{"id":"7ec1004f-0f69-429d-8358-9590c77fab6e","arxiv_id":"2504.20489","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The filtered A∞-operations in the Bott-Morse de Rham model for collections of relatively spin Lagrangians satisfy the A∞-relations with the explicitly computed signs.","lead":"A technical note in symplectic topology works out the sign and orientation conventions that make the filtered A∞-relations hold in a de Rham-model Fukaya category over a collection of Lagrangian submanifolds. The paper supplies a definition of the operations and a proof of the relations, clearing a known bookkeeping hurdle for Lagrangian Floer theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 3.3 forces m0=0 without an unobstructedness hypothesis; the theorem therefore proves only an uncurved A∞ structure, not the claimed filtered A∞-category structure for general relatively spin Lagrangians.","rationale":"The paper defines filtered A∞-operations in a de Rham model for a collection of relatively spin Lagrangians and claims to verify the filtered A∞-relations. The most load-bearing defect is not the Kuranishi-perturbation formalism but the handling of m_0. In the filtered A∞ category, m_0 is an operation counting holomorphic disks with one boundary marked point; it is generally nonzero and is a central part of the structure. Definition 3.3 simply sets m_0=0. Consequently, Theorem 4.4 only proves the uncurved A∞ relations for the truncated operations; it does not prove that the geometric moduli spaces yield a filtered A∞-category structure unless every Lagrangian is unobstructed, a hypothesis never stated. This is visible already at k=1, where the filtered relation contains m_2(x,m_0) and m_2(m_0,x) terms that the proof never encounters. The boundary decomposition of Proposition 4.1 does include components with a single marked point, but the definition m_0=0 makes their contributions vanish identically. A concrete example (Clifford torus in CP^2, whose disk potential is nonzero) shows the assumption fails in general. The reader's weakest_assumption about integration along fibers over K-spaces is a valid technical caveat, but the m_0 gap is more directly connected to the central claim: even with a perfectly established virtual chain formalism, the theorem as stated would not describe the filtered A∞-category of a general relatively spin collection. The appropriate verdict remains CONDITIONAL, because the sign computations and the A∞ relations for k≥1 with m_0=0 are internally plausible and would be correct under an explicit unobstructedness assumption; the paper needs to state that assumption (or handle bounding cochains) before the claimed categorical structure is established.","tokens_in":16371,"tokens_out":24694,"duration_ms":251782,"concrete_test":"Take the Clifford torus L in CP^2, which is relatively spin, and compute the virtual pushforward (ev_0)_!(1) over the moduli M_1([D^2];L) of Maslov-index-2 holomorphic disks. The known disk potential of the Clifford torus is nonzero, so m_0≠0. If so, the operations defined in Definition 3.3 with m_0=0 fail the full k=1 filtered A∞ relation, which would require d^2x ± m_2(x,m_0) ± m_2(m_0,x)=0. This check settles that the paper's m_0=0 is an extra geometric hypothesis, not a harmless convention.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 3.3 sets m_{0,0}=0 and defines no m_{0,B} for B≠0, so m_0≡0. The boundary of each moduli space M_{k+1}(B;L;R) includes strata of the form M_{k'+1}(B') ×_{ev} M_1(B'') with k''=0, i.e., a disk with a single marked point; in the filtered A∞ category these strata produce the terms m_{k'}(...,m_0,...). Because m_0 is set to zero, the proof of Theorem 4.4 has no need to cancel them. But for a general relatively spin collection the virtual pushforward from M_1(B'';L) to L is the disk obstruction m_0, which is typically nonzero (e.g., a Lagrangian torus with nontrivial disk potential). Thus Theorem 4.4 establishes an uncurved A∞ structure under an implicit, unstated unobstructedness assumption. The k=1 filtered A∞ relation is d^2x ± m_2(x,m_0) ± m_2(m_0,x)=0; the paper verifies only d^2=0. Proposition 3.1(1) even prepares the orientation data for M_1(B;L;R) but no operation is defined from it, confirming that the m_0 contribution is dropped by fiat. This is a substantive geometric gap, independent of the Kuranishi-integration subtleties flagged by the reader.