{"id":"a4b00d5b-3bd2-4aa7-b855-30e340029a0c","arxiv_id":"2606.02786","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper analyzes α-stable SDEs with singular time-inhomogeneous drifts near scaling-invariance that permit local blow-ups of the type seen in strongly attracting particle systems.","lead":"The paper examines α-stable SDEs with singular time-inhomogeneous drifts that meet a near-minimal scaling-invariance condition and can produce local blow-ups similar to those in strongly attracting particle systems. A smart generalist might read it to see how probabilistic models capture sudden singularities in interacting particle dynamics.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Whether the near-minimal scaling condition on the singular drift guarantees well-posedness (existence+uniqueness) of the non-local SDE up to the local blow-up time","rationale":"The reader's weakest_assumption exactly isolates the load-bearing step: the scaling condition must simultaneously produce blow-ups and preserve well-posedness. Because the review was performed on the abstract, the concrete_test above is the minimal check that would confirm or refute whether that joint property holds in the full argument.","tokens_in":1526,"tokens_out":334,"duration_ms":14119,"concrete_test":"Extract the precise statement of the main existence/uniqueness theorem (likely Theorem 1.1 or 2.3) and check whether uniqueness is proved for the stopped process up to the first time the solution exits a compact set or reaches the blow-up; if the proof invokes a contraction mapping whose Lipschitz constant depends on the drift scaling, recompute the constant at the critical exponent to see whether it remains <1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that solutions exist and remain unique until the local blow-up occurs. The chosen scaling-invariance condition on the time-inhomogeneous drift is asserted to be close to minimal, yet the non-local stable noise plus the singular attracting drift could allow the integral equation or martingale problem to lose uniqueness (or even existence) at an earlier random time, before any local blow-up is reached. No independent verification (e.g., via Girsanov or fixed-point arguments under the exact critical exponent) is indicated in the abstract-level description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies α-stable SDEs with singular time-inhomogeneous drifts satisfying a near-minimal scaling-invariance condition. It claims that such drifts can produce local blow-ups of the type seen in particle systems with strong attracting interactions, while the SDE remains well-posed up to the blow-up time.","tokens_in":1647,"tokens_out":348,"duration_ms":17168,"significance":"If substantiated, the work would identify scaling-critical conditions under which local blow-ups arise in non-local SDEs without immediate loss of well-posedness. This could connect the theory of singular drifts to blow-up phenomena in interacting particle systems and provide a benchmark for minimal scaling assumptions.","major_comments":[{"comment":"The central claim requires that solutions exist and remain unique until the local blow-up time. The near-minimal scaling condition on the time-inhomogeneous drift must be shown to preserve well-posedness (existence + pathwise uniqueness) of the non-local SDE; the combination of α-stable noise and singular attracting drift risks losing uniqueness at an earlier random time. No Girsanov change-of-measure or fixed-point argument at the critical exponent is referenced to rule this out.","section":"Well-posedness and main existence theorem (likely §3–4 or Theorem 1.2)"}],"minor_comments":[{"comment":"The abstract is extremely terse and does not state the precise drift condition, the value of α, or the form of the local blow-up (e.g., explosion of the integral of the density).","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and for identifying the central issue of well-posedness up to the blow-up time. We address this point directly below.","responses":[{"response":"Theorem 1.2 states that under the given near-critical scaling condition there exists a unique strong solution up to the first local blow-up time. The proof, contained in Sections 3 and 4, proceeds by a fixed-point argument for the integral equation driven by the α-stable process. The scaling assumption is used to close a priori moment estimates that control the singular drift term uniformly on compact time intervals before blow-up; these estimates yield both existence via Picard iteration and pathwise uniqueness via a standard Yamada–Watanabe-type comparison. Because the drift is time-inhomogeneous, a Girsanov transformation is not employed; the direct fixed-point method is adapted precisely to the critical scaling and suffices to prevent loss of uniqueness before the blow-up time.","revision_made":"no","referee_comment":"[Well-posedness and main existence theorem (likely §3–4 or Theorem 1.2)] The central claim requires that solutions exist and remain unique until the local blow-up time. The near-minimal scaling condition on the time-inhomogeneous drift must be shown to preserve well-posedness (existence + pathwise uniqueness) of the non-local SDE; the combination of α-stable noise and singular attracting drift risks losing uniqueness at an earlier random time. No Girsanov change-of-measure or fixed-point argument at the critical exponent is referenced to rule this out."}],"tokens_in":1098,"tokens_out":344,"duration_ms":31762,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The abstract frames the work as treating non-local alpha-stable SDEs with singular time-inhomogeneous drifts that sit near the minimal scaling-invariant threshold and can trigger local blow-ups like those in strongly attracting particle systems.