{"id":"dfe1fded-3694-4476-8356-c479c8eb7f8c","arxiv_id":"2404.08578","paper_version":5,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Tensor t-structures on D^b_coh(X) and variants for suitable Noetherian schemes X are classified, with existence on perfect complexes detecting regularity and local-to-global principles established.","lead":"The paper classifies tensor t-structures on the bounded derived category of coherent sheaves on suitable Noetherian schemes, including variants with prescribed support. It further shows existence of such structures on perfect complexes detects regularity and develops local-to-global principles.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the scope conditions on the scheme. Since the full text is now available per the prompt and no further technical flaw (e.g., in the local-to-global argument or the nonexistence part) can be identified without manufacturing one, the unverdicted status is unchanged.","tokens_in":1578,"tokens_out":257,"duration_ms":14471,"concrete_test":"Extract the precise definition of 'suitable Noetherian scheme' from the introduction or §1 of the full paper and check whether the main classification theorem (likely Theorem A or equivalent) is stated exactly under those conditions with no additional unlisted hypotheses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim classifies tensor t-structures on D^b_coh(X) (and variants) for a suitable Noetherian scheme X and shows that their restriction to perfect complexes detects regularity, recovering Neeman's affine result. The local-to-global principles are presented as tools. No internal inconsistency, hidden assumption in an equation, or unsupported step is visible from the abstract and claim description; the 'suitable' conditions are the explicit scope restriction rather than an unstated gap.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript classifies tensor t-structures on the bounded derived category of coherent sheaves D^b_coh(X) (and variants with prescribed support) for a suitable Noetherian scheme X. It shows that the existence of such t-structures restricting to perfect complexes detects regularity of X, recovering Neeman's affine result by different methods, and establishes local-to-global principles for tensor t-structures.","tokens_in":1647,"tokens_out":300,"duration_ms":31104,"significance":"If the classification and detection results hold, the work provides a useful extension of t-structure theory to coherent sheaves with support conditions and a new approach to regularity detection. The local-to-global principles are a potential strength for applications in derived algebraic geometry.","major_comments":[{"comment":"Abstract and §1: the classification is stated for a 'suitable Noetherian scheme' but the precise conditions (quasi-compactness, separatedness, finite Krull dimension, etc.) that make the statements hold are not listed explicitly at the outset; this is load-bearing for the scope of the main theorems.","section":"Abstract and §1"}],"minor_comments":[{"comment":"Ensure that each main theorem statement includes a self-contained list of hypotheses on X rather than relying solely on the global 'suitable' qualifier.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and positive assessment of the manuscript. We address the single major comment below.","responses":[{"response":"We agree that the current phrasing leaves the precise hypotheses implicit at the outset. In the revised version we will explicitly list the standing assumptions (Noetherian, quasi-compact, separated, finite Krull dimension) both in the abstract and at the opening of §1, while retaining the detailed discussion already present in §2.","revision_made":"yes","referee_comment":"[Abstract and §1] Abstract and §1: the classification is stated for a 'suitable Noetherian scheme' but the precise conditions (quasi-compactness, separatedness, finite Krull dimension, etc.) that make the statements hold are not listed explicitly at the outset; this is load-bearing for the scope of the main theorems."}],"tokens_in":1097,"tokens_out":201,"duration_ms":10052,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper classifies tensor t-structures on D^b_coh(X) and its supported variants for suitable Noetherian schemes, then shows that existence of such structures restricting to perfect complexes detects regularity. It recovers Neeman's affine result by different methods and supplies local-to-global principles as tools. That is the core output. The classification with explicit support conditions is the new piece; the regularity detection is presented as a consequence that re-proves a known theorem without relying on the original approach. The local-to-global statements look like a practical addition for working with these structures on non-affine schemes. The abstract and claim description give no sign of circularity or hidden fitting, and the recovery of an external theorem counts as independent grounding. The main soft spot is the scope: everything rests on the scheme being 'suitable' Noetherian, and the precise list of conditions (quasi-compact, finite dimension, etc.) will determine how widely the classification applies. If those conditions turn out narrow or require extra hypotheses in the proofs, the result shrinks. No load-bearing gaps are visible from the given material, but the absence of proof sketches in the abstract means the actual verification has to happen in review. This is for readers already working on t-structures, tensor triangulated categories, or regularity criteria in algebraic geometry. A specialist in the area would get direct value from the classification statements and the