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper addresses sign and orientation conventions in the construction of filtered A∞-operations in the de Rham model of Lagrangian Floer theory in the Bott–Morse (clean intersection) setting. The author defines operations m_{k,B} as integrations along fibers of moduli spaces of bordered stable polygons with Kuranishi structures (Definition 3.3), setting m_{0,0}=0 and m_{1,0}=d, and then states that these operations satisfy the filtered A∞-relations (Theorem 4.4). The proof computes boundary contributions and checks that the sign κ in Proposition 4.1 and the composition sign κ′ in Lemma 4.3 cancel to yield the Stokes boundary term. The paper relies on the Kuranishi structure constructions of [1,3] and on the integration-along-fibers formalism of [6].","tokens_in":16712,"tokens_out":5730,"duration_ms":57612,"significance":"If the sign verification is correct and the underlying Kuranishi constructions are available, the paper would provide a useful de Rham model for the filtered Fukaya category for a collection of relatively spin Lagrangian submanifolds, complementing the singular chain treatment of [4,5]. The explicit sign bookkeeping in Propositions 3.1 and 4.1 is a genuine contribution. However, as it stands the theorem covers only the uncurved case m0=0, and the paper's reduction of Kuranishi integration to the smooth submersion case is asserted rather than proved.","major_comments":[{"comment":"The definition sets m_{0,0}=0 and defines no m_{0,B} for B≠0, so m_0≡0. In the filtered A∞-relation written in Section 4, the sum over k′+k″=k+1 includes terms with k″=0 (and k′=0) in which m_0 is inserted; for a general relatively spin collection the disk potential m_0 is generally nonzero (e.g., a Lagrangian torus with nontrivial disk potential in the sense of Fukaya–Oh–Ohta–Ono). The proof of Theorem 4.4 therefore never cancels the boundary strata of the form M_{k′+1}(B′) ×_{ev} M_1(B″), because m_0 was set to zero by fiat. Consequently Theorem 4.4 establishes only an uncurved A∞ structure under an implicit unobstructedness assumption, not the filtered A∞-category relations for general relatively spin Lagrangians. Proposition 3.1(1) prepares orientation data for M_1(B;L) but no operation is defined from it, confirming that the m_0 contribution is omitted. The theorem should either state an explicit unobstructedness hypothesis or include the m_0 terms in the verification.","section":"Definition 3.3 and Theorem 4.4"},{"comment":"The paper states: 'For the verification of the sign convention in the filtered A∞-relations, it is sufficient to treat the integration along fibers of a proper weakly submersive strongly smooth map as if the one for proper submersion between smooth manifolds.' This is the basis for suppressing CF-perturbation and obstruction-bundle data in all sign computations, including the orientation comparison in Proposition 4.1 and Lemma 4.2. No proof or detailed reference is given for this reduction. Since the sign κ entering the boundary correspondence depends on orientations of fiber products of Kuranishi spaces, the sufficiency claim is load-bearing for Theorem 4.4. Please either prove the reduction from the formalism of [6, Sections 9 and 10] or state precisely which statements in [6] imply it.","section":"Section 2"},{"comment":"The moduli spaces M_{k+1}(B;L;R) are assumed to carry tree-like K-systems, and their construction is postponed to [1,3], with [1] listed as 'in preparation'. The verification in Theorem 4.4 is therefore conditional on an unpublished construction. Please clarify whether [3] alone supplies the needed Kuranishi structures for the boundary decomposition used in Proposition 4.1, and update [1] if it is now available.","section":"References [1] and [3]"}],"minor_comments":[{"comment":"The phrase 'appearing the filtered A∞-formulae' should read 'appearing in the filtered A∞-formulae'.","section":"Abstract"},{"comment":"The sign ϵ(ξ_1,...,ξ_k) contains a trailing '+1' in its definition; please state explicitly whether this is part of the convention and how it is fixed.","section":"Equation (3.4)"},{"comment":"The paper uses the coefficient ring Λ0 rather than Λ_{0,nov}; as stated in Remark 3.4, the operations are not exactly of degree 1 in this ring. Please clarify how the degree property (3.5) is used in the verification of the A∞-relations.","section":"Remark 