{"id":"72796952-b226-4a67-8f5c-7df2c7c43115","arxiv_id":"2607.06415","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Sharp stability estimates with optimal exponents are established for the affine Sobolev inequality and its critical points for p≥2, including a new affine spectral gap inequality.","lead":"The paper proves sharp quantitative stability for the affine Sobolev inequality for p≥2, showing that functions nearly saturating the inequality are quantitatively close to optimizers, with best-possible exponents. It also proves an analogous stability result for critical points of the affine Sobolev functional in the single-bubble regime.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The key spectral gap computation (Prop. 4.6(iv)) checks out under independent verification.","rationale":"The reader correctly identifies the affine spectral gap inequality as the load-bearing ingredient and flags Proposition 4.6(iv) as requiring verification. I performed an independent check of the two key claims: (1) λ_{2,p}=1, which follows from direct substitution and is correct; (2) the strict monotonicity of {λ_{2k,p}}, which reduces to showing n(n+p) < 2k(2k+n)(n+2p−2) for k≥1, n≥2, p∈[2,n)—this holds because 2k(2k+n) ≥ 2(n+2) and n+2p−2 ≥ n+2, giving a lower bound of 2(n+2)² > 2n² ≥ n(n+p). The ratio formula itself is a direct algebraic consequence of the definitions. The nonlinear extension (Theorem 4.11) uses a well-structured contradiction argument with appropriate convergence estimates. The sharpness constructions in Section 6 follow established methods. The paper follows the Figalli–Zhang framework with a genuinely new ingredient (the variance term and its spectral gap control), and a concurrent work [19] obtained a weaker result, confirming novelty. I do not find a specific error or logical gap that would warrant changing the ACCEPT verdict. The reader's MODERATE confidence due to not having line-by-line checked the computations is reasonable, but my targeted verification of the most critical computation supports the result.","tokens_in":59234,"tokens_out":5861,"duration_ms":218687,"concrete_test":"Independently verify the ratio computation in Proposition 4.6(iv): expand (μ_{2k+2}+p−n)(2k+n+p)² − (μ_{2k}+p−n)(2k−p)² and confirm it equals (4k+n)[n(n+p) − 2k(2k+n)(n+2p−2)]. If this identity fails for any k≥1, n≥2, p∈[2,n), the monotonicity argument collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new ingredient is the affine spectral gap inequality (Theorem 4.2), whose proof reduces via spherical harmonic decomposition to Proposition 4.6. Parts (i)–(iii) reduce to the classical spectral gap of the linearized p-Laplacian (Lemma 4.1). Part (iv), for even ℓ≥4, hinges on showing λ_{ℓ,p} < 1, where λ_{2k,p} = (n+p)/p² · (μ_{2k}+p−n) · (∏_{j=0}^{k-1} (2j−p)/(2j+n+p))². The paper verifies λ_{2,p}=1 by direct substitution (the j=0 factor gives p²/(n+p)², and μ₂+p−n = n+p, yielding exactly 1) and proves strict monotonicity of {λ_{2k,p}} by computing the ratio λ_{2k+2,p}/λ_{2k,p} − 1 = (4k+n)[n(n+p) − 2k(2k+n)(n+2p−2)] / [(μ_{2k}+p−n)(2k+n+p)²]. The numerator is negative because for k≥1, n≥2, p∈[2,n): 2k(2k+n)(n+2p−2) ≥ 2(n+2)(n+2) = 2(n+2)² > 2n² ≥ n(n+p). This confirms λ_{2k,p} < 1 for all k≥2. The nonlinear extension (Theorem 4.11) uses a standard contradiction/compactness argument with careful passage to the limit in the eL(2) terms via Newton–Leibniz decomposition and dominated convergence. The overall structure follows the Figalli–Zhang framework with the key new element being the variance term and its control. No specific error or gap is identified.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The paper proves sharp quantitative stability for the affine Sobolev inequality for $p�[2,n)$, including both stability of the deficit (Theorem 1.1) and stability of critical points in the absence of bubbling (Theorem 1.3). The right-hand side of the main stability inequality (1.3) contains two terms: a $p$-th power distance to the optimizer manifold $M_{aff}$ and a weighted quadratic term $|∇v|^{p-2}|∇(f∘B−v)|^2$. Both exponents ($p$ and $2$) are shown to be sharp in Section 6. The proof follows the Figalli–Zhang framework for the classical Sobolev inequality, adapted to the affine setting. The central new ingredient is an affine spectral gap inequality (Theorem 4.2), which controls a variance term on $S^{n-1}$ arising from the nonlinear outer average in the affine energy. The spectral gap is proved via spherical harmonic decomposition, reducing to a classical gap for $ℓ≤2$ and to an explicit monotonicity computation for even $ℓ≥4$ (Proposition 4.6). A nonlinear extension (Theorem 4.11) is obtained by a contradiction/compactness argument.","tokens_in":60035,"tokens_out":860,"duration_ms":143730,"significance":"The paper makes a substantial contribution to the stability theory of sharp functional inequalities. The affine Sobolev inequality is a strengthening of the classical Sobolev inequality with a larger symmetry group, and its stability analysis requires handling the nonlocal structure of the affine energy. The key technical achievement is the affine spectral gap inequality (Theorem 4.2/4.11), which is the load-bearing new ingredient and is verified by an explicit computation involving Gegenbauer polynomial coefficients (Proposition 4.6(iv)). The sharpness constructions in Section 6 are concrete and falsifiable. The result improves upon the concurrent work [19] by including the quadratic term, which is shown to be sharp and relevant for applications. The paper also provides a complete proof of stability for critical points (Theorem 1.3) with sharp exponent $p-1$. The overall structure is clear and the proofs are detailed.","major_comments":[],"minor_comments":[{"comment":"Appendix A, line containing 'deinfed': typo, should be 'defined'.","section":null},{"comment":"Remark 1.4(b): 'be definition' should be 'by definition'.","section":null},{"comment":"The notation switches between $T_{λ,S,x}$ and $T^{(q)}_{λ,A,x}$; a brief remark in Section 1.3 or Section 2 clarifying the relationship once more would help the reader.","section":null},{"comment":"In the proof of Theorem 4.11, the passage from (4.27) to the liminf estimates for $I_j$ and $II_j$ involves several applications of dominated convergence and weak convergence. The logic is correct but dense; adding a sentence summarizing why the cross-terms vanish would improve readability.","section":null},{"comment":"The acknowledgment of AI tools is appropriate; no action needed.","section":null},{"comment":"Some cross-references to equations in Section 3 could be made more precise (e.g., the reference to (3.13) in the proof of Proposition 3.4). This is a minor presentation issue.","section":null},{"comment":"In Section 6.2, the condition (6.12) is introduced after the construction of $f_ε$; stating it earlier or more prominently would improve clarity.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a strong contribution. The main technical novelty (affine spectral gap) checks out under scrutiny. The concurrent work [19] by Fan et al. obtains a weaker form (without the quadratic term); the authors are transparent about this. I see no novelty or citation concerns. The restriction to $p≥2$ is acknowledged as technical; extending to $p<2$ would require different tools (the Taylor expansion in Proposition 3.1 uses $p≥2$) and is a natural open problem. The paper is well within the journal's scope."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and the positive assessment. The referee's report recommends minor revision but does not list specific major comments requiring changes. We address the substance of the report below.","responses":[{"response":"We thank the referee for the careful and accurate summary of our paper. The description of the main results, the role of the affine spectral gap inequality as the central new ingredient, the spherical harmonic decomposition and Gegenbauer polynomial computation in Proposition 4.6, and the sharpness constructions in Section 6 are all faithfully represented. We confirm that the referee's characterization of the paper's contribution and structure is correct.","revision_made":"no","referee_comment":"The referee provides a detailed and accurate summary of the paper's structure, main results (Theorems 1.1 and 1.3), the affine spectral gap inequality (Theorems 4.2/4.11), the sharpness constructions in Section 6, and the overall proof strategy following the Figalli–Zhang framework adapted to the affine setting."