{"id":"5d376766-4809-45ac-9b3c-8921c34acd03","arxiv_id":"2508.08614","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Near-optimizers of the nonlocal Sobolev inequality on the Heisenberg group are linearly close to the optimizer family U_{λ,ζ} in gradient L² norm, with a companion statement for weakly interacting bubbles in dimension Q=4.","lead":"This paper proves a quantitative stability estimate for the nonlocal Sobolev inequality on the Heisenberg group: functions nearly attaining equality are close to the known optimizer family, with distance controlled linearly by the Euler-Lagrange error. The result extends a widely used stability framework to a sub-Riemannian setting and includes a weak-interaction version for sums of bubbles in dimension four.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed linear bound δ(u) ≤ C Γ(u) depends on an unstated spectral-gap/nondegeneracy property of the linearized nonlocal operator at each optimizer; a spurious zero mode would degrade the rate to at best δ ≲ √Γ, and the abstract does not supply the required coercivity proof.","rationale":"Read in good faith, the paper aims to prove a local quantitative stability estimate for the nonlocal Sobolev inequality on Heisenberg groups, with an additional multi-bubble statement. The central condition for any such estimate is coercivity of the linearized operator modulo the exact symmetry group. This condition is not visible in the abstract, cannot be checked in the corrupted full text, and the Q = 4 limitation of the bubble-sum statement signals that the argument is not dimension-robust. The concern is internal to the claimed theorem rather than a disagreement with consensus: if the spectrum check fails, the linear rate is genuinely wrong; if it passes, the missing ingredient is supplied. This aligns with the reader's weakest_assumption about nondegeneracy and coercivity. I would not reject the paper outright, because the statement is plausible and the decisive condition can be checked explicitly, but the paper should not be accepted as a complete proof until the distance definition is fixed and the coercivity/nondegeneracy estimate is proved or cited. Hence I recommend a conditional acceptance, pending that verification.","tokens_in":13675,"tokens_out":10568,"duration_ms":130192,"concrete_test":"Locate the theorem in the manuscript and first check that δ is defined with an infimum over λ,ζ; without it, a small translation of any optimizer gives a counterexample to the stated inequality. Then compute the spectrum of the linearized operator L on the orthogonal complement of the conformal directions for the Heisenberg bubble, for example by mapping U to the CR sphere and diagonalizing L in terms of Folland–Stein spherical harmonics for n = 1 (Q = 4) and n = 2 (Q = 6). If the smallest eigenvalue of L on T_U^⊥ is strictly positive, the coercivity step is valid and the linear bound has the required support; if a spurious zero mode appears, the linear stability theorem as stated is false and only a Hölder-type bound can hold.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof schema is standard but hides the decisive hypothesis. Let E(u) denote the left-hand side of the displayed Euler equation. The theorem asserts ||∇H(u−U)||_{L^2} ≤ C ||E(u)||_{(S^{1,2})^{-1}}. Expanding E around an optimizer U, if the linearization L = dE(U) has a nontrivial null vector φ orthogonal to the tangent space of the optimizer manifold, then u = U + tφ gives Γ(u) = O(t^2) while δ(u) = O(t); no positive C can make the claimed linear bound true, and only a square-root bound is plausible. Thus the result needs a coercivity estimate ⟨Lφ,φ⟩ ≥ γ ||∇Hφ||_{L^2}^2 on the orthogonal complement of the conformal directions, together with a proof that the kernel of L is exactly the finite-dimensional conformal parameter space. The abstract states neither, and the provided full text is too corrupt to verify such a proof. The unexplained restriction of the weakly interacting bubble result to Q = 4 reinforces that dimension-specific interaction and coercivity estimates are load-bearing; for Q ≠ 4 the same linear control may fail or require different constants. In addition, the displayed δ is written as ||∇H u − ∇H U_{λ,ζ}||_{L^2} with no infimum over λ,ζ, so literally it is not a distance to the optimizer set; the invariant formulation must be δ(u) = inf_{λ,ζ} ||∇H u − ∇H U_{λ,ζ}||_{L^2}.