components: the explicit diagonalization of the two bilinear forms in Section 4, the two-bubble expansion in Section 5.2 leading to the bound H_NS < 2 - 2^{1/Q*_mu}, the high-mode argument in Section 7 for the sharp universal upper bound, and the overall profile-decomposition framework. However, the central spectral strictness step (3.4) is explicitly admitted in Section 5.1 to be unproved, and the proposed repair is invalid; since Theorems 3.2 and 3.4 rely on that step, the main claims are not supported in the stated form.","major_comments":[{"comment":"The proof of the strict spectral upper bound (3.4) reduces to the sign condition (5.2) for a spherical harmonic omega in H^{n+1}_{2,0}. The manuscript itself states that 'the cubic strictness argument in its present form does not establish (3.4)', and the proposed repair omega = zeta1*zeta2 + zeta1*zeta3 + zeta2*zeta3 does not work. Under the rotation zeta -> e^{i theta} zeta on S^{2n+1}, omega^3 transforms by e^{6 i theta}, so the standard rotation-invariant measure forces int_{S^{2n+1}} omega^3 = 0 for every omega in H^{n+1}_{2,0}; in particular the displayed value 6 * int zeta1^2 zeta2^2 zeta3^2 is itself zero because that monomial has holomorphic degree 6. Therefore condition (5.2) fails for every (2,0)-harmonic, and the derivation of (3.4) collapses.","section":"Section 5.1, Eq. (5.2) and Theorem 3.1-(3.4)"},{"comment":"The mu <= 4 branch of Theorem 3.2 uses Lemma 4.1 together with the strict inequality H_NS < H_spec_NS from (3.4) to exclude minimizing sequences approaching the bubble manifold. Since (3.4) is unproved, the compactness barrier for mu <= 4 is missing, and the attainment conclusion of Theorem 3.2 and the strict comparison H_NS > H_BE in Remark 3.1 are unsupported in that parameter range. The same invalid sign condition (5.2) is invoked in the proof of Theorem 3.4 in Section 8, so the critical-point strict bound is also unsupported.","section":"Section 6, Proposition 6.1 and Theorem 3.2"},{"comment":"The manuscript explicitly flags the gap in the cubic strictness argument and then attempts to repair it with a computation that contradicts the symmetry statement made on the same page. This internal inconsistency shows that the issue is not a minor omitted detail: the chosen test function cannot produce the required definite sign, and a corrected perturbation scheme or fourth-order expansion would be a substantial new ingredient rather than a local fix.","section":"Section 5.1, internal consistency"}],"minor_comments":[{"comment":"For n = 1 the proposed harmonic omega = zeta1*zeta2 + zeta1*zeta3 + zeta2*zeta3 is not defined on S^3, since S^{2n+1} = S^3 has only two complex coordinates; the proof of (3.4) for n = 1 would require a separate argument in any case.","section":"Section 5.1, proposed test function"},{"comment":"The sentence admitting that the cubic strictness argument does not establish (3.4) should not appear in a submitted proof without a complete resolution; the subsequent 'repair' should either be verified or removed, as the current text contradicts itself.","section":"Section 5.1, exposition"},{"comment":"The limiting argument for the sharp upper bound 1 would be easier to follow if the asymptotic behavior nu_{i,j}, tau_{i,j} -> infinity as i+j -> infinity were stated explicitly before letting i+j tend to infinity.","section":"Section 7, proof of Theorem 3.3"},{"comment":"The paper imports the basic positive lower bound H_NS > 0 from the companion manuscript [12]; this dependence should be stated explicitly in the statement of Theorem 3.1 or the beginning of Section 6, since the reader cannot verify that imported result from the present text alone.","section":"Section 3 and Remark 3.1"}],"recommendation":"reject","confidential_remarks":"The manuscript's central spectral strictness step is explicitly acknowledged by the authors to be unproved, and the proposed repair is false by U(1)-invariance on S^{2n+1}. Because Theorems 3.2 and 3.4 rest on this step, the main attainment and strict-comparison claims are currently unsupported. The two-peaks bound and the universal upper bound appear to be independent and viable, but they do not compensate for the failure of the spectral barrier. Given the paper's own admission of the gap, I do not see how a routine revision can address it within the scope of this manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has genuine substance, but the central attainment result is unsupported. The author's own Section 5.1 admits that the cubic strictness argument does not establish (3.4), and the proposed repair with omega = zeta1 zeta2 + zeta1 zeta3 + zeta2 zeta3 fails cleanly: every (2,0) spherical harmonic satisfies omega(e^{i theta} zeta) = e^{2 i theta} omega(zeta), so omega^3 transforms by e^{6 i theta}, and the U(1)-invariant measure on S^{2n+1} forces the cubic integral to vanish. The displayed positive value 6 integral zeta1^2 zeta2^2 zeta3^2 is also zero by the same phase covariance. So condition (5.2) fails for every (2,0) mode, not just the displayed one. Since (3.4) is the compactness barrier for the mu <= 4 branch of Theorem 3.2 and is an input to Theorem 3.4, those results are unsupported on the claimed parameter range.