{"id":"2ccefb35-62d6-4a05-b33d-4d5d4411026e","arxiv_id":"2508.20964","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Each closed exact Lagrangian gives a finite-dimensional dg-module over the Chekanov-Eliashberg algebra, and Lagrangian Floer cohomology is recovered as derived Hom between such modules.","lead":"The authors construct finite-dimensional representations of the Chekanov-Eliashberg algebra from closed exact Lagrangians in Weinstein manifolds, and show that Lagrangian Floer homology is the derived Hom between these representations. This extends a result of Ekholm and Lekili and is a step toward a full A-infinity functor from the compact Fukaya category.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Deferred immersed-cobordism Floer theory is load-bearing: Theorem 7.3 and Lemma 11.5 rely on it, and Section 6 leaves the details to future work.","rationale":"The reader’s weakest assumption correctly identifies the sketchy immersed-cobordism Floer theory of Section 6 as the load-bearing point. My independent pass finds the same concern, and no more serious or more specific flaw. The central chain of implications is: Theorem 8.1 deforms L0,L1 to immersed caps/fillings; Lemma 11.5 converts HF(L0,L1) into Cthulhu homology of caps using Theorem 7.3; Theorem 7.3 depends on the Section 6 complex and its unproved properties. Since the paper explicitly defers the full construction of this Floer theory, the proof of Theorem 1.2 is conditional on future work. The paper’s statement that the main result can be derived from existing generation and surgery results is reassuring but not a substitute for the proof presented here. Thus no change to the reader’s CONDITIONAL verdict is needed.","tokens_in":64460,"tokens_out":6583,"duration_ms":79663,"concrete_test":"Write out the proof of Theorem 7.3 in the special case of a single critical handle (e.g. W = T*S^n, n ≥ 3) with C0 and C1 standard caps in standard position, and verify the differential matrix (14) by directly computing the moduli-space counts in the stretched-neck limit, especially the composite terms d+0−∘d−+0 and d+−+∘d−+−. If these counts do not reproduce (14), the Section 6/7 theory is internally inconsistent; if they do, the remaining task is to supply the missing general compactness/gluing proof for immersed cobordisms before Lemma 11.5 is fully justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 6 introduces Floer theory for immersed exact Lagrangian cobordisms and states: “Our presentation will be rather sketchy, leaving the details to a future work.” This is not a peripheral gap. Lemma 11.5 identifies HF(L0,L1) with H Cth*(C0,C1) using Theorem 7.3, and Theorem 7.3 is proved from the Section 6 Cthulhu complex. The proof depends on several unproved assertions: d^2=0 for the Cthulhu differential with “tentacles” asymptotic to self-intersections, continuation/invariance under the regular exact homotopies of Theorem 8.1, and the stretched-neck chain homotopy in Lemma 7.8 whose differential matrix (14) is asserted rather than derived. In particular, Lemma 7.8 requires a two-level building decomposition for immersed boundary conditions with double points; the relevant SFT compactness and gluing are not written. If any of these fail—for example, if a broken configuration with multiple positive punctures is not canceled, or if the neck-stretching limit contributes additional terms—then Lemma 11.5 and hence Theorem 1.2 lack a proof. The authors note an alternative route via generation and surgery, but they do not carry it out in this paper, so the paper’s own argument stands or falls with Section 6.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves, at the cohomological level, a duality between closed exact Lagrangians in a Weinstein domain and finite-dimensional modules over the Chekanov–Eliashberg algebra of the attaching Legendrian link. To each closed exact Lagrangian L satisfying a transversality condition with all cocores it associates a dg module V_L over A_S, and for two such Lagrangians it claims an isomorphism HF^*(L_0,L_1) ≅ H^* Rhom_{A_S}(V_{L_0},V_{L_1}). The proof combines an algebraic short resolution of the diagonal bimodule over semi-projective dgas, a geometric deformation of L into an immersed Lagrangian L = Σ ∪ C consisting of a filling in the subcritical part and a cap made of multiple copies of the cores, and a neck-stretching comparison of the resulting Floer/Cthulhu complexes. A corollary gives strong restrictions on intersection numbers when A_S is Z-graded with H^{≤0}(A_S)=k_S.","tokens_in":64808,"tokens_out":4895,"duration_ms":57794,"significance":"If correct, the result