{"id":"bc1d42a5-4169-4cd7-b0a9-9adfcd45e18d","arxiv_id":"2605.27244","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces residual regularity for tensor triangulated categories, proves descent/ascent under finite separable extensions, and classifies all finite groups with residually regular derived permutation module categories.","lead":"The paper introduces residual regularity as a new notion for tensor triangulated categories, proves it descends and ascends under finite separable extensions, and classifies finite groups whose derived permutation module categories satisfy it. A smart generalist might read it for insight into new classification tools in derived categories arising in representation theory and algebraic geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the only point that could render the classification claim empty. No further technical gap in the stated theorems is apparent once that point is granted.","tokens_in":1490,"tokens_out":247,"duration_ms":20799,"concrete_test":"Verify that at least one finite group (e.g., C_2) satisfies the definition of residual regularity while another (e.g., a non-abelian group of order 8) does not, using the exact definition and the classification statement in the paper; agreement between the two checks confirms the definition is non-vacuous and the classification is non-trivial.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims introduce a new definition and then establish its stability under finite separable extensions together with a classification for permutation-module derived categories. Because the definition is new, its non-vacuousness is part of what the classification is meant to demonstrate; the descent/ascent statements are internal to the definition and do not rely on external data that could falsify them. No internal inconsistency or hidden assumption in the argument structure is visible from the given material.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces a new notion of residual regularity for tensor triangulated categories. It proves that this property descends and ascends along finite separable extensions and classifies all finite groups whose derived category of permutation modules satisfies residual regularity.","tokens_in":1554,"tokens_out":196,"duration_ms":19493,"significance":"If residual regularity is a meaningful and non-vacuous property, the stability results under finite separable extensions and the classification for permutation-module categories would constitute a useful contribution to tensor triangular geometry, particularly for understanding regularity phenomena in derived categories of group representations. The classification result would serve as evidence that the definition is not vacuous.","major_comments":[],"minor_comments":[{"comment":"The abstract does not indicate whether the definition of residual regularity is accompanied by concrete examples or computations that would allow readers to verify non-vacuousness independently of the classification theorem.","section":null}],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary of the manuscript. The recommendation is marked uncertain, which appears to hinge on whether residual regularity is a meaningful notion. The classification of all finite groups whose derived permutation module categories are residually regular provides concrete evidence that the property is non-vacuous and distinguishes interesting examples, supporting the utility of the descent/ascent results under finite separable extensions.","responses":[],"tokens_in":960,"tokens_out":94,"duration_ms":21296,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this paper defines a new notion of residual regularity in tensor triangulated categories, establishes its stability under finite separable extensions, and classifies the finite groups for which the derived category of permutation modules satisfies the property.\n\nWhat is new is the definition and the two theorems that follow from it. The descent and ascent results are the sort of technical statements that allow the notion to be applied more broadly, and the classification provides specific content by identifying which groups work.\n\nThe paper does well in setting up a property that behaves nicely with respect to extensions. This is a common technique in the field, and applying it here gives the definition some structure right away. The classification is also a positive step because it turns the abstract definition into something that can be checked for particular examples.\n\nOn the soft spots, the main one is the motivation for the new notion. Without more context on why residual regularity is needed beyond existing concepts like regularity in tt-geometry, it is not clear how much this reorganizes the area. The classification will be key to showing it is non-vacuous, but if it turns out to coincide with groups having semisimple or regular representation categories in a known way, the contribution is more limited. The abstract does not indicate any computational or external verification, so the strength rests entirely on the internal proofs.\n\nThis paper is aimed at researchers in tensor triangular geometry and related areas of algebraic K-theory or representation theory. A reader who follows papers on properties of triangulated categories with tensor structure would get the most out of it. It is worth sending to peer review because the results are stated clearly enough to be evaluated by experts in the subfield, even if the ultimate impact stays within that community.","headline":"Van Rooy defines residual regularity, proves its stability under finite separable extensions, and classifies the groups with residually regular permutation module categories.","tokens_in":1991,"tokens_out":421,"would_cite":false,"duration_ms":32051,"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":"Residual regularity descends and ascends via finite separable extensions, classifying all finite groups whose derived category of permutation modules satisfies the property.","keywords":["residual regularity","tensor triangulated categories","finite separable extensions","permutation modules","derived categories","finite groups","classification"],"falsifier":"A finite separable extension where residual regularity fails to descend or ascend, or a finite group outside the classified list whose derived category of permutation modules is residually regular.","tokens_in":2389,"feed_emoji":"","tokens_out":393,"duration_ms":33635,"temperature":0.7,"pith_summary":"The paper defines residual regularity as a new notion of regularity for tensor triangulated categories. It proves that this property descends and ascends along finite separable extensions. It applies the result to classify all finite groups for which the derived category of permutation modules is residually regular. A sympathetic reader would care because the stability gives a practical way to move the property between categories and the classification identifies exactly when it holds in this representation-theoretic setting.","feed_headline":"Residual regularity descends and ascends via finite separable extensions","feed_subtitle":"A new regularity property for tensor triangulated categories transfers across finite separable extensions, yielding a classification of all","key_machinery":"Residual regularity, the newly defined property of tensor triangulated categories that is shown to transfer along finite separable extensions and to classify the relevant groups.","core_discovery":"We introduce residual regularity as a new notion of regularity for tensor triangulated categories. We show that residual regularity descends and ascends via finite separable extensions and we classify all finite groups whose derived category of permutation modules is residually regular.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Residual regularity transfers via finite separable extensions","Classifying finite groups with residually regular permutation modules","Residual regularity ascends and descends via finite separable extensions","Finite groups classified by residual regularity in derived categories"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The definition of residual regularity must pick out a meaningful and non-vacuous property on the tensor triangulated categories under study.","fun_headline_variants_meta":{"raw":{"variants":["Residual regularity transfers via finite separable extensions","Classifying finite groups with residually regular permutation modules","Residual regularity ascends and descends via finite separable extensions","Finite groups classified by residual regularity in derived categories"]},"model":"grok-4.3","cost_usd":0.009586,"raw_usage":{"total_tokens":4158,"prompt_tokens":432,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":95862000,"prompt_tokens_details":{"text_tokens":432,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3669,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":432,"tokens_out":57,"duration_ms":40412,"temperature":1.0,"reasoning_tokens":3669,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T14:19:34.848298+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A finite separable extension where residual regularity fails to descend or ascend, or a finite group outside the classified list whose derived category of permutation modules is residually regular.","supporting_citations":[],"review_version":1}