{"id":"0eb0a483-8803-464f-8654-6192881f1002","arxiv_id":"2606.09555","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For 2 ≤ p < n, the affine L^p-Sobolev deficit controls the Ẇ^{1,p}-distance to the affine bubble manifold with the sharp exponent p: no better power law is possible.","lead":"Functions that nearly achieve equality in the affine p-Sobolev inequality are shown to be close to the family of optimal bubbles, with the distance controlled by the p-th power of the energy deficit — and this exponent p is best possible. The affine inequality is a stronger, more symmetric refinement of the classical Sobolev inequality, so its stability underpins rigidity and convergence arguments in geometric analysis.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniform directional decoupling (4.6) is asserted via a hand-waved η↓0 limit; near the singular projector P_ξ the uniformity is unproved, and if it fails the no-splitting step of Proposition 4.1 collapses.","rationale":"The reader's weakest assumption identifies the same load-bearing point: the uniform-in-ξ directional decoupling (4.6). This is the only step in the global argument that has no detailed proof and whose failure would break Proposition 4.1 and hence Theorem 1.1. I agree with the reader's assessment that this is a genuine gap in the manuscript. However, the gap appears repairable: pointwise-in-ξ decoupling follows from applying the standard L^p profile decomposition to ∂_ξ u_k, and the uniform-in-ξ convergence then follows from the uniform Lipschitz property of A_ξ on bounded subsets of Ẇ^{1,p}. The paper's two-sentence argument via M_{ξ,η} is insufficient as written, but the underlying claim is likely true. The second defect noted by the reader — the transposed identities in Lemma 4.1 — is real but localized and immediately fixed by choosing M=L instead of M=L^{-1}; it does not threaten the central theorem once corrected. Given these localized, repairable issues, the reader's CONDITIONAL verdict remains appropriate; no change to the verdict is warranted.","tokens_in":27811,"tokens_out":35512,"duration_ms":373444,"concrete_test":"Re-derive (4.6) without invoking M_{ξ,η}. For a fixed ξ, write f_k = ∂_ξ u_k = Σ_j g_{j,k}(∂_ξ W_j) + ∂_ξ r_k^F; the profiles ∂_ξ W_j inherit the same dilations/translations and dislocation conditions as W_j, so the standard L^p decoupling lemma should give A_ξ(u_k) = Σ_j A_ξ(W_j) + A_ξ(r_k^F) + o_k(1) pointwise in ξ. Then use the estimate |A_ξ(v) − A_η(v)| ≤ p ∥∇v∥_p^p |ξ−η|, valid uniformly for the bounded sequence, to upgrade to sup_{ξ∈S^{n-1}}. If the pointwise step cannot be completed, test the critical case with a two-profile sequence u_k = U + λ_k^{-n/p}U(·/λ_k), λ_k→∞, for p=3, n=4, and compute sup_ξ |A_ξ(u_k) − A_ξ(U) − A_ξ(g_k U)|; a nonzero limit would invalidate Proposition 4.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The global no-splitting argument in Proposition 4.1 relies on eq. (4.6): for every affine near-extremizing sequence, A_ξ(u_k) = Σ_j A_ξ(W_j) + A_ξ(r_k^F) + o_k(1) uniformly in ξ ∈ S^{n-1}. The proof applies the usual gradient decoupling to the degenerating family M_{ξ,η} = P_ξ + η(I − P_ξ) and then lets η↓0, asserting uniformity 'by compactness of S^{n-1}'. This is not a valid two-sentence argument: the decoupling error for a fixed invertible M generally depends on the condition number of M, and as η→0 the family degenerates at the equator. The order of the limits η↓0 and k→∞ matters, and no uniform-in-ξ, uniform-in-η error estimate is supplied. If (4.6) fails, the chain (4.7) cannot be run, and the exclusion of multi-bubble splitting — hence the global step of Theorem 1.1 — is unsupported. The concern is an analytic gap in the proof rather than a contradiction with the claimed result; a repair is plausible via pointwise decoupling for each fixed ξ followed by the uniform Lipschitz bound |A_ξ(v) − A_η(v)| ≤ p∥∇v∥_p^p |ξ−η|, which holds for every v with bounded gradient.