}],"tokens_in":58666,"tokens_out":224,"duration_ms":20393,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The paper proves sharp stability for the affine Sobolev inequality for p in [2,n), with optimal exponents on both the distance term (power p) and the weighted quadratic term (power 2), plus a corresponding critical-point stability result with exponent p-1. The main new ingredient is the affine spectral gap inequality (Theorem 4.2), which handles a variance term on the sphere that has no classical counterpart. This variance term arises because the affine energy involves a negative-power spherical average of directional derivatives, and controlling it is the central difficulty the paper resolves. The sharpness constructions in Section 6 are clean and match the exponents claimed in the main theorems. The overall strategy follows the Figalli-Zhang framework adapted to the affine setting, which is the natural approach, and the execution is careful throughout. The compactness argument in Section 2, the Taylor-type expansions in Section 3, and the nonlinear extension of the spectral gap (Theorem 4.11) via contradiction/compactness are all present and internally consistent. The concurrent work [19] obtained a weaker result without the quadratic term, which confirms the novelty here. The stress-test concern about Proposition 4.6(iv) checks out: the key computation showing the sequence {lambda_{2k,p}} is strictly decreasing with lambda_{2,p}=1 is verified by an explicit ratio calculation, and the bound is correct for k>=1, n>=2, p in [2,n). The nonlinear extension in Theorem 4.11 uses a standard contradiction argument with Newton-Leibniz decomposition and dominated convergence, which is handled correctly. The restriction to p>=2 is acknowledged as technical, and removing it is left as an open problem. This is a genuine limitation but not a flaw — the p<2 case likely requires different tools. I did not find errors in the proofs I checked in detail. The paper is written for specialists in sharp Sobolev inequalities and geometric analysis. It deserves a serious referee who can verify the spectral gap computations line by line, particularly the Sturm-Liouville analysis and the passage to the limit in Theorem 4.11.","headline":"Sharp stability for the affine Sobolev inequality, p>=2, with a new affine spectral gap inequality as the key ingredient","tokens_in":60301,"tokens_out":505,"would_cite":true,"duration_ms":50233,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35","26D10"],"pacs":[],"model":"glm-5.2","headline":"Sharp stability for affine Sobolev inequality proved for p≥2","keywords":["affine Sobolev inequality","quantitative stability","sharp exponent","spectral gap","p-Laplacian","spherical harmonics","Gegenbauer polynomials","critical points"],"falsifier":"Construct a sequence of functions f_epsilon = U + epsilon * phi for perturbations phi in the orthocomplement of the tangent space, showing that the deficit is of order epsilon^2 while the p-th power distance is of order epsilon^p with p > 2, confirming that the quadratic term is the leading contribution and cannot be replaced by any higher power.","tokens_in":59480,"feed_emoji":"📐","tokens_out":1213,"duration_ms":147835,"temperature":0.7,"pith_summary":"This paper proves that the affine Sobolev inequality admits sharp quantitative stability for exponents p at least 2, meaning that any function whose affine Sobolev ratio is close to the best constant must be close, in a precise and optimal sense, to the family of extremal functions. The affine Sobolev inequality is a strengthening of the classical Sobolev inequality that replaces the L^p norm of the gradient with the affine Sobolev energy E_p, a quantity obtained by averaging directional gradient norms over the unit sphere with a negative power. This energy is invariant under all volume-preserving affine transformations, a much larger symmetry group than the classical inequality enjoys. The authors show that the deficit E_p(f)^p - S_aff^p ||f||_{p*}^p controls two terms simultaneously: the p-th power of the distance from f to the nearest extremal function, and a weighted quadratic term involving |∇v|^{p-2} |∇(f-v)|^2. Both exponents p and 2 are proved to be sharp. A parallel result establishes stability for near-critical points of the affine Sobolev functional in the single-bubble regime, with the optimal exponent p-1 on the distance. The central new ingredient is an affine spectral gap inequality that controls a variance-type term on the sphere arising from the nonlinear outer average in the affine energy, a term with no classical counterpart.","feed_headline":"Sharp stability for affine Sobolev inequality proved for p≥2","feed_subtitle":"Deficit controls distance to optimizers with optimal exponents p and 2, plus a new affine spectral gap inequality on the sphere.","key_machinery":"The proof combines a compactness reduction via profile decomposition, a nonlinear Taylor expansion of the affine energy E_p that produces a variance term Var(f,g) on the sphere, and an affine spectral gap inequality proved by decomposing perturbations into spherical harmonics and verifying strict monotonicity of a sequence of eigenvalue ratios involving Gegenbauer polynomial coefficients.","core_discovery":"The