{"id":"b6cea3ce-d43f-4562-ad71-f347e51db6a2","arxiv_id":"2607.03234","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The compact subcategory W(M,D) of the CO(β)-deformed wrapped Fukaya category is filtered quasi-equivalent to Seidel's relative Fukaya category F(M,D).","lead":"This paper proves that the full subcategory of closed Lagrangians in a certain deformed wrapped Fukaya category is filtered quasi-equivalent to Seidel's relative Fukaya category. The result settles a conjecture that identifies two standard constructions of deformed Fukaya categories for monotone symplectic manifolds with orthogonal divisors.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript supplies a complete, self-contained proof of the stated conjecture by combining gauge equivalence of Maurer-Cartan elements with an identification of structure constants. The only non-trivial analytic input is the existence of a common residual set of adapted perturbation data making all relevant moduli spaces regular; this is precisely the assumption already used (and accepted) for each category separately in the literature the paper cites. Because residual sets are dense and the adaptation conditions (preservation of the divisors, contact-type on the neck) are closed under the operations needed for both constructions, the simultaneous-genericity claim is the expected one rather than a new soft spot. No free parameters, no circular appeal to the conjecture itself, and no contradiction with known results appear. Consequently the reader’s ACCEPT verdict with moderate confidence and medium correctness risk remains appropriate; no adjustment is required.","tokens_in":33572,"tokens_out":550,"duration_ms":6555,"concrete_test":"Verify that the forgetful map from the joint space of adapted (K,J) pairs (those that are contact-type on the neck and preserve each Dj) to the product of the two separate Baire spaces used for F(M,D) and for the CO-moduli spaces is continuous and open on a residual set; if residual sets remain residual under this map, simultaneous genericity holds and the strict isomorphism of Step 2 is justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.2 / 5.1) rests on two steps: (i) gauge equivalence of the Maurer-Cartan elements α and β via the L∞-morphism property of CO, producing a zig-zag of filtered quasi-equivalences between W(M,D) and W^α(M,D), and (ii) a strict isomorphism W^α(M,D) ≅ F(M,D) obtained by matching structure constants of the A∞-operations once the same adapted perturbation data are used. Both steps are standard once the moduli spaces are smooth of expected dimension. The reader correctly flags simultaneous genericity of those data as the weakest assumption, but this is the ordinary transversality hypothesis of the subfield (already asserted for each construction separately in the cited works [BAS24], [SBEAS24], [PS22], [AS10]); the paper does not introduce a new analytic gap. No internal inconsistency, missing estimate, or circular reduction appears in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves Conjecture 1.3 of [SBEAS24]: for a closed monotone symplectic manifold (M,ω) with orthogonal simple normal-crossing divisors D satisfying 2c1(M)=∑λjDj (λj≤2 rational), the full subcategory W(M,D) of the CO(β)-deformed wrapped Fukaya category of X=M∖D, whose objects are closed exact Lagrangian branes, is filtered quasi-equivalent to Seidel’s relative Fukaya category F(M,D). After recalling Liouville structures on X (McLean), domain moduli spaces with Fulton–MacPherson/aligned framings, and the L∞-structure on symplectic cohomology together with the L∞-morphism CO, the proof in §5 proceeds in two steps: (i) gauge equivalence of the Maurer–Cartan elements α=∑xDj qj and β via the homotopy model gΛt yields a zig-zag of filtered quasi-equivalences W(M,D)≃Wα(M,D); (ii) under identical adapted perturbation data the structure constants of the A∞-operations of Wα(M,D) and F(M,D) match, giving a strict isomorphism.","tokens_in":33808,"tokens_out":1090,"duration_ms":9417,"significance":"The result supplies the missing identification between two independently constructed deformations of the compact Fukaya category of the complement of a symplectic divisor. It therefore makes the Maurer–Cartan deformation of wrapped Floer theory constructed in [SBEAS24] available for computations that have previously been performed only with Seidel’s relative Fukaya category, and conversely. The argument is short once the analytic foundations of [BAS24], [SBEAS24] and [PS22] are granted, and it cleanly separates the algebraic gauge-equivalence step from the geometric matching of moduli spaces. The paper does not claim new transversality or compactness theorems; its contribution is the identification itself.","major_comments":[{"comment":"The opening sentence of the proof of Theorem 5.1 (and the constructions throughout §4) invokes a single generic consistent universal choice of adapted perturbation data that simultaneously renders all moduli spaces appearing in both F(M,D) and the CO(β)-deformed wrapped category smooth oriented manifolds of the expected dimension. While each construction separately asserts such genericity in the cited works, the simultaneous statement is not proved here and is load-bearing for the strict isomorphism of Step 2. A short paragraph confirming that the Baire-category intersection of the two residual sets remains residual (or an explicit reference to a joint transversality result) would close the gap.","section":null},{"comment":"Assumption 4.69 (Q-span of the relative Poincaré duals generates H2(M,X;Q) and each class appears at least n+1 