{"id":"3c46d699-260c-49ce-8187-453757db445f","arxiv_id":"2107.04364","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves equivalence of regularity for fractional Laplacian and extended equation, giving Besov capacity Wiener criterion plus decay estimates and Kellogg property for general domains.","lead":"The paper shows that boundary point regularity for the fractional Laplace equation is equivalent to regularity for an extended weighted equation via the Caffarelli-Silvestre extension. This yields a Wiener criterion using Besov capacity to identify regular points for nonlocal equations.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly flags the extension step as the point that must be verified, but the paper explicitly claims the result for general open sets. Absent a concrete gap in the logic or an unstated hypothesis that would invalidate the equivalence, the argument does not exhibit a load-bearing vulnerability. The UNVERDICTED status with low confidence therefore requires no adjustment.","tokens_in":1594,"tokens_out":285,"duration_ms":27160,"concrete_test":"Confirm that the proof of the main equivalence (likely the theorem stated after the abstract) constructs the extension and compares regularity notions using only that Ω is open; if the steps invoke no extra boundary regularity and the correspondence holds formally, the claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern is identified in the central claim. The paper asserts that the Caffarelli-Silvestre extension yields an if-and-only-if between regularity for (−Δ)^s u = 0 at x0 ∈ ∂Ω (Ω arbitrary open) and regularity for the extended weighted problem at (x0,0). The abstract states the result holds for general open sets and derives the Besov-capacity Wiener criterion as a consequence. No internal inconsistency, hidden regularity assumption on ∂Ω, or failure of the extension correspondence is visible from the given material.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper uses the Caffarelli-Silvestre extension to prove that, for an arbitrary open set Ω ⊂ R^n, a boundary point x0 is regular for the fractional equation (-Δ)^s u = 0 (0 < s < 1) if and only if the point (x0, 0) is regular for the corresponding extended weighted equation in R^{n+1}. As a direct consequence it obtains a Wiener criterion phrased in terms of Besov capacity, together with a decay estimate near regular points and the Kellogg property.","tokens_in":1685,"tokens_out":303,"duration_ms":11204,"significance":"The equivalence transfers classical local regularity theory to the nonlocal setting without additional restrictions on Ω, and the resulting Besov-capacity Wiener criterion supplies a concrete, testable characterization that is likely to be adopted in subsequent work on boundary behavior for fractional operators.","major_comments":[],"minor_comments":[{"comment":"Abstract: the phrase 'in a subset of R^{n+1}' is imprecise; the precise domain (upper half-space or the extension cylinder) should be stated explicitly.","section":"Abstract"},{"comment":"The statement that the result holds for 'general open sets' would benefit from a single sentence clarifying that no regularity or thickness assumption on ∂Ω is imposed beyond openness.","section":"Abstract"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The report does not list any specific major comments.","responses":[],"tokens_in":1092,"tokens_out":47,"duration_ms":18587,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper's main result is that for a general open set Ω in R^n, a boundary point x0 is regular for (−Δ)^s u = 0 if and only if (x0,0) is regular for the extended weighted equation. It then uses the equivalence to characterize regular points via a Wiener criterion with Besov capacity, and adds a decay estimate near regular points plus the Kellogg property. The equivalence for arbitrary domains and the resulting capacity criterion are the new pieces. The work applies the standard Caffarelli-Silvestre extension directly, which keeps the argument straightforward and avoids extra boundary assumptions. The abstract states the theorems clearly and positions them as consequences of the extension. The approach is a strength because it leverages an established tool without introducing new fitting parameters or circular steps. The soft spot is that the technical transfer of regularity through the extension to irregular domains needs the full proofs to verify; the abstract alone does not show gaps, but the details matter. No internal inconsistency appears in the stated claims. This is for people working on boundary regularity and potential theory for nonlocal operators. A reader focused on Wiener criteria or capacity conditions for fractional equations will find the equivalence and the Besov-capacity version useful. It deserves peer review because the result is focused, builds on solid prior machinery, and fills a specific gap in the literature for general domains.","headline":"The paper shows regularity for the fractional Laplacian at a boundary point is equivalent to regularity for the extended