{"id":"b47ab3a4-0ce7-42c9-9d46-e34ed5ecec13","arxiv_id":"2410.23801","paper_version":4,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves solvability of the L^p regularity problem for parabolic operators under a parabolic Carleson condition on coefficients, for p in (1, p0) for some p0>1.","lead":"The paper proves that the regularity problem for parabolic PDEs with elliptic bounded measurable coefficients satisfying a Carleson condition is solvable in L^p for p in (1, p0) with some p0>1. A smart generalist might read it to understand progress on boundary value problems for time-dependent diffusion equations on irregular domains.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flagged the result as unverified due to abstract-only access. With the full text now in principle available, the argument structure matches the claim without an evident load-bearing flaw that can be isolated without further computation or counter-example construction. No adjustment to UNVERDICTED is warranted on the basis of an identifiable technical defect.","tokens_in":1680,"tokens_out":272,"duration_ms":26580,"concrete_test":"Reproduce the small-Carleson-norm case (the previously open subcase) by verifying that the key Carleson-measure estimate in the proof yields a uniform bound on the nontangential maximal function independent of the smallness parameter; if the constant blows up, the extension to the large-Carleson regime fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a proof that the L^p regularity problem is solvable in (1,p0) for some p0>1 under ellipticity, bounded measurable coefficients, and the parabolic Carleson condition. No internal inconsistency, hidden assumption, or gap in the stated hypotheses is visible from the abstract or the description of the approach. The elliptic precedent is cited as motivation with explicit acknowledgment of new parabolic difficulties, but the manuscript asserts these are overcome.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to fully resolve the L^p regularity problem for the parabolic PDE −∂_t u + div(A ∇ u) = 0 on a Lipschitz cylinder O × R. Under the assumptions that A is elliptic with bounded measurable coefficients satisfying a parabolic Carleson condition (the natural parabolic analog of the DKP condition), the authors prove that the regularity problem is solvable for all p ∈ (1, p_0) for some p_0 > 1. The result is asserted to hold even when the Carleson norm is small, a case previously open; the proof strategy is motivated by one of the recent elliptic resolutions but must overcome new parabolic difficulties.","tokens_in":1755,"tokens_out":378,"duration_ms":15589,"significance":"If the central proof is correct, the result would constitute a substantial advance in the theory of parabolic equations with rough coefficients. It completes the parabolic counterpart to the recent elliptic resolutions of the regularity problem (one of which is due to two of the present authors together with Hofmann) and supplies the first proof even in the small-perturbation regime. The manuscript explicitly identifies the new parabolic obstacles and claims to surmount them, which, if verified, would be a notable technical contribution.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the result holds 'for some p0 > 1' but does not indicate whether an explicit value or dependence on the Carleson norm is obtained; a brief remark in the introduction on the size of p0 would be helpful for context.","section":null},{"comment":"Notation for the parabolic Carleson condition is introduced without an equation number in the abstract; assigning a numbered display equation when the condition is first written would improve readability.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the significance of resolving the parabolic L^p regularity problem, including in the small Carleson norm regime, is recognized as a substantial advance.","responses":[],"tokens_in":1287,"tokens_out":67,"duration_ms":10356,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper claims to fully resolve the L^p regularity problem for the parabolic equation -∂t u + div(A ∇u)=0 on a Lipschitz cylinder. Under ellipticity, bounded measurable coefficients, and the parabolic Carleson condition, they show the regularity problem is solvable in (1, p0) for some p0>1. They stress that this remained open even when the Carleson norm is small. The elliptic version was settled only recently by two groups, and this work adapts the method from the group that includes two of the authors plus Hofmann while handling the extra time-dependent issues in the parabolic setting. The paper states the result and the assumptions cleanly and notes the new challenges explicitly. The main soft spot is that the abstract gives no proof steps or estimates, so it is impossible to check whether the adaptation actually works without the full manuscript. The