3.4"},{"comment":"The notation m_{1,0}=d for the de Rham differential may be confused with the operation for B=0 in the same family; consider using a separate symbol such as m_{1}^{dR} for the differential.","section":"Definition 3.3"},{"comment":"In the display after equation (4.6), the expression ev_{B′}^{∗}∘(ev_{B″}^{0})_! should be explained as composition of pullback with integration along fibers in the correspondence sense; as written it is easy to misread as an operator on a single manifold.","section":"Lemma 4.3"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this note is exactly what it says on the tin—a sign/orientation check for filtered A∞ operations in the de Rham Bott–Morse model—but the version on arXiv proves less than it claims. The operations in Definition 3.3 set m0 ≡ 0, and no m_{0,B} for B ≠ 0 is ever defined. That choice is not harmless. The boundary of M_{k+1}(B) includes strata where one component is a disk with a single marked point (a disk bubble). Those strata are responsible for the m_{k'}(..., m0(B''), ...) terms in the filtered A∞ relation. They are absent from Proposition 4.1 and from the proof of Theorem 4.4. So the theorem only establishes an uncurved A∞ structure, and only under an implicit unobstructedness assumption. For a Lagrangian torus with a nontrivial disk potential, m0 is nonzero and the claim as stated fails. This is the main problem.\n\nWhat is genuinely useful: the paper writes out the m_{k,B} operations, the sign ε, and the boundary orientation comparison for the polygon–polygon strata with care. Proposition 4.1's κ sign and Lemma 4.3's κ′ are the kind of bookkeeping that is easy to get wrong, and having them on record is valuable for anyone working in the de Rham model. The proof of the relation for the strata that are considered is coherent: equations (4.1)–(4.2) and (4.8) match.\n\nThe other caveats are more minor. The paper explicitly says it treats Kuranishi pushforwards as smooth submersions for sign purposes; for a sign paper this deserves more than a sentence, but it is a stated assumption, not a hidden one. And it depends on the unpublished Kuranishi construction [1] for the moduli spaces, which makes the statement conditional on that work. Neither of these is load-bearing if the m0 gap is fixed.\n\nBottom line: the signs for the m_k, k ≥ 1, are probably right, but the filtered A∞-category statement is not correct as written for general relatively spin Lagrangians. A serious referee should ask the author to either include the m0 terms and the disk-bubble strata, or restrict the theorem to an unobstructed collection and say so. The note could then be a useful reference. As it stands, I would not cite it for the filtered A∞-category claim.","headline":"Useful sign-bookkeeping for the de Rham Fukaya category, but it silently drops m0/disk-bubble terms, so the filtered A∞ theorem is only proven for unobstructed collections.","tokens_in":17219,"tokens_out":10976,"would_cite":false,"duration_ms":107416,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D40","53D37","58A99"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the filtered $A_{\\infty}$-relations hold for the Bott–Morse de Rham model once the sign convention of Definition 3.3 is fixed, and identifies the boundary sign $\\kappa$ that makes the cancellation work.","keywords":["filtered A∞-relation","Bott-Morse","de Rham model","Lagrangian Floer theory","sign convention","Kuranishi structure","bordered stable maps","relatively spin"],"falsifier":"Compute the $k=2$ filtered $A_{\\infty}$-relation in the simplest nontrivial Bott–Morse example: two relatively spin Lagrangian submanifolds intersecting cleanly in a circle, with a single holomorphic disk class, so that $\\mathcal{M}_3$ is a compact oriented manifold with boundary fibering over $S^1$; evaluating the left side of the relation on explicit bump forms supported in the interior of the clean intersection should give zero, and a nonzero result or a sign in disagreement with Proposition 4.1 would falsify the theorem.","tokens_in":16150,"feed_emoji":"♾️","tokens_out":13553,"duration_ms":123041,"temperature":0.7,"pith_summary":"The paper addresses a sign and orientation problem in Lagrangian Floer theory in the Bott–Morse setting, where Lagrangian submanifolds are allowed to intersect cleanly along positive-dimensional components rather than only at isolated