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a sharp quantitative stability estimate for the affine L^p-Sobolev inequality of Lutwak--Yang--Zhang for 2≤p<n. For every nonzero u∈Ẇ^{1,p}(R^n), the affine-Sobolev deficit controls the p-th power of the Ẇ^{1,p}-distance to the full affine extremal manifold M_aff, and the exponent p is shown to be optimal. The proof combines a local second-variation analysis of the affine energy, a spectral-gap result for the affine Hessian whose kernel is enlarged by trace-free degree-two modes, and a global compactness argument based on profile decomposition after a John-ellipsoid normalization. The sharpness is tested on translated bumps.","tokens_in":27938,"tokens_out":18137,"duration_ms":173497,"significance":"If correct, this is the first sharp gradient-type stability theorem for the affine Sobolev inequality, with the optimal exponent p. It extends the classical Figalli--Zhang theory to the SL(n)-invariant affine setting and identifies explicitly the enlarged extremal manifold. The paper is substantial: it contains a detailed second-variation computation, a Funk--Hecke/Saalschütz evaluation of the spectral coefficients, a calibration identity for the degree-two sector, and a self-contained sharpness construction. The overall architecture is sound, and the main unresolved point is a uniformity issue in the directional profile decomposition used in the global compactness step.","major_comments":[{"comment":"The directional decoupling A_ξ(u_k)=Σ_j A_ξ(W_j)+A_ξ(r_k^F)+o_k(1) uniformly in ξ is load-bearing for the no-splitting step and is not proved. The stated derivation applies the usual gradient decoupling to M_{ξ,η}=P_ξ+η(I-P_ξ) and lets η↓0, claiming uniformity 'by compactness of S^{n-1}'. This is insufficient: for each fixed invertible M the decoupling error depends on the condition number of M, which degenerates as η→0 at the equator, and no uniform-in-(ξ,η) estimate is supplied. If (4.6) fails, the chain (4.7) and the exclusion of multi-bubble splitting collapse. A repair appears plausible: prove pointwise-in-ξ decoupling using standard directional profile decomposition, then use the uniform Lipschitz bound |A_ξ(v)-A_η(v)|≤C∥∇v∥_p^p |ξ-η| to pass to uniformity in ξ. The manuscript should either provide this argument or a different honest proof of (4.6).","section":"§4, Eq. (4.6)"}],"minor_comments":[{"comment":"Typo: 'hlod' should be 'hold'.","section":"§4, proof of Lemma 4.2"},{"comment":"The notation eε_j = ε_j λ_j is introduced but never used after (3.24); consider simplifying the normalization step.","section":"§3.2"},{"comment":"The sentence 'Then, by (2.8) yields' is ungrammatical; the reference to (2.8) in (4.12) should be clarified.","section":"§4, proof of Proposition 4.1"},{"comment":"The expression 'cbδaff,p(uε,R)' in (4.21) mixes notations; clarify the role of c and the subscript p.","section":"§4.1, sharpness proof"},{"comment":"Reference [17] is given only as 'arXiv preprint'; provide a full arXiv number or journal data if available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is solid in its local analysis and spectral computations; I checked the main algebra and found it internally consistent. The single serious concern is the unproved uniform directional decoupling (4.6), which is central to the global compactness argument. The authors should be asked to supply a rigorous proof of that uniformity before acceptance. I do not see a fundamental obstruction, so major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this is a serious paper, the result is likely true and new, but the global step has two unpatched seams. The sharp exponent p for the affine p-Sobolev stability is a real advance. The mechanism — a variance term in the second variation that enlarges the classical Sobolev kernel by trace-free quadratic modes x·B∇U — is explicit and correct, and the Funk–Hecke sector analysis checks out. I verified the second-variation algebra, the calibration identity in §5, the monotonicity ν_{m+1}/ν_m<1, and the translated-bump sharpness construction; they are internally consistent. So my conditional recommendation is not doubt about the main claim.\n\nTwo things need repair. First, Lemma 4.1 has an indexing error. From (T_{M,0}u)(x)=u(M^{-1}x) one gets A_ξ(Su)=A_{M^{-1}ξ}(u) and K_{Su}=M K_u, the opposite of the displayed identities. With M=L^{-1}, the asserted B_r⊂K_{Su} does not follow; taking M=L fixes it. This is a repairable local error, but it sits under the affine normalization, so the global step is not justified as printed.