{"id":"54b79f79-82c6-41be-9279-30e464b3517b","arxiv_id":"1908.00525","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper extends nonlocal ABP, Harnack and Hölder regularity to arbitrary anisotropic exponents b_i and 0<s<4/bmax, but the written proof has a gap in the barrier construction.","lead":"This paper claims interior Hölder and C^{1,\\gamma} regularity for solutions of a class of anisotropic nonlocal equations, extending earlier work to more general directional exponents. The proof has a load-bearing gap: a key barrier estimate relies on treating the anisotropic norm as rotation invariant, which is not valid.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.9 assumes the anisotropic norm is rotation invariant; the barrier behind the Harnack chain and Theorem 4.19 is not proved as written.","rationale":"The reader's weakest assumption is exactly the step I find load-bearing: the only proof of an anisotropic barrier for M^- rotates the coordinate system and then treats the anisotropic norm as Euclidean-rotation invariant. Since the kernel is defined by the anisotropic norm, this is not a cosmetic issue; the later lemmas in Section 4.3 rely on that barrier at every level, from Psi in Lemma 4.12 to Lemma 4.13, the Harnack inequality, and Theorem 4.19. I also checked the smooth part: Lemma 2.3 is simply asserted, and Lemma 3.1 cites it for the key contradiction, so the C^gamma theorem for smooth solutions also has an unproved ingredient. I do not claim the theorems are false; the approach is plausible and the direct expansion of delta(f,x,y) might supply a rotation-free proof. But as written, the proof has a false identity in a central lemma and an omitted proof in the smooth case. Since there is no machine-checked verification or reproducible code providing independent support, I concur with REJECT.","tokens_in":26803,"tokens_out":13132,"duration_ms":138429,"concrete_test":"Check the claimed identity (4.15)-(4.17) numerically for n=2, b=(2,1), s=1, x=(1/sqrt(2),1/sqrt(2)), and T the rotation with T e_1=x, comparing integral_{B_{1/4}} y_1^2/||T y||^4 dy with integral_{B_{1/4}} y_1^2/||y||^4 dy. If the values are unequal, the rotation step in Lemma 4.9 is false. Then test whether Lemma 4.9 can be repaired by replacing the rotation with the direct lower bound min_{|e|=1} integral_{B_{1/4}} (e dot y)^2/||y||^4 dy > 0 and a suitable choice of p; if that bound holds and the negative terms are controlled, the barrier may be salvageable, otherwise the Harnack chain lacks a valid barrier.","verdict_should_be":"REJECT","load_bearing_attack":"Section 4.2, Lemma 4.9: to prove that f(x)=min(2^p,|x|^{-p}) is a barrier for M^- on 1<=|x|<=R, the proof fixes x and chooses a Euclidean rotation T_x with x=|x|T_x e_1. After the change of variables, equations (4.15)-(4.17) replace the denominator sum_i |(|x|T_x y)_i|^{b_i} by ||y||^{c+s} and then replace T_x^{-1} with the Euclidean scalar product with x. This is exactly the assertion that ||T_x y||=||y|| for every rotation, which is true only when b_1=...=b_n. For example, for n=2, b=(2,1) and a 45-degree rotation, ||y||=y_1^2+|y_2| while ||T y||=((y_1-y_2)^2/2)+|(y_1+y_2)/sqrt(2)|, so the two integrands differ. The lower bound for I_1 and the conclusion M^-f>=0 therefore do not follow. The barrier is used in Corollaries 4.10-4.11 and Lemma 4.12 to construct Psi, which is the input to Lemma 4.13; the measure estimate, Theorem 4.18 Harnack, and the central Theorem 4.19 all depend on it. Thus the viscosity C^gamma theorem is not established by this manuscript. A separate gap is Lemma 2.3: the Silvestre inequality is stated without proof and is used in Lemma 3.1, so the smooth C^gamma theorem also lacks a complete argument. These are proof gaps, not evidence that the results are false; a direct estimate using (x dot y)^2/||y||^{c+s} or a barrier adapted to the anisotropic norm might repair them.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies Hölder regularity for integro-differential equations with anisotropic kernels K(y) comparable to ‖y‖^{-(c+s)}, where ‖y‖^2 = Σ_{i=1}^n |y_i|^{b_i}, under the restriction 0 < s < 4/b_max. The authors claim a C^γ estimate for smooth solutions of the anisotropic fractional Laplacian (Theorem 3.3) and for viscosity solutions of fully nonlinear integro-differential equations (Theorem 4.19). The smooth case is approached through Silvestre's De Giorgi technique, relying on an anisotropic Silvestre inequality (Lemma 2.3). The viscosity case follows Caffarelli–Leitão–Urbano: a nonlocal ABP estimate (Theorem 4.7), a barrier construction (Lemma 4.9), a measure estimate (Lemma 4.13), and a Harnack inequality (Theorem 4.18). The paper also states C^{1,γ} estimates under additional kernel assumptions.","tokens_in":27210,"tokens_out":20386,"duration_ms":180309,"significance":"If the proofs were correct, the results would provide a meaningful extension of the regularity theory for anisotropic nonlocal operators to kernels with different directional homogeneities, with explicit dependence on the lower bound s0. The geometric framework—anisotropic ellipses, the Caffarelli–Calderón covering lemma, and the nonlocal ABP structure—is well chosen and the theorem statements are precise. However, the central claims are not currently supported: Lemma 2.3 is unproved, and the proof of Lemma 4.9 contains an invalid change of variables that is load-bearing for the viscosity regularity theorem. These are not cosmetic issues.","major_comments":[{"comment":"The proof of Lemma 4.9 is not sound as written. In equations (4.15)–(4.17), the change of variables treats the anisotropic norm ‖y‖ (defined by ‖y‖^2 = Σ_{i=1}^n |y_i|^{b_i}) as if it were invariant under Euclidean rotations and homogeneous under