points. The filtered $A_{\\infty}$-operations are defined by integrating differential forms along evaluation maps from moduli spaces of holomorphic polygons, and the $A_{\\infty}$-relation only holds if the signs are fixed with care. The author defines operations $m_k$ on the de Rham complexes of the clean-intersection components, with coefficients in certain $O(1)$-local systems, and proves (Theorem 4.4) that with the chosen sign $\\epsilon$ and boundary sign $\\kappa$ they satisfy $\\sum_{k'+k''=k+1} m_{k'}\\circ \\hat{m}_{k''}=0$ for all $k\\ge 1$. If correct, this fixes the sign convention needed for a well-defined filtered $A_{\\infty}$-category in the de Rham model.","feed_headline":"Filtered A∞-relations verified for Bott-Morse de Rham model","feed_subtitle":"A sign κ is pinned down so the de Rham-model operations mk satisfy the full filtered A∞-equation.","key_machinery":"The central object is the moduli space $\\mathcal{M}_{k+1}(B;\\mathbf{L};\\mathbf{R})$ of bordered stable maps of genus zero with connected boundary and $k+1$ boundary marked points, carrying a Kuranishi structure (a virtual manifold-chart system with obstruction bundles that supports integration along fibers). The argument is carried by sign bookkeeping with orientation bundles of fiber products: the relative orientation $o(\\sigma_{\\alpha_0};\\sigma_{\\alpha_1},\\ldots,\\sigma_{\\alpha_k})$ of the evaluation map $ev_0$ is compared with the orientation of the fiber product $\\mathcal{M}_{k'+1}\\times_{ev}\\mathcal{M}_{k''+1}$ appearing in the boundary of $\\mathcal{M}_{k+1}$. The factor $(-1)^\\kappa$ in Proposition 4.1 is the precise compensation for this comparison, and the factor $(-1)^\\epsilon$ in Definition 3.3 compensates the exchange of differential forms and local systems; together they make the Stokes boundary term cancel all composition terms.","core_discovery":"The central claim is that the naive push-forward definition, without the sign $\\epsilon$, would not satisfy the $A_{\\infty}$-relations, and the paper identifies exactly what sign is needed. Definition 3.3 sets $m_{k,B}(\\xi_1,\\ldots,\\xi_k)=(-1)^{\\epsilon(\\xi_1,\\ldots,\\xi_k)}(ev_0)_!(ev_1^*\\xi_1\\wedge\\cdots\\wedge ev_k^*\\xi_k)$, where $\\epsilon$ records each form's shifted degree and the Maslov indices of the relevant clean-intersection components. Theorem 4.4 states that the resulting operations satisfy the filtered $A_{\\infty}$-relation for every $k=1,2,\\ldots$. The proof is a sign-counting argument: Proposition 4.1 computes the sign $\\kappa$ that compares the orientation of the boundary of the moduli space with the fiber product of two smaller moduli spaces, Lemma 4.3 shows how compositions of operations carry that sign, and the Stokes-type formula of Proposition 2.5 turns the boundary term into the required cancellation.","pith_inferences":["A natural next step, not taken in the note, would be to carry out the same sign verification for filtered $A_{\\infty}$-bimodules and $A_{\\infty}$-functors in the de Rham model; the paper notes the bimodule construction only in the singular-chain model.","Because the signs $\\epsilon$ and $\\kappa$ are given by explicit formulas, the $k=2$ relation could be checked by computer algebra for a moduli space that is a smooth fiber bundle over the clean intersection, providing an independent test of the bookkeeping.","The paper's treatment of signs suggests that the same orientation comparison should govern the obstruction-bundle degrees in more general Kuranishi models; tracking those degrees explicitly could reveal whether the 'ignore CF-perturbations' reduction used here remains valid beyond the present setting."],"forward_implications":["For any finite collection of relatively spin Lagrangian submanifolds with clean intersections, the filtered $A_{\\infty}$-category in the de Rham model is well defined with the stated signs.","The same sign convention extends to clean self-intersections and, as the paper notes, to immersed Lagrangian submanifolds with clean self-intersections.","With the coefficient ring replaced by the universal Novikov ring $\\Lambda_{0,nov}$, the operations $m_k$ have degree $1$ and hence give a graded filtered $A_{\\infty}$-structure, as noted in Remark 3.4.","The $A_{\\infty}$-relations imply the usual chain-level identities, in particular that $m_1=d$ together with $m_2$ forms a differential graded algebra up to the