{"id":"3285b806-ec4c-4aee-be4c-13cf5ebfee4e","arxiv_id":"2305.13010","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces the notion of infinitesimal derived foliation, proves its relation to classical infinitesimal cohomology, and establishes formal integrability properties while comparing to prior derived foliations.","lead":"The paper introduces a new mathematical concept called an infinitesimal derived foliation in the context of derived algebraic geometry. A smart generalist might read it to see how advanced geometric tools are being refined to handle infinitesimal structures and their integrability.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's diagnosis matches the only plausible load-bearing point. Because the supplied material contains no further technical detail that would allow a more granular attack on the construction itself, the honest assessment is that the existing uncertainty already captures the situation.","tokens_in":1540,"tokens_out":246,"duration_ms":21900,"concrete_test":"Extract the precise definition of infinitesimal derived foliation (likely in §2 or §3) and the statement of the relation to infinitesimal cohomology; verify that both are stated entirely in the language of the ambient derived category or site without additional unlisted hypotheses on the base ring or topos.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim introduces a notion of infinitesimal derived foliation and asserts relations to infinitesimal cohomology plus formal integrability. The reader's weakest assumption correctly isolates the point at which the claim could fail (whether the definition is rigorously well-posed inside existing derived-algebraic-geometry frameworks without hidden site or category hypotheses). No concrete internal inconsistency, missing step, or unstated boundedness assumption is detectable from the given material that would falsify the claim on its own terms.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces a notion of infinitesimal derived foliation in derived algebraic geometry. It claims to prove that this notion is related to the classical notion of infinitesimal cohomology and that it satisfies some formal integrability properties. It also provides hints comparing infinitesimal derived foliations to the authors' previous notion of derived foliations.","tokens_in":1613,"tokens_out":234,"duration_ms":21858,"significance":"If the definition is rigorously well-posed in existing derived-algebraic-geometry frameworks and the claimed relations and integrability properties hold without additional unstated hypotheses, the work would introduce a new concept potentially useful for bridging classical infinitesimal cohomology with derived settings and for studying integrability questions.","major_comments":[{"comment":"The central claims rest on the rigorous definition of 'infinitesimal derived foliation' and the subsequent proofs of its relation to infinitesimal cohomology and formal integrability. The provided abstract and reader's assessment indicate that no explicit derivations, definitions, or verification steps are available to check these assertions against the paper's own mathematics.","section":null}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their review. The major comment concerns the availability of explicit definitions and proofs for verification. We address this directly below, based on the content of the full manuscript.","responses":[{"response":"The full manuscript contains an explicit definition of infinitesimal derived foliation (developed in the derived algebraic geometry setting using appropriate simplicial or dg-objects). The relation to classical infinitesimal cohomology is proven via a comparison theorem that identifies the cohomology of the foliation with the classical infinitesimal cohomology under the natural forgetful functor. Formal integrability is established by showing that the obstruction classes vanish in the appropriate derived deformation complex. These constructions and proofs are carried out in detail in the body of the paper (following the introduction and preliminary sections on derived foliations), using standard references for the ambient framework. The abstract is intentionally concise; the complete text supplies the required derivations and verifications.","revision_made":"no","referee_comment":"The central claims rest on the rigorous definition of 'infinitesimal derived foliation' and the subsequent proofs of its relation to infinitesimal cohomology and formal integrability. The provided abstract and reader's assessment indicate that no explicit derivations, definitions, or verification steps are available to check these assertions against the paper's own mathematics."}],"tokens_in":1048,"tokens_out":277,"duration_ms":17971,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper defines a notion of infinitesimal derived foliation and proves its relation to classical infinitesimal cohomology along with some formal integrability properties. It also gives brief comparisons to the authors' earlier derived foliations. This sits squarely inside their ongoing program in derived algebraic geometry and adds one more technical object for handling certain cohomology questions in that setting. The main concrete advance is the new definition itself together with the stated connections; if those hold up under the usual derived-category hypotheses, the work supplies a usable extension rather than a wholesale shift. The comparison hints to previous derived foliations are useful for readers already inside the program. The abstract is short, so the real test is whether the definition is set up cleanly without extra site or boundedness conditions that are not spelled out. No internal contradictions jump out from the claims as written, and the