hypotheses are standard and there is no sign of circularity or hidden fitting in the stated claim. If the details hold up, the result is a direct advance on a question left open in the parabolic case. This is for specialists working on boundary value problems and harmonic analysis techniques for parabolic PDEs. A reader tracking progress on L^p solvability for divergence-form operators would get value from seeing how the elliptic ideas transfer. It deserves a serious referee because it targets an open question with a concrete claim. I would send it to peer review to have the argument checked rather than desk reject it.","headline":"Claims to resolve the open parabolic L^p regularity problem for some p>1 under the Carleson condition on coefficients.","tokens_in":2207,"tokens_out":359,"would_cite":false,"duration_ms":20402,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Classical parabolic PDE regularity theory; no RS-shaped structure","alignment":"orthogonal","rationale":"The paper proves L^p solvability of the regularity problem for parabolic divergence-form operators under a parabolic Carleson/DKP-type condition on coefficients (Theorems 1.6, 1.8). Its machinery consists of energy estimates, nontangential maximal functions, square/area functions, block-form reductions, and duality between Dirichlet and regularity problems (Lemmas 3.19–3.21, Sections 5–11). None of this invokes J-cost, ratio symmetry, golden-ratio identities, 8-tick periodicity, or parameter-free constant derivations. The domain (math.AP, boundary-value problems on Lipschitz cylinders) lies outside the scope of the RS forcing chain.","tokens_in":68548,"confidence":"high","tokens_out":179,"duration_ms":4792,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The L^p regularity problem for parabolic operators with Carleson coefficients is solvable for p in (1, p0) for some p0 > 1.","keywords":[],"falsifier":"A specific example of an elliptic bounded measurable coefficient matrix satisfying the Carleson condition for which no p>1 allows solvability of the regularity problem would falsify the result.","tokens_in":2588,"feed_emoji":"","tokens_out":363,"duration_ms":42229,"temperature":0.7,"pith_summary":"This paper shows that the regularity problem for the parabolic equation with divergence form operator is solvable in L^p spaces for p close to 1. The coefficients are only assumed to be elliptic, bounded and measurable but must satisfy a Carleson-type condition that measures their oscillation in a scale invariant manner. This result holds on Lipschitz cylinders in space-time and resolves the question even when the Carleson norm is small.","feed_headline":"Parabolic regularity problem solvable for some p>1","feed_subtitle":"Under a Carleson condition the L^p range (1,p0) works for the regularity problem on Lipschitz cylinders.","key_machinery":"The natural parabolic Carleson condition on the coefficients, a scale-invariant bound on their oscillations over cylinders, which enables the L^p estimates for the solution gradient.","core_discovery":"Under the assumptions that the coefficient matrix A is elliptic with bounded measurable coefficients satisfying a parabolic Carleson condition, the regularity problem for -∂_t u + div(A ∇u) = 0 is solvable in the range (1, p0) for some p0 > 1 on Lipschitz cylinders.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Parabolic regularity solvable in Lp for some p>1 under Carleson","Lp regularity holds for parabolic PDEs with Carleson coefficients","Parabolic PDE regularity in range (1,p0) on Lipschitz cylinders","Carleson condition solves Lp regularity for parabolic operators"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The coefficients satisfy a natural parabolic Carleson condition in addition to being elliptic with bounded measurable entries.","fun_headline_variants_meta":{"raw":{"variants":["Parabolic regularity solvable in Lp for some p>1 under Carleson","Lp regularity holds for parabolic PDEs with Carleson coefficients","Parabolic PDE regularity in range (1,p0) on Lipschitz cylinders","Carleson condition solves Lp regularity for parabolic operators"]},"model":"grok-4.3","cost_usd":0.005527,"raw_usage":{"total_tokens":2643,"prompt_tokens":649,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":55274500,"prompt_tokens_details":{"text_tokens":649,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1922,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":649,"tokens_out":72,"duration_ms":17786,"temperature":1.0,"reasoning_tokens":1922,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-23T18:58:07.981696+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A specific example of an elliptic bounded measurable coefficient matrix satisfying the Carleson condition for which no p>1 allows solvability of the regularity problem would falsify the result.","supporting_citations":[],"review_version":1}