\n\nSecond, the uniform directional decoupling (4.6) is asserted in two sentences: apply gradient decoupling to M_{ξ,η}=P_ξ+η(I−P_ξ) and let η↓0, with uniformity by compactness of S^{n-1}. The stress-test is right: the decoupling error generally depends on the condition number of M, which blows up as η→0 near the equator. The interchanging of η↓0 and k→∞ is not backed by an estimate. If (4.6) fails, the chain (4.7) and the exclusion of multi-bubble splitting collapse. A repair is plausible via pointwise decoupling plus a uniform Lipschitz bound |A_ξ(v)−A_η(v)| ≤ p∥∇v∥_p^p |ξ−η|, but it is not in the paper.\n\nThe citation pattern is honest, the external benchmarks are standard, and the paper does not overclaim. It deserves a serious referee; an editor should send it out rather than desk reject. I would not cite it until the gaps are patched, but I would bring it to the reading group.","headline":"Sharp affine p-Sobolev stability, likely true and locally solid, but Lemma 4.1's indexing and the uniform decoupling (4.6) need repair before the global step is airtight.","tokens_in":28692,"tokens_out":2671,"would_cite":false,"duration_ms":26125,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35","26D10","35A23"],"pacs":[],"model":"deepseek-v4-flash","headline":"For 2 ≤ p < n, the affine p-Sobolev inequality is quantitatively stable: the normalized deficit controls the distance to the affine extremal manifold to the power p, and this exponent cannot be improved.","keywords":["affine p-Sobolev inequality","quantitative stability","sharp stability exponent","affine extremal manifold","SL(n) invariance","affine Hessian second variation","profile decomposition","directional energy decoupling"],"falsifier":"Take two copies of the bubble U placed at distance R apart, u_R(x) = U(x - R e_1) + U(x + R e_1), and compute the directional energies A_ξ(u_R) for ξ nearly perpendicular to e_1 as R grows. If the coupling between the two bubbles in such directions does not vanish uniformly in ξ, the key decoupling step (4.6) fails and the global argument would need a different proof; if it does vanish uniformly, one can test the exponent-p inequality directly on these sequences.","tokens_in":27445,"feed_emoji":"🎯","tokens_out":8701,"duration_ms":83343,"temperature":0.7,"pith_summary":"Quantitative stability for inequalities asks how close a near-optimizer must be to the set of exact optimizers. This paper answers that question for the affine L^p-Sobolev inequality, a sharp inequality that refines the classical Sobolev inequality and is invariant under all volume-preserving linear changes of coordinates. The authors prove that for 2 ≤ p < n, if a function u has affine energy close to the optimal value, then u is close, in a gradient L^p sense and after an optimal affine renormalization, to some affine image of the standard bubble — with distance raised to the p-th power controlling the deficit. They also show that p is the smallest exponent that can work. The proof identifies the second variation of the affine energy as the classical Sobolev Hessian minus a variance term, and shows that this term enlarges the zero-modes precisely to the tangent space of the affine extremal manifold.","feed_headline":"Affine Sobolev stability proven sharp with exponent p","feed_subtitle":"Any near-extremizer must lie within distance^p of an affine bubble, and no weaker bound can hold.","key_machinery":"The engine is the second variation of the affine energy E at the normalized bubble U. The affine energy is a negative mean over directions ξ of directional L^p energies, so its Hessian splits as the classical Sobolev Hessian minus a variance term R_p that measures how much the directional linear responses L_ξ(ϕ) fluctuate over the sphere. A spherical harmonic decomposition and a standard angle-averaging identity show that R_p affects only the even angular sectors; in the degree-two sector it cancels the positive classical contribution exactly, creating the new zero directions x·B∇U for trace-free symmetric matrices B. The kernel of the affine Hessian is thereby identified as the tangent spac","core_discovery":"The