local-to-global tools. It is coherent on its own terms and engages the literature by re-deriving a known theorem differently, so it deserves a serious referee even if revisions are needed on the exact hypotheses or on filling in the proofs.","headline":"Classification of tensor t-structures on coherent derived categories for suitable Noetherian schemes, plus an alternative proof that their restriction to perfect complexes detects regularity.","tokens_in":2132,"tokens_out":399,"would_cite":false,"duration_ms":10708,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Classification of tensor t-structures on derived categories of schemes","alignment":"orthogonal","rationale":"The paper's machinery (Thomason filtrations, weak Cousin conditions, restriction of t-structures from Dqc,Z(X) to PerfZ(X) or Db_coh,Z(X), detection of regularity via bounded t-structures on perfect complexes) lies entirely in algebraic geometry/homological algebra. RS framework derives spacetime, J-cost = ½(x + x⁻¹) − 1, φ, 8-tick periodicity and constants from a single distinction via reality_from_one_distinction and related modules (AbsoluteFloorClosure, Cost.FunctionalEquation, DimensionForcing, etc.). No shared concepts, no ratio-symmetric cost, no ladder spacings, no parameter-free constant derivations. Domain mismatch is total.","tokens_in":57490,"confidence":"high","tokens_out":186,"duration_ms":8529,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Tensor t-structures on the bounded derived category of coherent sheaves are classified by support data for suitable Noetherian schemes, and their existence on perfect complexes detects regularity.","keywords":["tensor t-structures","derived categories","coherent sheaves","perfect complexes","scheme regularity","Noetherian schemes","support conditions"],"falsifier":"A concrete counterexample would be a suitable Noetherian scheme that is not regular yet admits a tensor t-structure on its perfect complexes.","tokens_in":2476,"feed_emoji":"","tokens_out":602,"duration_ms":13463,"temperature":0.7,"pith_summary":"The paper classifies all tensor t-structures with prescribed support on the bounded derived category of coherent sheaves and related variants. It further proves that the existence of any such t-structure that restricts to the subcategory of perfect complexes is equivalent to the scheme being regular. The classification rests on local-to-global principles that reduce questions about global t-structures to local data. A reader would care because the result gives an explicit description of these structures and recovers a known detection theorem for regularity in the affine case by new methods.","feed_headline":"Tensor t-structures on coherent derived categories classified by support","feed_subtitle":"Existence of those restricting to perfect complexes detects regularity of suitable Noetherian schemes.","key_machinery":"Tensor t-structures (t-structures compatible with the derived tensor product) on derived categories of coherent sheaves, classified via support conditions.","core_discovery":"Given a suitable Noetherian scheme X, the tensor t-structures on D^b(Coh(X)) with prescribed support are completely classified; moreover, the existence of a tensor t-structure on Perf(X) detects that X is regular, recovering Neeman's theorem in the affine case by different methods, while the same tools yield local-to-global principles for tensor t-structures.","pith_inferences":["The support classification may extend to unbounded derived categories if additional finiteness conditions are imposed.","The detection of regularity via t-structures could be tested on explicit non-affine examples such as projective space or singular curves.","Local-to-global principles might apply to other compatibility conditions beyond the tensor product."],"forward_implications":["All tensor t-structures with given support on D^b(Coh(X)) can be listed explicitly from local data.","Regularity of X is equivalent to the existence of any tensor t-structure restricting to Perf(X).","Questions about global tensor t-structures reduce to local questions via the established principles.","The same classification applies to variants of the category with prescribed support."],"fun_headline_variants":["Support classifies tensor t-structures on coherent derived categories","Existence on perfects detects regularity of Noetherian schemes","Local-to-global principles for tensor t-structures on schemes","Tensor t-structures with prescribed support classified on schemes"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The scheme must be suitable Noetherian so that the support-based classification and local-to-global reduction apply.","fun_headline_variants_meta":{"raw":{"variants":["Support classifies tensor t-structures on coherent derived categories","Existence on perfects detects regularity of Noetherian schemes","Local-to-global principles for tensor t-structures on schemes","Tensor t-structures with prescribed support classified on schemes"]},"model":"grok-4.3","cost_usd":0.005971,"raw_usage":{"total_tokens":2745,"prompt_tokens":498,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":59712000,"prompt_tokens_details":{"text_tokens":498,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2183,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":498,"tokens_out":64,"duration_ms":11889,"temperature":1.0,"reasoning_tokens":2183,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T02:29:24.396430+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counterexample would be a suitable Noetherian scheme that is not regular yet admits a tensor t-structure on its perfect complexes.","supporting_citations":[],"review_version":1}