times) is used for positivity of intersection (Lemma 4.71) and for the absence of sphere bubbles of codimension 1. The paper treats it as standing, but it is not automatic for every orthogonal simple normal-crossing divisor satisfying (1.1). Either a brief verification that the assumption holds under the hypotheses of Theorem 1.2, or an explicit restriction of the main theorem to divisors satisfying Assumption 4.69, is needed.","section":null}],"minor_comments":[{"comment":"The abstract and introduction are extremely terse (“We give a proof of Conjecture 1.3 of [SBEAS24]”). A one-sentence statement of the geometric content of the conjecture would help non-specialist readers.","section":null},{"comment":"Notation for the two Novikov rings and the two filtrations (P-filtration versus the q-adic filtration) is introduced in several places; a short “Notation” paragraph at the beginning of §4 would reduce the risk of confusion.","section":null},{"comment":"In Definition 4.46 the partial L∞-operations are declared zero when all inputs are of the form xDj; a parenthetical remark explaining why this is compatible with the Maurer–Cartan equation would be useful.","section":null},{"comment":"Typographical inconsistencies appear (e.g., “adpated”, “campatible”, “equivanlent”, “homoligically”). A careful proof-reading pass is recommended.","section":null},{"comment":"The appendix on Gromov’s graph trick is standard; a one-line pointer to the corresponding statements in [MS12] or [AS10] would suffice and free space.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is essentially a short identification paper that relies almost entirely on the analytic foundations already published by overlapping authors. That is legitimate, but the journal should confirm that the simultaneous-genericity claim is regarded as routine by the community; if not, a more substantial revision or a joint note with the authors of [BAS24]/[SBEAS24] may be preferable. The result is correct and useful once the two minor analytic points above are clarified."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles Conjecture 1.3 of SBEAS24: the full subcategory of the CO(β)-deformed wrapped Fukaya category whose objects are closed exact Lagrangians is filtered quasi-equivalent to Seidel’s relative Fukaya category F(M,D). That identification is the only new result; everything else is assembly of existing pieces.\n\nWhat it does well is keep the argument short and transparent. Section 5 has two clean steps. First, α and β are gauge-equivalent via the homotopy model and the L∞ property of the closed-open map, so the two deformations of the wrapped category are quasi-equivalent by a zig-zag. Second, once the same adapted perturbation data are used, the structure constants of the A∞ operations match exactly (by the maximum principle and positivity of intersection already in PS22), giving a strict isomorphism between the α-deformed category and F(M,D). The supporting material in §2–4 is just the necessary recollections of Liouville structures, domain moduli, and the constructions from BAS24/SBEAS24/AS10/Sei08; it is dense but accurate.\n\nThe soft spot the reader flags is real but ordinary: the paper asserts rather than re-proves the existence of a single generic consistent universal choice of adapted (H,J) that works simultaneously for both constructions. That is the usual transversality hypothesis of the subfield, already claimed separately in the cited preprints; no new analytic estimate is missing and no circularity appears. Dependence on recent overlapping preprints is heavy, but those papers supply independent definitions of the two sides and of CO, so the identification itself is not tautological.\n\nThis is for people already working with relative Fukaya categories or Maurer–Cartan deformations of wrapped categories in the monotone/Fano setting. It will be cited whenever one needs to move freely between the two models. The math is solid enough that a serious editor should send it to referees; the usual subfield caveats about genericity and citation of recent preprints will be handled in revision. I would engage with it.","headline":"Clean proof of the SBEAS24 conjecture identifying the CO(β)-deformed compact wrapped subcategory with Seidel’s relative Fukaya category; standard technology, no new analytic gaps.","tokens_in":34360,"tokens_out":559,"would_cite":true,"duration_ms":7165,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D37","53D40","18G70"],"pacs":[],"model":"grok-4.5","headline":"The wrapped Fukaya category of the complement, deformed by a Maurer-Cartan element from the divisors, is filtered quasi-equivalent to Seidel's relative Fukaya category.","keywords":["relative Fukaya category","wrapped Fukaya category","Maurer-Cartan element","Closed-Open map","monotone symplectic manifold","filtered quasi-equivalence","Lagrangian branes","Novikov ring"],"falsifier":"Exhibit a concrete monotone pair $(M, D)$ and a closed Lagrangian for which the structure constants of the two $A_\\\\infty$-structures (the relative counts versus the $\\\\mathrm{CO}(\\\\alpha)$-deformed wrapped counts) differ by a non-zero power of the Novikov variables, or show that no common regular perturbation data can exist for both moduli problems simultaneously.","tokens_in":34448,"feed_emoji":"🔗","tokens_out":1019,"duration_ms":11266,"temperature":0.7,"texified_at":"2026-08-05T21:13:59.244305+00:00","pith_summary":"The paper proves that two constructions of deformed Fukaya categories of a monotone symplectic manifold relative to a divisor are the same up to filtered quasi-equivalence. One construction starts from the wrapped Fukaya category of the complement of the divisor and deforms it by a Maurer-Cartan element coming from the Closed-Open map applied to a class built from the divisor components. The other is Seidel's relative Fukaya category, which counts holomorphic polygons that meet the divisor. The proof shows that the deformation by the simpler Maurer-Cartan element already recovers the relative category, and that this simpler deformation is gauge-equivalent to the more complicated one used in the wrapped setting. A sympathetic reader cares because the equivalence identifies two natural ways of packaging the same geometric data, so calculations or invariants defined in one language can be transferred to the other.