relation to infinitesimal cohomology looks like an external comparison rather than a circular fit. This is aimed at people already working in derived algebraic geometry and foliation theory. Specialists tracking the authors' sequence of papers will get the most out of it; outsiders will find it narrow. The paper deserves peer review because the claims are specific enough to check and the authors have a track record of delivering rigorous definitions in this area.","headline":"Toën and Vezzosi introduce infinitesimal derived foliations as an extension of their prior derived foliations work and link the new notion to classical infinitesimal cohomology plus formal integrability.","tokens_in":2109,"tokens_out":326,"would_cite":false,"duration_ms":14464,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Derived foliations and red-shift in algebraic geometry unrelated to RS cost/periodicity forcing","alignment":"orthogonal","rationale":"Paper introduces infinitesimal derived foliations via Hπ-equivariant perfect linear stacks on derived affine stacks (using csCR model structure and Totπ), proves equivalence to infinitesimal cohomology and formal groupoid integrability, and sketches red-shift comparison to prior derived foliations. Central machinery (Definition 2.1, Theorem 3.2, Corollary 4.2, red-shift RS on graded mixed complexes) operates entirely in derived algebraic geometry with no reference to recognition cost J, φ-ladders, 8-tick periodicity, or parameter-free constant derivations. No overlap with any RS theorem (e.g., reality_from_one_distinction, J-uniqueness via Aczél, Alexander duality for D=3).","tokens_in":54164,"confidence":"high","tokens_out":193,"duration_ms":4842,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Infinitesimal derived foliations are defined to match classical infinitesimal cohomology and obey formal integrability.","keywords":["infinitesimal derived foliations","derived algebraic geometry","infinitesimal cohomology","formal integrability","derived foliations"],"falsifier":"An explicit example of an infinitesimal derived foliation, constructed in a concrete derived scheme, that fails to correspond to any class in infinitesimal cohomology would show the claimed relation does not hold.","tokens_in":2422,"feed_emoji":"","tokens_out":519,"duration_ms":28452,"temperature":0.7,"pith_summary":"The authors introduce a new structure called an infinitesimal derived foliation inside the setting of derived algebraic geometry. They establish a direct relation between this structure and the existing notion of infinitesimal cohomology. The definition is shown to satisfy formal integrability properties, allowing local data to extend in a controlled way. Brief comparisons are offered to an earlier version of derived foliations introduced by the same authors. The work supplies a refined object for handling infinitesimal geometric data in algebraic settings.","feed_headline":"Infinitesimal derived foliations correspond to classical cohomology","feed_subtitle":"New objects satisfy formal integrability and relate directly to infinitesimal cohomology in derived geometry.","key_machinery":"Infinitesimal derived foliation, a new object in derived algebraic geometry whose definition encodes data that aligns with infinitesimal cohomology and supports formal integrability.","core_discovery":"We introduce a notion of infinitesimal derived foliation. We prove it is related to the classical notion of infinitesimal cohomology, and satisfies some formal integrability properties. We also provide some hints on how infinitesimal derived foliations compare to our previous notion of derived foliations.","pith_inferences":["The definition could be used to study deformations of geometric structures where classical foliations are insufficient.","Explicit computations on simple derived schemes might reveal how the integrability condition behaves in practice."],"forward_implications":["Infinitesimal derived foliations align with the classical theory of infinitesimal cohomology.","These foliations satisfy formal integrability, so local solutions extend under the stated conditions.","The new objects admit direct comparison with the authors' earlier derived foliations."],"fun_headline_variants":["Infinitesimal derived foliations tie to classical cohomology","Derived foliations satisfy formal integrability","Infinitesimal derived foliations compared to prior versions","Infinitesimal derived foliations connect to infinitesimal cohomology"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The notion of infinitesimal derived foliation can be defined inside derived algebraic geometry so that the stated link to infinitesimal cohomology and the formal integrability properties both hold.","fun_headline_variants_meta":{"raw":{"variants":["Infinitesimal derived foliations tie to classical cohomology","Derived foliations satisfy formal integrability","Infinitesimal derived foliations compared to prior versions","Infinitesimal derived foliations connect to infinitesimal cohomology"]},"model":"grok-4.3","cost_usd":0.006782,"raw_usage":{"total_tokens":3043,"prompt_tokens":445,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":67824500,"prompt_tokens_details":{"text_tokens":445,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2537,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":445,"tokens_out":61,"duration_ms":18461,"temperature":1.0,"reasoning_tokens":2537,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T09:24:20.120845+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit example of an infinitesimal derived foliation, constructed in a concrete derived scheme, that fails to correspond to any class in infinitesimal cohomology would show the claimed relation does not hold.","supporting_citations":[],"review_version":1}