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims a quantitative stability theorem for the nonlocal Sobolev inequality on the Heisenberg group: when a function u is close to solving the Euler-Lagrange equation, the natural L^2-gradient distance to the optimizer family U_{\\lambda,\\zeta} is linearly controlled by the dual norm \\Gamma(u) of the Euler-Lagrange expression. The abstract further claims the analogous linear stability for weakly interacting sums of bubbles when the homogeneous dimension is Q=4. The proof is said to rest on importing the sharp constant and optimizer classification from earlier work. However, the supplied full text is severely corrupted and largely unreadable, so the proof and the precise hypotheses cannot be verified from the manuscript as provided.","tokens_in":13994,"tokens_out":3627,"duration_ms":42543,"significance":"If the linear stability bound is correct, it would be a meaningful quantitative refinement of the known qualitative compactness and stability theory for the nonlocal Sobolev inequality in the Heisenberg group, and the Q=4 bubble-sum statement would add a new multi-bubble stability result. The statement is explicit and does not involve fitted constants, which is a strength. The use of the previously established sharp constant and optimizer classification is external grounding rather than circular reasoning. Nevertheless, because the body of the paper is unreadable and no complete proof, lemma statement, or theorem statement can be checked, I cannot certify soundness from the present submission.","major_comments":[{"comment":"The supplied body consists of corrupted text with replacement characters and broken equations; I could not read any complete theorem, lemma, or proof. Since the central claims rest entirely on these unreadable arguments, this is a blocking issue. The authors must provide a clean, readable PDF or LaTeX source before a substantive content review can be completed.","section":"Full text (all sections after the Abstract)"},{"comment":"The displayed definition δ(u)=||∇_H u-∇_H U_{\\lambda,\\zeta}||_{L^2} depends on an arbitrary choice of the parameters (\\lambda,\\zeta), so it is not a distance to the optimizer family. As written, the linear bound δ(u)≤C Γ(u) cannot have the stated geometric meaning. The theorem should either define δ(u)=inf_{\\lambda,\\zeta} ||∇_H u-∇_H U_{\\lambda,\\zeta}||_{L^2} or specify an equivalent gauge-fixing procedure and then prove the bound for that quantity.","section":"Abstract, definition of δ(u)"},{"comment":"The restriction of the weakly interacting bubble result to Q=4 is unexplained in the abstract and in the readable fragments. If the proof relies on dimension-specific interaction estimates, the paper should either prove the result for every admissible Q or state precisely which estimate fails for Q≠4. As it stands, the restriction indicates a load-bearing hypothesis whose verification is not visible.","section":"Abstract, Q=4 restriction"},{"comment":"The claimed linear bound δ(u)≤C Γ(u) requires a coercivity or spectral-gap estimate for the linearized nonlocal Euler-Lagrange operator at each optimizer, on the complement of the finite-dimensional parameter manifold. Neither the abstract nor the readable text states or proves such an estimate. Without it, a spurious approximate null direction would reduce the best possible rate to at most δ(u)≲√Γ(u). This is the decisive analytical hypothesis and must be stated and proved explicitly.","section":"Abstract, linear-rate claim"}],"minor_comments":[{"comment":"There is a spelling error 'Hiesenberg' in the first sentence; it should be 'Heisenberg'.","section":"Abstract"},{"comment":"The phrase 'the the set of optimizers' contains a duplicated article and should be corrected.","section":"Abstract"},{"comment":"The equation label 'non-critical-n' appears in the abstract but no corresponding numbered display is visible in the readable text; the numbering should be made consistent.","section":"Abstract"},{"comment":"The notation Q^*_μ is used without a reminder of its dependence on Q and μ beyond the first occurrence; a brief restatement near the Euler equation would improve readability.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The submission appears to be a corrupted text extraction rather than a mathematical manuscript; the proof is not assessable in this form. I recommend asking the authors to resubmit a clean PDF/source and to revise the abstract so that δ(u) is genuinely an infimum and the Q=4 restriction is justified. I do not see evidence of circularity in the claim itself."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper announces a linear stability estimate delta(u) <= C Gamma(u) for the nonlocal Sobolev inequality on the Heisenberg group, plus a multi-bubble version for Q=4. That is a natural and worthwhile extension of results known in Euclidean and fractional settings, and the statement is clean. The authors import the sharp constant and optimizer classification, so the circularity burden is low. I see no invented entities or fitted constants.