\n\nWhat is genuinely new and good: the coupled CR-HLS spectral diagonalization on complex spherical harmonics is a useful piece of work, and the explicit constants H_NS^{spec} and H_NS^{2-peak} are new. The two-peaks expansion in Section 5.2 appears to be derived carefully and stands independent of the spectral argument; it gives the strict bound (3.3) for the two-bubble threshold. The universal upper bound H_UB = 1 in Section 7 also looks correct and is a nice complement to the local Heisenberg result. The parameter discussion in Remark 3.1 and the appendix are fine.\n\nThe soft spot is not minor: it is the load-bearing step of the whole compactness argument. The paper is honest about the gap, which is to its credit, but honesty does not fill the gap. The reliance on the companion paper [12] for the basic positivity H_NS > 0 is worth remembering, but it is not the main problem here.\n\nWho is this for? Specialists in quantitative stability and nonlocal Sobolev inequalities on the Heisenberg group. The spectral machinery and two-peaks bound could be useful even if the main theorem needs a corrected fourth-order expansion or a different perturbation scheme.\n\nRecommendation: a serious referee should engage with this paper, because the partial results are valuable and the flaw is localized. I would send it to peer review, but with the expectation of major revision: either repair the strict spectral bound or restrict the claims to the range where the two-peaks bound suffices.","headline":"Real spectral and two-bubble machinery, but the paper's own text concedes the strict spectral bound (3.4) is unproved, and the proposed repair vanishes by U(1) symmetry, so the attainment theorem does not stand.","tokens_in":38662,"tokens_out":2142,"would_cite":false,"duration_ms":21450,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B35","35P30","35A23","45E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the optimal stability constant for a critical nonlocal Sobolev inequality on the Heisenberg group is attained and strictly exceeds the local one.","keywords":["Heisenberg group","nonlocal Sobolev inequality","quantitative stability","stability constant","Jerison-Lee bubbles","Hardy-Littlewood-Sobolev inequality","critical exponent","variational methods"],"falsifier":"Compute $\\int_{S^{2n+1}} \\omega_{2,0}(\\zeta)^3\\,d\\zeta$ for the proposed perturbation $\\omega_{2,0}=\\zeta_1\\zeta_2+\\zeta_1\\zeta_3+\\zeta_2\\zeta_3$. The rotation $\\zeta\\mapsto e^{i\\theta}\\zeta$ multiplies the integrand by $e^{6i\\theta}$, so the integral is zero; hence condition (5.2) fails and the strict inequality $H_{\\mathrm{NS}} < H_{\\mathrm{NS}}^{\\mathrm{spec}}$ is not established by the displayed perturbation.","tokens_in":37477,"feed_emoji":"📐","tokens_out":9080,"duration_ms":77524,"temperature":0.7,"pith_summary":"This paper studies the sharp quantitative stability of the critical nonlocal Sobolev inequality on the Heisenberg group $\\mathbb{H}^n$: it asks how much the Sobolev deficit controls the squared distance from a function to the manifold $\\mathfrak{M}$ of Jerison--Lee bubbles. Its central aim is to prove that the optimal stability constant $H_{\\mathrm{NS}}$ is attained by a function outside $\\mathfrak{M}$, and that this value is strictly larger than $H_{\\mathrm{BE}}$, the optimal constant for the local Folland--Stein--Sobolev inequality. The proof couples the Hardy--Littlewood--Sobolev interaction with the noncommutative geometry of $\\mathbb{H}^n$, diagonalizes the second variation on complex spherical harmonics, and uses two compactness thresholds---a spectral threshold and a two-peaks threshold---to force minimizing sequences to converge. If the argument is right, the nonlocal interaction genuinely raises the optimal lower stability constant, while the universal upper comparison constant remains $1$.","feed_headline":"Nonlocal term lifts the sharp stability constant on Heisenberg group","feed_subtitle":"If true, the optimal lower stability constant is attained and