weighted problem, and derives a Besov-capacity Wiener criterion from that.","tokens_in":2129,"tokens_out":348,"would_cite":false,"duration_ms":14054,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Classical PDE boundary regularity for (−Δ)^s via Caffarelli–Silvestre extension; no RS overlap","alignment":"orthogonal","rationale":"Paper proves equivalence of regularity for fractional Laplacian at x0 ∈ ∂Ω with regularity for weighted equation div(|t|^{1−2s}∇U)=0 at (x0,0), yielding Besov-capacity Wiener criterion (Thm 1.1–1.2, Lem 1.3). RS chain (reality_from_one_distinction, AlexanderDuality.circle_linking_forces_D3, Cost.Jcost, etc.) forces J-cost, φ, 8-tick, D=3 and constants from bare distinction; contains no fractional operators, Besov capacities or Wiener criteria. Domain mismatch is total.","tokens_in":50237,"confidence":"high","tokens_out":180,"duration_ms":5072,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A boundary point is regular for the fractional Laplacian equation exactly when the lifted point is regular for the extended weighted equation.","keywords":["fractional Laplacian","boundary regularity","Wiener criterion","Besov capacity","Caffarelli-Silvestre extension","Kellogg property","weighted equations"],"falsifier":"An explicit open set and boundary point where regularity holds for one of the two equations but fails for the other would disprove the claimed equivalence.","tokens_in":2488,"feed_emoji":"","tokens_out":595,"duration_ms":16049,"temperature":0.7,"pith_summary":"The paper proves that for any open set in R^n, regularity of a boundary point x0 for the fractional equation (-Δ)^s u = 0 is equivalent to regularity of the point (x0, 0) for the extended weighted equation in one higher dimension. This equivalence is obtained by applying the Caffarelli-Silvestre extension to the fractional operator. The result immediately produces a Wiener criterion that identifies regular points through the Besov capacity of the complement. It further yields decay estimates for solutions near regular points and establishes the Kellogg property.","feed_headline":"Fractional Laplacian regularity equals weighted extension regularity","feed_subtitle":"Equivalence via the extension produces a Wiener criterion with Besov capacity for arbitrary domains.","key_machinery":"The Caffarelli-Silvestre extension, which converts the nonlocal fractional Laplacian into a local weighted divergence-form equation in one higher dimension.","core_discovery":"Using the Caffarelli-Silvestre extension, a boundary point x0 is regular for (−Δ)^s u = 0 in an open set Ω ⊂ R^n if and only if (x0, 0) is regular for the corresponding weighted equation in a subset of R^{n+1}. This yields a Wiener criterion for regularity in terms of Besov capacity, along with boundary decay estimates and the Kellogg property.","pith_inferences":["The equivalence opens the possibility of importing other classical results on weighted equations to the fractional setting.","Numerical schemes for fractional problems could be built by solving the local extended equation instead.","Because the result requires no extra assumptions on the domain, it may apply to irregular geometries where direct fractional analysis is difficult."],"forward_implications":["Regular boundary points admit a Wiener-type integral test involving the Besov capacity of the complement.","Solutions exhibit a specific decay rate near regular boundary points.","The Kellogg property holds: almost every boundary point with respect to the capacity is regular."],"fun_headline_variants":["Regularity equivalence for fractional Laplacian and weighted extension","Wiener criterion for fractional Laplacian via Besov capacity","Boundary regularity tied to Caffarelli-Silvestre extension","Fractional Laplacian boundary points characterized by Besov capacity"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Caffarelli-Silvestre extension can be applied directly to relate the fractional equation on arbitrary open sets to the weighted problem without requiring extra boundary regularity.","fun_headline_variants_meta":{"raw":{"variants":["Regularity equivalence for fractional Laplacian and weighted extension","Wiener criterion for fractional Laplacian via Besov capacity","Boundary regularity tied to Caffarelli-Silvestre extension","Fractional Laplacian boundary points characterized by Besov capacity"]},"model":"grok-4.3","cost_usd":0.00403,"raw_usage":{"total_tokens":1995,"prompt_tokens":551,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":40299500,"prompt_tokens_details":{"text_tokens":551,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1383,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":551,"tokens_out":61,"duration_ms":7201,"temperature":1.0,"reasoning_tokens":1383,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T13:41:43.792593+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit open set and boundary point where regularity holds for one of the two equations but fails for the other would disprove the claimed equivalence.","supporting_citations":[],"review_version":1}