scalar multiplication. Specifically, the denominator (Σ_{i=1}^n ||x|(T_x y)_i|^{b_i})^{(c+s)/2} is converted into |x|^{-n}‖y‖^{c+s} with no additional factor, which is valid only when all b_i coincide. Moreover, the replacement of ⟨T_x^{-1} y, |x|^{-1} e_1⟩^2 by ⟨y, x⟩^2 omits a factor |x|^{-4}, and the integration domain is changed from B_{1/4} to another ball without justification. Consequently, the lower bound for the term I_1 and the conclusion M^- f(x) ≥ 0 for 1 ≤ |x| ≤ R do not follow. Since this barrier is used in Corollaries 4.10–4.11 and Lemma 4.12 to construct Ψ, which is the input to Lemma 4.13, the measure estimate, the Harnack inequality (Theorem 4.18), and the central Theorem 4.19 all rest on this lemma. The viscosity regularity theorem is therefore not established by the manuscript.","section":"§4.2, Lemma 4.9"},{"comment":"Lemma 2.3 (the anisotropic Silvestre inequality) is stated without proof. It is a key ingredient in the proof of Lemma 3.1 (Growth lemma), which in turn is used in Lemma 3.2 and Theorem 3.3. The introduction indicates that the inequality should follow from the anisotropic scaling T_{β,r} and radial barriers, but no derivation appears in Section 2.1. As a consequence, the smooth C^γ regularity theorem (Theorem 3.3) also lacks a complete proof.","section":"§2.1, Lemma 2.3"}],"minor_comments":[{"comment":"The notation for the norm is confusing: the paper writes ‖y‖^2 = Σ_{i=1}^n |y_i|^{b_i} but then uses ‖y‖^{c+s} in the kernel. This should be clarified by explicitly defining ‖y‖ = (Σ|y_i|^{b_i})^{1/2} and consistently using that throughout.","section":"Introduction and Section 2"},{"comment":"In the proof of Lemma 3.1, the last displayed inequality does not match the form of (2.7): the term (3/4+|y|)^τ appears instead of a scaled variable such as |8y|^τ, suggesting a missing rescaling of the integration variable.","section":"§3, Lemma 3.1"},{"comment":"The statement 'without loss of generality, we can assume that x ∈ {y : x_i ≥ 0} and x_1 ≥ 1/n' after equation (4.17) is not justified, since a Euclidean rotation does not preserve the anisotropic kernel and an arbitrary vector cannot be rotated into the positive orthant.","section":"§4.2, Lemma 4.9"},{"comment":"There are numerous typographical errors, including 'Leito' in the abstract, inconsistent use of 'qmax,s' versus 'q_{max,s}', and undefined centers in the balls in Lemma 4.2. These should be corrected in any revision.","section":"General"}],"recommendation":"reject","confidential_remarks":"The two major gaps are substantial: the anisotropic Silvestre inequality is not proved, and the barrier computation in Lemma 4.9 contains a false change of variables. Together they invalidate the proofs of both main theorems. If the authors can supply a correct proof of Lemma 2.3 and repair the barrier argument using careful estimates based on the equivalence of anisotropic norms rather than rotation invariance, a resubmission could be considered. As it stands, however, the manuscript does not establish its advertised results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper genuinely extends the anisotropic nonlocal regularity theory from the special case in Caffarelli-Leitão-Urbano to arbitrary b_i > 0 and 0 < s < 4/bmax, and it adds a smooth-solution De Giorgi argument. But as written it does not prove its advertised theorems: the smooth theorem relies on Lemma 2.3, a Silvestre inequality that is stated without proof, and the viscosity theorem relies on Lemma 4.9, whose barrier computation treats the anisotropic norm as rotation invariant and Euclidean homogeneous. Both are load-bearing. The central claims may well be true, but the manuscript needs repair.\n\nThe extension is real. The paper says clearly that [8] covered b_i = n + sigma_i, s = 2 - c, and it pushes the method to the full range. The geometric setup with ellipses E_{r,l}, the anisotropic scaling T_{\\beta,r}, and the ABP/covering structure are recognizable and mostly clean. I found no sign of circularity: it leans on [8], coauthored by one of the present authors, but for a special case, not for the conclusion.\n\nSoft spots, in order of seriousness. Lemma 2.3 is a missing proof, not a wrong statement; a referee can ask for the argument. Lemma 4.9 is worse. To show M^-f >= 0 for 1 <= |x| <= R, the proof chooses a Euclidean rotation with x = |x|T_x e_1 and then replaces || |x|T_x y || by ||y||. That requires ||·|| to be invariant under rotations and homogeneous under Euclidean scaling, which fails unless all b_i coincide. The change of variables also seems to mix c and n in the determinant. Since this barrier drives the construction of Psi, Lemma 4.13, and the Harnack inequality, Theorem 4.19 is not established by this text. These are proof gaps rather than evidence of falsity; a barrier built with the anisotropic scaling T_{\\beta,r} might repair the argument.\n\nThe paper is honest: it flags the s restriction, mentions possible extension via a metric, and cites competing work. If the gaps are fixable, the result is a useful unification within the regularity-theory subfield. As is, I would not cite it or rely on it. I would send it to a serious referee, with a request to focus on Lemma 4.9 and the missing proof of Lemma 2.3. A major-revision verdict, not a terminal reject, seems right if the authors can supply the missing pieces.","headline":"A genuine but unproved extension: the smooth and viscosity C^gamma theorems both rest on gaps, the worse being a barrier lemma that treats the anisotropic norm as rotation invariant.","tokens_in":27740,"tokens_out":4532,"would_cite":false,"duration_ms":45537,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:50:55.475420+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}