exceeds the local Sobolev one.","key_machinery":"The load-bearing object is the simultaneous diagonalization of two nonlocal bilinear forms that appear in the second variation after the Cayley transform from $\\mathbb{H}^n$ to $S^{2n+1}$. On each bidegree space $H^{n+1}_{i,j}$ of complex spherical harmonics the two forms act by eigenvalues $\\nu_{i,j}$ and $\\tau_{i,j}$, and the lowest normal mode $(2,0)$ yields the spectral threshold $H_{\\mathrm{NS}}^{\\mathrm{spec}} = \\frac{(4n+8-2\\mu)(\\mu+4)}{2(n+4)(4n+8-\\mu)}$, which equals $1 - \\left(\\frac{Q^*_\\mu-1}{\\nu_{2,0}} + \\frac{Q^*_\\mu}{\\tau_{2,0}}\\right)$. A separate construction, a two-bubble expansion using Heisenberg inversion and the distance formula, gives the two-peaks threshold $2 - 2^{\\frac{2n}{4n+4-\\mu}}$. Both thresholds are proved to be strict upper bounds for $H_{\\mathrm{NS}}$, and these strict inequalities are the barriers that exclude concentration on the bubble manifold and splitting into two bubbles.","core_discovery":"The paper's central claim is that the optimal stability constant $H_{\\mathrm{NS}}$ for the critical nonlocal Sobolev inequality on the Heisenberg group is attained, not merely approached, under the conditions $\\mu\\le 4$ or (3.5), and that this attained value is strictly larger than $H_{\\mathrm{BE}}$, the optimal constant for the local Folland--Stein--Sobolev inequality. The route is a two-threshold compactness argument: a spectral threshold produced by the lowest coupled eigenmode of the nonlocal second variation, and a two-peaks threshold produced by the interaction of two separated Jerison--Lee bubbles. The paper also establishes that the sharp universal upper comparison constant is $1$, with equality only on the bubble manifold, and that the associated Euler--Lagrange residual quotient obeys a strict single-bubble upper bound. In Section 5.1 the text itself records that the cubic strictness step, as written, does not establish the spectral threshold and that the proposed repair has zero cubic integral; that caveat is the load-bearing open point in the proof.","pith_inferences":["A repaired fourth-order expansion or a different choice of $(2,0)$ mode could extend the attainment theorem beyond $\\mu\\le 4$; the two-peaks barrier already holds for every $0<\\mu<Q$.","The comparison $H_{\\mathrm{NS}}>H_{\\mathrm{BE}}$ is a purely nonlocal effect: since both problems share the same bubble manifold, any difference in the optimal constant must come from the Hardy--Littlewood--Sobolev kernel rather than from the extremals.","The same simultaneous-diagonalization strategy should apply to analogous critical nonlocal inequalities on other groups with spherical-harmonic decompositions, such as H-type groups or CR spheres of higher dimension.","A numerical scan of the sign condition $(4-\\mu)(4n+8-\\mu)+3\\mu(\\mu+4)>0$ over integer $n$ and $\\mu\\in(0,Q)$ would immediately show which parameter ranges survive a corrected perturbation."],"forward_implications":["Under $\\mu\\le 4$ or condition (3.5), a minimizing sequence can neither approach the bubble manifold (spectral barrier) nor split into two bubbles (two-peaks barrier), so $H_{\\mathrm{NS}}$ is attained.","The attained minimizer lies outside $\\mathfrak{M}$, and the sharp Hardy--Littlewood--Sobolev inequality then forces the strict comparison $H_{\\mathrm{NS}} > H_{\\mathrm{BE}}$.","Every admissible function satisfies deficit $\\le \\mathrm{dist}(u,\\mathfrak{M})^2$, the constant $1$ is optimal, and equality holds exactly on the bubble manifold.","The Euler--Lagrange residual quotient has a strict single-bubble upper bound equal to the same spectral constant, giving quantitative stability for critical points independent of functional attainment.","Both compactness thresholds for the nonlocal problem exceed the corresponding thresholds of the local Folland--Stein--Sobolev problem, so the nonlocal term changes the quantitative structure despite the common extremal manifold."],"supporting_citations":[{"why":"The authors' companion preprint proving quantitative stability of the same inequality; it supplies the starting lower bound and notation.","marker":"[12]"},{"why":"Establishes the base remainder estimate for the nonlocal Sobolev inequality that this paper sharpens to optimality.","marker":"[18]"},{"why":"Derives the sharp Hardy--Littlewood--Sobolev constants and the Funk--Hecke eigenvalue identities on the Heisenberg group; supplies the diagonalization machinery.","marker":"[24]"},{"why":"Introduces the one-bubble spectral perturbation strategy used to prove the strict spectral upper bound.","marker":"[29]"},{"why":"Provides