key discovery is that the affine Sobolev energy, despite being a nonlinear nonlocal functional involving a negative-power average of directional gradient norms over the sphere, admits a local second-order expansion whose leading correction includes a variance term on S^{n-1} that has no analogue in the classical Sobolev inequality. This variance term detects the extra trace-free affine directions in the tangent space of the extremal manifold. The authors prove an affine spectral gap inequality showing that this variance term, combined with the weighted quadratic form from the directional derivatives, is bounded below by a positive multiple of the weighted L^2 norm of the perturbation, as","pith_inferences":["The sharpness of the quadratic exponent 2 for p > 2 suggests that the affine Sobolev inequality has a fundamentally different local geometry than the classical Sobolev inequality near its extremals, where the exponent is p rather than 2 for the leading distance term.","The affine spectral gap inequality on spherical harmonics may have independent interest in spectral theory, as it relates the spectrum of the linearized p-Laplacian to the variance operator coming from the nonlinear sphere average, potentially connecting to representation-theoretic properties of Gegenbauer polynomials.","The parallel between the exponents (p, 2) in the stability theorem and (p-1) in the critical point theorem mirrors the classical Sobolev pattern, suggesting a universal relationship between the stability exponent and the critical point exponent for inequalities with similar variational structure."],"forward_implications":["The stability result with the sharp quadratic term may enable quantitative convergence rates for numerical approximations of the affine Sobolev constant, paralleling applications of classical Sobolev stability to finite element methods.","The affine spectral gap inequality established here could serve as a foundation for studying multi-bubble stability of the affine Sobolev inequality, where the interaction between multiple extremal functions must be controlled.","The techniques for handling the nonlinear outer average and variance term may extend to other affine-invariant inequalities, such as the affine Moser-Trudinger or affine Morrey-Sobolev inequalities, where similar nonlocal averaging structures appear.","The restriction to p at least 2 is technical; the spectral gap argument for odd spherical harmonics is immediate but the even case requires an intricate monotonicity verification, suggesting that new ideas may be needed for the range 1 < p < 2."],"fun_headline_variants":["Affine Sobolev inequality: sharp stability with optimal exponents for p≥2","Variance on sphere yields sharp stability for affine Sobolev inequality","Optimal deficit bounds for affine Sobolev inequality when p≥2","Affine spectral gap inequality controls stability of Sobolev extremals","Trace-free affine directions detected via spherical variance in Sobolev stability"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing new ingredient is an affine spectral gap inequality that requires, for even spherical harmonic degrees at least 4, the strict monotonicity of a sequence of eigenvalue ratios. This monotonicity is verified by an explicit computation showing the ratio lambda_{2k+2,p}/lambda_{2k,p} is strictly less than 1, with lambda_{2,p} = 1, but the verification is intricate and specific to the structure of Gegenbauer polynomial coefficients.","fun_headline_variants_meta":{"raw":{"variants":["Affine Sobolev inequality: sharp stability with optimal exponents for p≥2","Variance on sphere yields sharp stability for affine Sobolev inequality","Optimal deficit bounds for affine Sobolev inequality when p≥2","Affine spectral gap inequality controls stability of Sobolev extremals","Trace-free affine directions detected via spherical variance in Sobolev stability"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":466,"prompt_tokens":387,"completion_tokens":79,"prompt_tokens_details":null},"tokens_in":387,"tokens_out":79,"duration_ms":43699,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T06:25:07.035936+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Construct a sequence of functions f_epsilon = U + epsilon * phi for perturbations phi in the orthocomplement of the tangent space, showing that the deficit is of order epsilon^2 while the p-th power distance is of order epsilon^p with p > 2, confirming that the quadratic term is the leading contribution and cannot be replaced by any higher power.","supporting_citations":[],"review_version":1}