specified signs, so the algebraic invariants of the Lagrangian intersection are well defined."],"supporting_citations":[{"why":"The companion construction, listed as in preparation, supplies the Kuranishi-structure (tree-like K-system) on moduli spaces of stable holomorphic polygons that the boundary decomposition in Proposition 4.1 presupposes.","marker":"[1]"},{"why":"Provides the tree-like K-system and filtered $A_{\\infty}$-functor framework cited alongside [1] as the source of the Kuranishi structures used in the note.","marker":"[3]"},{"why":"Supplies the orientation conventions for fiber products, the $O(1)$-local systems $\\Theta_{R_\\alpha}$, and the orientation of the two-point moduli space, which underlie Proposition 3.1 and the sign $\\kappa$ in Proposition 4.1.","marker":"[5]"},{"why":"Provides the Kuranishi-structure toolkit for integration along fibers—Stokes-type formula, base change, and composition formula (Propositions 2.4, 2.3, 2.6)—used in Lemma 4.2 and the final cancellation.","marker":"[6]"},{"why":"Gives the boundary description of holomorphic polygons in the Lagrangian immersion case, invoked to extend the sign and orientation argument to immersed Lagrangian submanifolds with clean self-intersections.","marker":"[2]"}],"fun_headline_variants":["Exact sign ensures A∞ relations in Bott-Morse de Rham","Missing sign κ found for filtered A∞ operations","A∞ equations hold in Bott-Morse after sign fix","Bott-Morse A∞ verified with correct sign ε","Sign convention resolved for de Rham A∞ in Floer"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that extra choices for integration over Kuranishi spaces (perturbations and obstruction-bundle data) do not affect signs, and that the Kuranishi structures from [1,3] exist as used; if either fails, the sign $\\kappa$ computed in Proposition 4.1 would not describe the actual moduli spaces and the filtered $A_{\\infty}$-relation could break.","fun_headline_variants_meta":{"raw":{"variants":["Exact sign ensures A∞ relations in Bott-Morse de Rham","Missing sign κ found for filtered A∞ operations","A∞ equations hold in Bott-Morse after sign fix","Bott-Morse A∞ verified with correct sign ε","Sign convention resolved for de Rham A∞ in Floer"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000283,"raw_usage":{"total_tokens":1606,"prompt_tokens":816,"completion_tokens":790,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":432,"completion_tokens_details":{"reasoning_tokens":706}},"tokens_in":432,"tokens_out":790,"duration_ms":7860,"temperature":1.0,"reasoning_tokens":706,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:27:12.649394+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $k=2$ filtered $A_{\\infty}$-relation in the simplest nontrivial Bott–Morse example: two relatively spin Lagrangian submanifolds intersecting cleanly in a circle, with a single holomorphic disk class, so that $\\mathcal{M}_3$ is a compact oriented manifold with boundary fibering over $S^1$; evaluating the left side of the relation on explicit bump forms supported in the interior of the clean intersection should give zero, and a nonzero result or a sign in disagreement with Proposition 4.1 would falsify the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The companion construction, listed as in preparation, supplies the Kuranishi-structure (tree-like K-system) on moduli spaces of stable holomorphic polygons that the boundary decomposition in Proposition 4.1 presupposes."},{"cited_title":"Part II, AMS/IP Stud","cited_arxiv_id":null,"evidence_quote":"Supplies the orientation conventions for fiber products, the $O(1)$-local systems $\\Theta_{R_\\alpha}$, and the orientation of the two-point moduli space, which underlie Proposition 3.1 and the sign $\\kappa$ in Proposition 4.1."},{"cited_title":"Math., Springer, Singapore, 2020","cited_arxiv_id":null,"evidence_quote":"Provides the Kuranishi-structure toolkit for integration along fibers—Stokes-type formula, base change, and composition formula (Propositions 2.4, 2.3, 2.6)—used in Lemma 4.2 and the final cancellation."},{"cited_title":"Immersed Lagrangian Floer Theory","cited_arxiv_id":"0803.0717","evidence_quote":"Gives the boundary description of holomorphic polygons in the Lagrangian immersion case, invoked to extend the sign and orientation argument to immersed Lagrangian submanifolds with clean self-intersections."}],"review_version":1}