the minimizer existence scheme, the two-bubble expansion, and the balancing lemmas adapted in Section 6.","marker":"[30]"},{"why":"Computes the optimal stability constant and the two thresholds for the local Folland--Stein--Sobolev inequality; these are the comparison baseline.","marker":"[43]"},{"why":"Proves nondegeneracy of the bubbles and the two coupled eigenvalue identities used in Proposition 4.1.","marker":"[45]"},{"why":"Solves the analogous Euclidean nonlocal minimizer problem; serves as the benchmark whose Heisenberg counterpart is established here.","marker":"[16]"},{"why":"Gives the sharp universal upper constant 1 for the local Heisenberg Sobolev inequality, used to show the same constant holds nonlocally.","marker":"[35]"}],"fun_headline_variants":["Nonlocal Heisenberg Sobolev stability constant attained, beats local","Sharp nonlocal Sobolev stability on Heisenberg: constant attained and larger","Stability constant for critical nonlocal Sobolev on Heisenberg: attained, beats local","Attainment of Heisenberg nonlocal Sobolev stability: proof gap at cubic step","Heisenberg nonlocal Sobolev stability: best constant attained but proof caveat"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The strict spectral upper bound needs a perturbation mode whose cubic integral on the unit sphere is nonzero; the mode displayed in the paper has zero cubic integral, and the paper's own text says the cubic strictness argument does not establish the bound as written.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal Heisenberg Sobolev stability constant attained, beats local","Sharp nonlocal Sobolev stability on Heisenberg: constant attained and larger","Stability constant for critical nonlocal Sobolev on Heisenberg: attained, beats local","Attainment of Heisenberg nonlocal Sobolev stability: proof gap at cubic step","Heisenberg nonlocal Sobolev stability: best constant attained but proof caveat"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000899,"raw_usage":{"total_tokens":4006,"prompt_tokens":1215,"completion_tokens":2791,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":831,"completion_tokens_details":{"reasoning_tokens":2685}},"tokens_in":831,"tokens_out":2791,"duration_ms":18828,"temperature":1.0,"reasoning_tokens":2685,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:17:07.625376+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\int_{S^{2n+1}} \\omega_{2,0}(\\zeta)^3\\,d\\zeta$ for the proposed perturbation $\\omega_{2,0}=\\zeta_1\\zeta_2+\\zeta_1\\zeta_3+\\zeta_2\\zeta_3$. The rotation $\\zeta\\mapsto e^{i\\theta}\\zeta$ multiplies the integrand by $e^{6i\\theta}$, so the integral is zero; hence condition (5.2) fails and the strict inequality $H_{\\mathrm{NS}} < H_{\\mathrm{NS}}^{\\mathrm{spec}}$ is not established by the displayed perturbation.","supporting_citations":[{"cited_title":"Chen, Z.X","cited_arxiv_id":null,"evidence_quote":"The authors' companion preprint proving quantitative stability of the same inequality; it supplies the starting lower bound and notation."},{"cited_title":"Deng, X.L","cited_arxiv_id":null,"evidence_quote":"Establishes the base remainder estimate for the nonlocal Sobolev inequality that this paper sharpens to optimality."},{"cited_title":"Frank, E.H","cited_arxiv_id":null,"evidence_quote":"Derives the sharp Hardy--Littlewood--Sobolev constants and the Funk--Hecke eigenvalue identities on the Heisenberg group; supplies the diagonalization machinery."},{"cited_title":"K¨ onig, On the sharp constant in the Bianchi-Egnell stability inequality,Bull","cited_arxiv_id":null,"evidence_quote":"Introduces the one-bubble spectral perturbation strategy used to prove the strict spectral upper bound."},{"cited_title":"Tang, B.W","cited_arxiv_id":null,"evidence_quote":"Computes the optimal stability constant and the two thresholds for the local Folland--Stein--Sobolev inequality; these are the comparison baseline."},{"cited_title":"Nondegeneracy of positive solutions for critical Hartree equation on Heisenberg group and it's applications","cited_arxiv_id":"2508.07719","evidence_quote":"Proves nondegeneracy of the bubbles and the two coupled eigenvalue identities used in Proposition 4.1."},{"cited_title":"Deng, X.L","cited_arxiv_id":null,"evidence_quote":"Solves the analogous Euclidean nonlocal minimizer problem; serves as the benchmark whose Heisenberg counterpart is established here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the sharp universal upper constant 1 for the local Heisenberg Sobolev inequality, used to show the same constant holds nonlocally."}],"review_version":1}