\\n\\nThe problem is that I cannot check the proof: the full text in the provided copy is garbled beyond use. So my verdict rests on the abstract and on what the argument must contain. The stress-test note is on target: the linear rate needs a spectral gap for the linearized operator around every optimizer, i.e. coercivity on the orthogonal complement of the conformal directions and exact identification of the kernel. Without that, a spurious zero mode would degrade the rate to square-root. The abstract says nothing about this, so the reader cannot tell whether the hard part is done. The unexplained Q=4 restriction for bubble sums reinforces that dimension-specific estimates are load-bearing.\\n\\nThere is also a small but real defect in the displayed definition: delta(u)=||nabla_H u - nabla_H U_{lambda,zeta}|| has no infimum over lambda,zeta, so literally it is not a distance to the optimizer set. The intended invariant formulation must be delta(u)=inf_{lambda,zeta} ||nabla_H u - nabla_H U_{lambda,zeta}||. I assume this is a typo, but as written it would make the theorem meaningless for a family of optimizers.\\n\\nThe citation pattern appears reasonable; the abstract cites the sharp constant and classification rather than claiming them. The spelling error 'Hiesenberg' is minor.\\n\\nWho is this for? Analysts working on sharp functional inequalities, stability of optimizers, and blow-up analysis in sub-Riemannian geometry. If the proof delivers the coercivity estimate, the paper is a useful tool. But the burden is on the authors to state and prove the spectral-gap property; the abstract hides it.\\n\\nMy recommendation: if the full text is readable, this deserves a serious referee. The result is important enough and the statement is precise enough to warrant referee time, even though the abstract alone cannot certify correctness. I would not cite it until the proof is verified.","headline":"Plausible and clearly stated extension of quantitative stability to the nonlocal Sobolev inequality on the Heisenberg group, but the supplied text is too corrupt to verify the proof, and the abstract omits the coercivity/nondegeneracy hypotheses that the linear rate depends on.","tokens_in":683,"tokens_out":840,"would_cite":false,"duration_ms":22665,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","43A80","35R09"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that near-solutions of the nonlocal Sobolev Euler-Lagrange equation on the Heisenberg group are linearly close to the explicit optimizer family, and that the same control holds for weakly interacting bubble sums when the…","keywords":["quantitative stability","nonlocal Sobolev inequality","Heisenberg group","Hardy-Littlewood-Sobolev inequality","Folland-Stein inequality","Euler-Lagrange equation","bubble solutions","stability of critical points"],"falsifier":"Take a one-parameter family $u_\\varepsilon = U_{\\lambda,\\zeta} + \\varepsilon v$ with $v$ orthogonal to the tangent directions of the optimizer family and compute the ratio $\\Gamma(u_\\varepsilon)/\\delta(u_\\varepsilon)$ as $\\varepsilon\\to 0$; if the ratio tends to zero for some $v$, the claimed linear bound is false. Such a $v$ would be an eigenfunction of the linearized Euler-Lagrange operator with eigenvalue approaching zero.","tokens_in":13476,"feed_emoji":"📐","tokens_out":10844,"duration_ms":109066,"temperature":0.7,"pith_summary":"The paper asks how rigid the nonlocal Sobolev inequality on the Heisenberg group is, once a function is almost a solution of its Euler-Lagrange equation. The central claim is that the gradient distance $\\delta(u)=\\|\\nabla_H u-\\nabla_H U_{\\lambda,\\zeta}\\|_{L^2}$ from $u$ to the explicit optimizer family is bounded by a constant times $\\Gamma(u)$, the dual norm of the Euler-Lagrange residual. This turns a compactness statement into a quantitative, linear one: functions with small residual cannot drift away from the optimizer manifold. The same linear control is claimed for sums of widely separated bubbles when the homogeneous dimension is $Q=4$. If true, the optimizer family is the only place where near-critical functions can concentrate, and the rate of approach is governed exactly by the equation's residual.","feed_headline":"Near-solutions to a Heisenberg inequality stay close to bubbles","feed_subtitle":"The paper proves the residual of the Euler-Lagrange equation controls the distance to the optimizer family.","key_machinery":"The argument is carried by the Euler-Lagrange residual and its