{"id":"a7210de4-56f0-4e21-9de7-a79a96689ba4","arxiv_id":"2607.01167","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"One-sided median porosity of E is necessary and sufficient for d_E to the minus alpha to be in one-sided A_p for some alpha greater than zero when 1 less than p less than infinity, with new median characterizations of A_p and BMO.","lead":"The paper defines one-sided median porosity for subsets of the real line and proves it is equivalent to the distance weight to the set belonging to a one-sided Muckenhoupt A_p class for some positive exponent when 1 less than p less than infinity. A generalist might read it to see how new porosity notions refine the theory of one-sided weights used in inequalities for operators.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly flags the interaction between the new median definitions and the distance weight. Because the abstract already announces that the median characterizations are proved en route and no counter-example or gap is indicated in the stated results, the assumption appears to be secured by the paper's own constructions rather than left open.","tokens_in":1750,"tokens_out":260,"duration_ms":28782,"concrete_test":"Verify that the median-based A_p condition (as defined in the paper) applied to d_E^{-α} recovers the standard one-sided A_p inequality for the specific weight; if the two characterizations coincide on this family, the necessity/sufficiency direction holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim equates one-sided median porosity of E with d_E^{-α} belonging to a one-sided A_p class for some α>0 (1<p<∞). The provided abstract states that new median characterizations of one-sided A_p and BMO are derived as part of the argument, and an explicit example distinguishes the new notion from prior ones (one-sided weak porosity). No internal inconsistency or missing step is visible from the stated equivalences and the reported range of exponents.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces one-sided median porosity for subsets E of R. It proves this condition is necessary and sufficient for the distance weight d_E^{-α} to belong to a one-sided Muckenhoupt A_p class for some α>0 and 1<p<∞. As part of the argument, new median-based characterizations of one-sided A_p weights and BMO functions are obtained. The precise range of α is determined for both the p=1 case (building on prior weak porosity results) and for 1<p<∞. It is also shown that E is median porous if and only if it is both left and right median porous, and an explicit example is given of a one-sided median porous set that is neither median porous nor one-sided weakly porous.","tokens_in":1850,"tokens_out":471,"duration_ms":43252,"significance":"If the equivalences hold, the work supplies a geometric characterization linking one-sided median porosity to membership of distance weights in one-sided A_p classes, extending the recent A_1/weak-porosity equivalence. The median characterizations of A_p and BMO provide potentially useful alternative tools in one-sided harmonic analysis. The range of α and the distinguishing example clarify the relationships among porosity notions. These are solid contributions to the study of weights and function spaces on the line.","major_comments":[],"minor_comments":[{"comment":"The abstract states that new median characterizations are derived but does not indicate their precise form (e.g., the median condition replacing the usual integral or supremum). Adding one sentence summarizing the characterization would improve readability for readers scanning the abstract.","section":"Abstract"},{"comment":"Notation for the one-sided median operator and the precise definition of one-sided median porosity should be introduced with a displayed equation or numbered definition in the introduction or §2 to avoid any ambiguity when the reader reaches the main theorems.","section":"Introduction"},{"comment":"In the example distinguishing the notions, confirm that the set is constructed so that the median porosity constant is positive while the weak porosity constant is zero; a short calculation or reference to the relevant inequality would strengthen the claim.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of the paper, the clear summary of its contributions, and the recommendation for minor revision. No major comments appear in the report.","responses":[],"tokens_in":1324,"tokens_out":53,"duration_ms":13394,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper introduces one-sided median porosity for subsets of R and proves it is necessary and sufficient for the distance weight d_E^{-α} to belong to a one-sided Muckenhoupt A_p class for some α>0 when 1<p<∞. They also obtain median-based characterizations of one-sided A_p weights and BMO functions, and they give the precise range of α that works for both the p=1 case and for p>1. This extends a recent result that used one-sided weak porosity for the A1 case.\n\nThe work does a clean job separating the new notion from earlier ones. They construct an explicit example of a set that is one-sided median porous but neither median porous nor one-sided weakly porous, and they prove that median porosity is equivalent to being both left and right median porous. The median characterizations appear independent of the prior self-citations and do not create circularity with the main equivalence.