central claim is Theorem 1.1: for 2 ≤ p < n there exists a constant c_{n,p} > 0 such that every nonzero u ∈ W^{1,p}(R^n) satisfies E(u)/(S_{n,p}‖u‖_{L^{p*}}) − 1 ≥ c_{n,p} · [inf_{a∈R, A∈SL(n), λ>0, x₀∈R^n} ‖∇(T_{λA,x₀}u − aU)‖_{L^p}/‖∇(T_{λA,x₀}u)‖_{L^p}]^p. The infimum measures, after optimal affine normalization, the gradient distance from u to the manifold M_aff of affine images of the bubble U. Thus near-extremizers of the affine Sobolev inequality are quantitatively close to M_aff at order p, and the exponent p is optimal: no α < p can replace it. The proof also establishes that the kernel of the affine Hessian at U is exactly the tangent space of M_aff, which is strictly larger th","pith_inferences":["If the uniform directional decoupling in (4.6) is made fully rigorous — the paper only sketches it — the same global argument would likely extend the stability theorem to endpoint cases or to related affine-invariant inequalities where the negative-mean structure appears.","The explicit ratio identity for the angle-averaging coefficients suggests a closed formula for the constants in the spectral gap; computing c_{n,p} explicitly would make the stability bound quantitative rather than existential.","The restriction p ≥ 2 is used in the convexity inequality and the sign of the variance correction; a natural test is whether the same stability statement, with perhaps a different exponent, holds for 1 < p < 2.","The construction testing optimality with distant translated bumps indicates that the sharp exponent is driven by 'splitting' a bubble off into the far field; a similar phenomenon might occur for fractional affine Sobolev stability, where the Hilbertian case p = 2 is already understood."],"forward_implications":["Near-extremizers with vanishing affine deficit converge, after affine renormalization and scaling, to an affine image of the bubble U.","The distance-to-extremals deficit controls the gradient distance to M_aff with power p, and no smaller power works.","The zero-directions of the affine second variation are exactly the infinitesimal affine reparametrizations of the bubble — the full tangent space of M_aff.","The local spectral-gap estimate is the quantitative content behind the stability inequality, so the proof reduces the global problem to a compactness statement."],"fun_headline_variants":["Affine Sobolev stability: exponent p is sharp","Optimal p-th power distance for affine Sobolev extremizers","Sharp affine Sobolev: near-extremizers are p-close to bubbles","Stability exponent p in affine Sobolev inequality is best"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on the claim that when a near-extremizing sequence is split into separate bumps, the total energy in every direction is essentially the sum of the energies of the bumps, with an error uniformly small over all directions; this uniformity is asserted by a compactness argument but not proved in detail near the most degenerate directions.","fun_headline_variants_meta":{"raw":{"variants":["Affine Sobolev stability: exponent p is sharp","Optimal p-th power distance for affine Sobolev extremizers","Sharp affine Sobolev: near-extremizers are p-close to bubbles","Stability exponent p in affine Sobolev inequality is best"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000721,"raw_usage":{"total_tokens":3041,"prompt_tokens":681,"completion_tokens":2360,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":425,"completion_tokens_details":{"reasoning_tokens":2297}},"tokens_in":425,"tokens_out":2360,"duration_ms":15122,"temperature":1.0,"reasoning_tokens":2297,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T12:02:15.490206+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two copies of the bubble U placed at distance R apart, u_R(x) = U(x - R e_1) + U(x + R e_1), and compute the directional energies A_ξ(u_R) for ξ nearly perpendicular to e_1 as R grows. If the coupling between the two bubbles in such directions does not vanish uniformly in ξ, the key decoupling step (4.6) fails and the global argument would need a different proof; if it does vanish uniformly, one can test the exponent-p inequality directly on these sequences.","supporting_citations":[],"review_version":2}