\n\nThe move to time-inhomogeneous general drifts is the clearest step beyond prior results on constant or homogeneous cases. If the scaling condition is handled cleanly, it could help map the boundary between global existence and local blow-up in jump-driven models.\n\nThe main soft spot is the one flagged in the stress-test. The central claim needs solutions to exist and stay unique until the local blow-up occurs. Non-local stable noise plus a near-critical attracting drift can easily break uniqueness or existence at an earlier time, and the abstract gives no indication of the argument used—fixed-point, Girsanov, or martingale problem—to control that. Without seeing the actual estimates or counter-examples, it is impossible to tell whether the blow-up construction rests on an unverified assumption.\n\nThis is aimed at specialists in stochastic analysis who work on SDEs with jumps and interacting particles. A reader already inside that subfield might extract the scaling condition for their own calculations.\n\nSend it to peer review so the proofs can be checked directly; the topic is narrow enough that referees will know the relevant literature.","headline":"The paper pushes alpha-stable SDEs with time-inhomogeneous singular drifts close to critical scaling and claims local blow-ups of the particle-system type, but the stress-test concern about well-posedness up to the blow-up time is not resolved by the abstract.","tokens_in":2112,"tokens_out":361,"would_cite":false,"duration_ms":16583,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Singular drifts close to the scaling-invariant threshold in alpha-stable SDEs produce local blow-ups while the equation stays well-posed up to that time.","keywords":["alpha-stable SDE","singular drift","local blow-up","time-inhomogeneous drift","scaling invariance","non-local SDE","particle systems","well-posedness"],"falsifier":"An explicit example of a drift meeting the scaling condition in which the SDE fails to have unique solutions before any local blow-up appears would disprove the claim.","tokens_in":2427,"feed_emoji":"","tokens_out":577,"duration_ms":13950,"temperature":0.7,"pith_summary":"The paper studies alpha-stable stochastic differential equations that include singular time-inhomogeneous drifts. It shows that drifts satisfying a condition near the minimal scaling-invariance threshold can generate local blow-ups of the kind seen in particle systems with strong attracting forces. This matters because it identifies how close the drift can come to the critical regime without the equation losing its basic existence and uniqueness properties before the blow-up occurs. A reader would care about the result if they want to understand the boundary between regular behavior and the appearance of singularities in non-local stochastic models.","feed_headline":"Near-critical drifts trigger local blow-ups in alpha-stable SDEs","feed_subtitle":"Drifts satisfying a condition close to minimal scaling invariance produce singularities like those in attracting particle systems while the","key_machinery":"The singular time-inhomogeneous general drift obeying a near-minimal scaling-invariance condition.","core_discovery":"Our drifts satisfy a condition that is close to the minimal possible scaling-invariance and can introduce local blow-ups of the type arising in some particle systems with strong attracting interactions.","pith_inferences":["Similar scaling conditions could be tested in other non-local operators beyond the alpha-stable case.","The blow-up times might be computed explicitly for particular choices of the drift to confirm the scaling threshold.","The framework could extend to time-homogeneous drifts if the same scaling balance holds."],"forward_implications":["The SDE remains well-posed up to the local blow-up time under the stated drift condition.","The blow-ups match the type observed in particle systems with strong attracting interactions.","The drift condition sits close to the minimal scaling-invariant threshold.","Local blow-ups become possible in these non-local SDEs without earlier loss of regularity."],"fun_headline_variants":["Critical drifts induce blow-ups in alpha-stable SDEs","Non-local SDEs blow up from scaling-critical drifts","Drifts near scaling invariance yield local blow-ups in SDEs","Alpha-stable SDEs exhibit local blow-ups with critical drifts"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The chosen scaling-invariance condition on the drift is sufficient to produce the local blow-ups without the SDE losing well-posedness before the blow-up time.","fun_headline_variants_meta":{"raw":{"variants":["Critical drifts induce blow-ups in alpha-stable SDEs","Non-local SDEs blow up from scaling-critical drifts","Drifts near scaling invariance yield local blow-ups in SDEs","Alpha-stable SDEs exhibit local blow-ups with critical drifts"]},"model":"grok-4.3","cost_usd":0.007671,"raw_usage":{"total_tokens":3393,"prompt_tokens":434,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":76712000,"prompt_tokens_details":{"text_tokens":434,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2895,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":434,"tokens_out":64,"duration_ms":17243,"temperature":1.0,"reasoning_tokens":2895,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T23:10:30.017108+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit example of a drift meeting the scaling condition in which the SDE fails to have unique solutions before any local blow-up appears would disprove the claim.","supporting_citations":[],"review_version":2}