\n\nThe central claims line up with the stated equivalences and the stress-test note finds no internal inconsistency. The median operator is applied in a way that matches the characterizations they derive. Soft spots are limited: the precise interaction of the median with the distance function would need checking in the proofs, but nothing in the abstract or stress-test suggests a load-bearing flaw.\n\nThis is a paper for people already working on one-sided weights, porosity conditions, and Muckenhoupt classes in harmonic analysis. A reader following that line would get concrete value from the exact exponent ranges and the distinguishing example. It refines existing characterizations rather than opening major new territory.\n\nSend it to peer review. The equivalences and example are sharp enough to deserve referee attention.","headline":"The paper defines one-sided median porosity and shows it exactly characterizes when d_E^{-α} sits in a one-sided A_p class for the right range of α, plus new median characterizations of A_p and BMO.","tokens_in":2365,"tokens_out":431,"would_cite":false,"duration_ms":29271,"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":"One-sided median porosity of E is necessary and sufficient for d_E^{-α} to be a one-sided Muckenhoupt A_p weight for some α>0 and 1<p<∞.","keywords":["one-sided median porosity","Muckenhoupt A_p weights","distance weights","one-sided BMO","porous sets","real line","weighted inequalities"],"falsifier":"A set E that is one-sided median porous but for which d_E^{-α} fails to satisfy the one-sided A_p condition for every α>0, or conversely a set where the weight condition holds but the set is not one-sided median porous.","tokens_in":2641,"feed_emoji":"","tokens_out":607,"duration_ms":43792,"temperature":0.7,"pith_summary":"This paper defines one-sided median porosity for subsets of the real line. It shows this geometric property is exactly equivalent to the distance weight d_E raised to a negative power belonging to a one-sided A_p class for appropriate α and p. The work also yields median-based characterizations of one-sided A_p weights and BMO functions. It determines the exact range of α for both p=1 and p>1 cases, and distinguishes one-sided median porosity from related notions with an example. Readers care because these conditions determine when distance weights can be used in one-sided weighted inequalities in real analysis.","feed_headline":"Median porosity sets when distance weights are one-sided A_p","feed_subtitle":"The new one-sided porosity condition on subsets of the line is equivalent to the distance function satisfying the Muckenhoupt integral condi","key_machinery":"one-sided median porosity, a median-based condition on subsets E of the real line that controls the distribution of E and its complement in intervals","core_discovery":"We introduce the notion of one-sided median porosity for subsets E of R. We prove that this condition is necessary and sufficient for the distance weight d_E^{-α} to belong to a one-sided Muckenhoupt A_p class for some α>0 and 1<p<∞. As part of the proof, we obtain new characterizations of one-sided A_p weights and one-sided BMO functions, in terms of medians. We find the precise range of exponents α>0 such that d_E^{-α} belongs to a one-sided A_p class, both for p=1 and for 1<p<∞. In addition, we show that E is median porous if and only if it is both left and right median porous, and we give an example of a one-sided median porous set which is neither median porous nor one-sided weakly poro","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["One-sided median porosity determines A_p class for distance weights","Equivalence of median porosity to one-sided A_p distance weights","Characterizations of A_p weights via one-sided median porosity","One-sided median porous sets and one-sided A_p distance functions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The definitions of one-sided median porosity and the one-sided Muckenhoupt classes are consistent with the median-based characterizations of A_p weights and BMO functions introduced in the paper.","fun_headline_variants_meta":{"raw":{"variants":["One-sided median porosity determines A_p class for distance weights","Equivalence of median porosity to one-sided A_p distance weights","Characterizations of A_p weights via one-sided median porosity","One-sided median porous sets and one-sided A_p distance functions"]},"model":"grok-4.3","cost_usd":0.00692,"raw_usage":{"total_tokens":3180,"prompt_tokens":770,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":69203000,"prompt_tokens_details":{"text_tokens":770,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2343,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":770,"tokens_out":67,"duration_ms":36093,"temperature":1.0,"reasoning_tokens":2343,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T02:27:54.654599+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A set E that is one-sided median porous but for which d_E^{-α} fails to satisfy the one-sided A_p condition for every α>0, or conversely a set where the weight condition holds but the set is not one-sided median porous.","supporting_citations":[],"review_version":1}