dual norm $\\Gamma(u)$, together with the local coercivity of the linearized operator around each optimizer $U_{\\lambda,\\zeta}$. Nondegeneracy means that on the subspace orthogonal to the tangent directions generated by varying the scale $\\lambda$ and the Heisenberg translation $\\zeta$, the linearized operator has a positive spectral gap; that gap is what converts a small residual $\\Gamma(u)$ into a small gradient gap $\\delta(u)$. The bubble family $U_{\\lambda,\\zeta}$ supplies the explicit reference set relative to which the distance $\\delta(u)$ is measured.","core_discovery":"The paper establishes the quantitative stability estimate $\\delta(u) \\le C\\, \\Gamma(u)$ for functions $u$ close to solving $-\\Delta_H u = \\left(\\int_{\\mathbb{H}^n} |u(\\eta)|^{Q^\\ast_\\mu}|\\eta^{-1}\\xi|^{-\\mu}\\,d\\eta\\right)|u|^{Q^\\ast_\\mu-2}u$, where $\\delta(u)$ is the $L^2$ gradient gap to the family $U_{\\lambda,\\zeta}$ of explicit optimizers and $\\Gamma(u)$ is the norm in the dual of $S^{1,2}(\\mathbb{H}^n)$ of the Euler-Lagrange expression $\\Delta_H u + \\left(\\int_{\\mathbb{H}^n} |u(\\eta)|^{Q^\\ast_\\mu}|\\eta^{-1}\\xi|^{-\\mu}\\,d\\eta\\right)|u|^{Q^\\ast_\\mu-2}u$. For sums $\\sum_{i=1}^\\nu U_{\\lambda_i,\\zeta_i}$ of weakly interacting bubbles, the same linear bound is claimed when $Q=4$.","pith_inferences":["The special role of $Q=4$ suggests a genuine dimensional threshold: for larger homogeneous dimensions, interaction between bubbles may be weaker, so the linear multi-bubble control could fail or require a different rate.","The same coercivity mechanism should yield explicit constants by computing the spectral gap of the linearized operator on the first nontrivial variation; such a computation would also identify the sharp stability threshold.","If the nondegeneracy assumption were to fail for some $\\mu\\in(0,Q)$, there should be a continuous family of near-critical functions with $\\Gamma(u)\\to 0$ but $\\delta(u)$ bounded away from zero; finding one would directly delimit the range of $\\mu$ where the theorem can hold."],"forward_implications":["Near-solutions of the Euler-Lagrange equation form a bounded neighborhood of the optimizer manifold, with the residual and the gradient gap equivalent up to a constant.","Minimizing sequences that approach equality in the nonlocal Sobolev inequality must converge to the optimizer family, at a rate controlled by their Euler-Lagrange residual.","At $Q=4$, multi-bubble near-solutions obey the same linear rigidity, so widely separated bubbles cannot hide a residual while the gradient profile drifts away.","The linear rate is the natural quantitative form of the Euler-Lagrange characterization of the sharp constant $C_{HL}(Q,\\mu)$."],"supporting_citations":[],"fun_headline_variants":["Heisenberg optimizers: residual bounds the gap","Residual dictates closeness to Heisenberg bubbles","Linear control for Heisenberg near-optimizers","Stability: residual controls distance to Heisenberg bubbles","Heisenberg nonlocal Sobolev: residual bounds deviation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs every optimizer function to be isolated up to its natural symmetries: the only nearly-free deformations should be changing the scale or the location. If some other deformation were almost free, a small residual $\\Gamma(u)$ could sit alongside a large gradient gap $\\delta(u)$, and the linear bound would break.","fun_headline_variants_meta":{"raw":{"variants":["Heisenberg optimizers: residual bounds the gap","Residual dictates closeness to Heisenberg bubbles","Linear control for Heisenberg near-optimizers","Stability: residual controls distance to Heisenberg bubbles","Heisenberg nonlocal Sobolev: residual bounds deviation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000574,"raw_usage":{"total_tokens":2878,"prompt_tokens":1279,"completion_tokens":1599,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":895,"completion_tokens_details":{"reasoning_tokens":1526}},"tokens_in":895,"tokens_out":1599,"duration_ms":12154,"temperature":1.0,"reasoning_tokens":1526,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:34:03.780229+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a one-parameter family $u_\\varepsilon = U_{\\lambda,\\zeta} + \\varepsilon v$ with $v$ orthogonal to the tangent directions of the optimizer family and compute the ratio $\\Gamma(u_\\varepsilon)/\\delta(u_\\varepsilon)$ as $\\varepsilon\\to 0$; if the ratio tends to zero for some $v$, the claimed linear bound is false. Such a $v$ would be an eigenfunction of the linearized Euler-Lagrange operator with eigenvalue approaching zero.","supporting_citations":[],"review_version":2}