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":6853,"prompt_tokens":634,"completion_tokens":6219,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":5628}},"feed_headline":"Wrapped and relative Fukaya categories match after deformation","feed_subtitle":"A Maurer-Cartan element from the divisors makes the two constructions filtered quasi-equivalent","key_machinery":"The Closed-Open $L_\\\\infty$-morphism $\\\\mathrm{CO}$ from symplectic cohomology of the complement to the Hochschild cochains of the wrapped Fukaya category, together with the Maurer-Cartan element $\\\\beta$ obtained by flowing the elementary class $\\\\alpha$ along a homotopy model. Gauge equivalence of $\\\\alpha$ and $\\\\beta$, combined with the fact that $\\\\mathrm{CO}$ is an $L_\\\\infty$-morphism, produces a zig-zag of filtered quasi-equivalences that identifies the deformed wrapped category with the relative category.","core_discovery":"The full subcategory of the $\\\\mathrm{CO}(\\\\beta)$-deformed wrapped Fukaya category whose objects are closed Lagrangian branes in the complement $X = M \\\\setminus D$ is filtered quasi-equivalent to Seidel's relative Fukaya category $\\\\mathcal{F}(M, D)$. The two categories have identical objects and, after a gauge equivalence that replaces $\\\\beta$ by the elementary Maurer-Cartan element $\\\\alpha = \\\\sum x_{D_j} q_j$, they have identical structure maps given by the same counts of holomorphic polygons meeting $D$.","pith_inferences":["The result suggests that many other relative or logarithmic Fukaya constructions may likewise arise as gauge-equivalent deformations of ordinary wrapped categories of the complement.","Once the common regular perturbation data are known to exist, numerical or computer-assisted counts of low-degree polygons in simple examples (e.g., projective space with a hyperplane) can be used to check the equality of structure constants directly.","The identification may simplify the study of mirror symmetry statements that currently switch between relative and absolute languages, by allowing both sides to be written in a single deformed-wrapped formalism."],"forward_implications":["Invariants or computations defined via Seidel's relative Fukaya category can be rewritten in the language of the deformed wrapped category of the complement, and vice versa.","The elementary Maurer-Cartan element built from the divisor components already produces the full relative deformation; the more complicated gauge-equivalent element β is not needed for the final category.","Any future comparison of relative and absolute Fukaya categories that uses either construction can freely switch to the other without changing the quasi-equivalence class.","The same zig-zag argument applies whenever a Closed-Open map takes a Maurer-Cartan element to a deformation of a Fukaya category whose coefficient moduli spaces match those of a relative construction."],"fun_headline_variants":["Deformed wrapped Fukaya matches Seidel relative category","Gauge-equivalent Maurer-Cartan equates wrapped and relative Fukaya","Closed branes of CO-deformed wrapped Fukaya equal relative Fukaya","Relative Fukaya recovered from deformed wrapped after gauge change","Filtered quasi-equivalence of deformed wrapped and relative Fukaya"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"There must exist a single generic choice of almost-complex structures and Hamiltonians, adapted to the divisor, that makes every relevant moduli space for both constructions into a smooth oriented manifold of the expected dimension at the same time.","fun_headline_variants_meta":{"raw":{"variants":["Deformed wrapped Fukaya matches Seidel relative category","Gauge-equivalent Maurer-Cartan equates wrapped and relative Fukaya","Closed branes of CO-deformed wrapped Fukaya equal relative Fukaya","Relative Fukaya recovered from deformed wrapped after gauge change","Filtered quasi-equivalence of deformed wrapped and relative Fukaya"]},"model":"grok-4.5","effort":"low","cost_usd":0.005692,"raw_usage":{"total_tokens":1326,"prompt_tokens":566,"num_sources_used":0,"completion_tokens":89,"cost_in_usd_ticks":56920000,"prompt_tokens_details":{"text_tokens":566,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":671,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":566,"tokens_out":89,"duration_ms":5587,"temperature":1.0,"reasoning_tokens":671,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T03:55:56.400848+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete monotone pair $(M, D)$ and a closed Lagrangian for which the structure constants of the two $A_\\\\infty$-structures (the relative counts versus the $\\\\mathrm{CO}(\\\\alpha)$-deformed wrapped counts) differ by a non-zero power of the Novikov variables, or show that no common regular perturbation data can exist for both moduli problems simultaneously.","supporting_citations":[],"review_version":1}