{"id":"bd52fdc4-2da0-4fe8-aa1a-b9c73e45d82e","arxiv_id":"2505.14431","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A Carleman estimate with a weight depending only on y shows that anisotropic exponential/polynomial decay forces u=0 for elliptic inequalities in exterior domains.","lead":"This paper proves that solutions of elliptic differential inequalities that decay extremely fast in some coordinate directions and only slowly in the others must be identically zero. It generalizes a classical uniqueness result of Meshkov and introduces a Carleman estimate with a weight depending on a subset of coordinates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Repairable gap: (2.14) controls |y|^{σ1}q using Lemma 2.4, which only bounds |q|; enlarging σ2 to 2σ1 keeps the theorem intact.","rationale":"I read the paper as a self-contained Carleman-based proof of an anisotropic Landis-type uniqueness theorem. The main construction is sound: the Carleman estimate in Section 5 is derived carefully, the cut-off geometry in Section 2 is coherent, and the final limit s→∞ correctly forces u = 0 on {|y|≥7}×R^{n−m}. The unique-continuation step flagged by the reader is less problematic than it appears: the singular hyperplane y = 0 never needs to be crossed, since local UCP in annuli {ε ≤ |y| ≤ K} with bounded coefficients propagates the zero set inward from |y| ≥ 7 to |y| > ε for every ε > 0, and then a measure-zero argument gives u = 0 everywhere in the exterior component. The genuine issue I found is a mismatch between Lemma 2.4 and its use in (2.14): Lemma 2.4 bounds |Δq| + |q|, while Lemma 2.2 and the application require |Δq| + |y|^{σ1}|q|. For choices of γ, θ with σ1 > 4α the displayed inequality in (2.14) is not justified. The gap is easily repaired by enlarging σ2, and the additional polynomial factors do not affect the convergence arguments because β > α and g satisfies (1.6). Thus the central mathematical claim is very likely correct, but the proof as submitted needs a local correction. Accordingly I recommend conditional acceptance rather than unconditional acceptance.","tokens_in":20812,"tokens_out":38939,"duration_ms":381026,"concrete_test":"Independently re-derive (2.14) without replacing |y|^{σ1}q_R^1 by |q_R^1|. Set σ2 := max{σ1 + 4α, 2σ1} in (2.15) and recompute S1, S2, S3. In particular, verify that ∫_{R−2≤|y|≤R+1} s^2 |y|^{2σ1} e^{−2C1|y|^{2β}+2s|y|^{2α}} dy tends to 0 as R→∞ for every admissible γ, θ, and that ∫_{|y|≥3} s^2 |y|^{2σ1} e^{−2C1|y|^{2β}+2s|y|^{2α}} dy is finite. If both remain true for β>α, the central conclusion of Theorem 1.1 survives the correction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most concrete soft spot is in the estimation of S1(S2) around (2.14). In the notation of Section 2, q_R^1 = χ_{R-1,R}(|y|) µ_R(|z|) |y|^{σ1} e^{2s|y|^{2α}}. Lemma 2.2 gives ∫ q_R^1 |∇u|^2 ≤ C∫ (|Δ q_R^1| + |y|^{σ1} q_R^1) u^2. On D1(R), q_R^1 = |y|^{σ1} e^{2s|y|^{2α}}, so the second integrand is |y|^{2σ1} e^{2s|y|^{2α}}. However Lemma 2.4, as stated and proved, gives only |Δq| + |q| ≤ C s^2 |y|^{σ1+4α} e^{2s|y|^{2α}}, not a bound for |y|^{σ1}|q|. When σ1 = max{2|θ|, 2|γ|, 4α} exceeds 4α — possible, for example, when γ < −4 or θ is sufficiently large — the displayed inequality in (2.14) is false as written because 2σ1 > σ1 + 4α. This is not a fatal flaw in the central claim. Replacing σ2 := σ1 + 4α by σ2 := max{σ1 + 4α, 2σ1} (or equivalently adding the extra polynomial factor |y|^{σ1−4α}) preserves the S1–S3 limits: the exponential decay e^{−2C1 R^{2β}} beats any polynomial in R since β > α, and the y-integrals in S2 remain finite because β > α. Thus the main theorem can be repaired, but the submitted proof contains an incorrect estimate that should be corrected.","agreement_with_reader":"partial"},"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nRead this paper. It proves a new uniqueness theorem for the elliptic inequality |Δu| ≤ C0(|y|^{-γ}|u| + |y|^{-θ}|∇u|) in an exterior domain, under exponential decay in m of the variables and polynomial decay in the rest. The anisotropic decay condition (1.7) and the Carleman weight |y|^{2α} are new as far as I can tell; the isotropic case m=n recovers known results but the partial-coordinate version is the contribution. The Carleman estimate in Section 5 is fully proven, self-contained, and the absorption conditions defining α come from the inequality itself, so there's no circularity.\n\nThe main theorem is likely correct, but the written proof contains a concrete error in the estimation of S1 (and similarly S2, S3). Around (2.14), the authors bound |y|^{σ1}q_R^1 by C s^2 |y|^{σ2} with σ2 = σ1 + 4α. Lemma 2.4 gives |Δq| + |q| ≤ C s^2 |y|^{σ1+4α} e^{2sφ}, but the second term in Lemma 2.2 is |y|^{σ1}q, which is |y|^{2σ1} e^{2sφ}. When σ1 > 4α (which happens, e.g., for γ < -4 or large |θ|), 2σ1 > σ1+4α and the displayed inequality is false as written. This is repairable: taking σ2 = max{σ1+4α, 2σ1} restores the estimate, and the limits as R→∞ still vanish because β > α dominates any polynomial. So the theorem survives, but the submitted proof needs a correction.\n\nOther notes: Theorem 1.4 is sketched in a few lines, relying entirely on the same argument; the unique continuation step from the unbounded strip to all of R^n\\U is imported from Choulli [2] without verifying its hypotheses here. These are legitimate soft spots but not fatal.\n\nThe paper deserves a serious referee. It is a genuine, checkable contribution to the Landis and unique continuation literature. I'd recommend engaging with it, and asking the authors to fix the σ2 definition and expand the proof of Theorem 1.4.\n\nBest","headline":"A new anisotropic decay uniqueness theorem with a genuinely proven Carleman estimate, but a repairable polynomial-weight error in the proof that the referee should catch.","tokens_in":21749,"tokens_out":6248,"would_cite":true,"duration_ms":52717,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-07T15:35:29.480937+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}