is significant: it gives finite-dimensional representations of Chekanov–Eliashberg algebras from closed exact Lagrangians and recovers Lagrangian Floer cohomology as derived Hom between these representations, generalizing Ekholm–Lekili beyond the single-intersection case. The algebraic machinery—idempotent dgas, the short resolution, expansion/omission of idempotents—is developed carefully and is of independent interest. The paper also contains substantial geometric material in Appendices B and C on flow trees and SFT-type compactness for varying Legendrian boundary conditions, and it explicitly records an alternative route through generation and Legendrian surgery. The main weakness is that the proof of the central theorem depends on a Floer theory for immersed exact Lagrangian cobordisms whose details are postponed to a future paper, and several key chain-level statements are asserted rather than proved.","major_comments":[{"comment":"The central technical tool of the paper is the Cthulhu complex for immersed exact Lagrangian cobordisms. The section begins: “Our presentation will be rather sketchy, leaving the details to a future work.” This is not a peripheral issue: Lemma 11.5 identifies HF(L_0,L_1) with H Cth*(C_0,C_1) using Theorem 7.3, and Theorem 7.3 is proved from the Section 6 complex. In particular, the proof of d^2=0 for the differential with tentacles asymptotic to pure Reeb chords and self-intersections is not supplied; Lemma 5.7 gives only a one-paragraph negative-energy argument. The compactness, gluing, and cancellation of broken configurations with multiple positive punctures must be established for immersed boundary conditions with double points. The paper’s own statement admits that these details are deferred, so the proof of Theorem 1.2 is incomplete as it stands.","section":"Section 6, Eq. (11)–(13)"},{"comment":"The stretched-neck comparison is load-bearing for Theorem 7.3, but the differential matrix in Eq. (14) is asserted rather than derived. The proof describes which degenerations are expected, but it does not prove that all other configurations cancel, nor that the two-level buildings with a mixed positive puncture and tentacles are the only contributions. In particular, the entries d^+_+- ∘ d^-_+0 and d^+_+- ∘ d^-_+- require a gluing theorem for buildings with immersed boundary conditions and double points; no such theorem is stated or proved. Remark 7.4 also notes that the positivity action condition needs an invariance result that “we have proved so far” does not cover. Consequently Theorem 7.3, and hence Lemma 11.5, are not established.","section":"Section 7, Lemma 7.8 and Eq. (14)"},{"comment":"The computation of the cap algebra and of the Cthulhu complex relies on rigid counts of holomorphic discs: Lemmas 9.5–9.8 and Propositions C.7, C.10, C.11. These counts are justified by Theorem B.1 and Theorem C.3, but both theorems are proved only in sketched form. For example, Theorem B.1 assumes a “no nodal disc” condition that is not verified in the applications, and Theorem C.3 proves only “partial SFT convergence” and explicitly ignores gradient-flow limit components. The paper states these are technical matters, but they are needed to identify D_C with A^+_/C and to compute Cth(C_0,C_1). Without a complete proof of these count identities, the definition of V_L and the isomorphism of Lemma 11.6 are not fully justified.","section":"Section 9 and Appendix C"},{"comment":"The passage from the original closed Lagrangians L_0,L_1 to the immersed decompositions L_i = Σ_i ∪ C_i uses regular exact homotopies whose Legendrian lifts are isotopic. The proof says Cthulhu homology for closed Lagrangians is Floer homology and is invariant under such homotopies by [7, Section 4.4]. However, the objects here are immersed and the homotopy is not shown to preserve the Cthulhu complex or the augmentations ε_{Σ_i}. Moreover, the identification H Cth*(Σ_0,Σ_1) ≅ HW^*((Σ_0,ε_0),(Σ_1,ε_1)) is imported from [24, Appendix B.1.1] in a setting with immersed fillings and augmentations, but no verification is provided that the hypotheses of that result hold. This is another load-bearing step in the proof of Theorem 1.2.","section":"Section 11, Lemma 11.5"}],"minor_comments":[{"comment":"“for two any such Lagrangian submanifolds” should read “for any two such Lagrangian submanifolds”; “out techniques” is a typo for “our techniques”.","section":"Abstract and Introduction"},{"comment":"“surgective” should be “surjective” (Lemma 2.5); “unnatrural” should be “unnatural”. In the proof of Lemma 3.1, the text says “it remains to prove that µ is injective”, but µ is the multiplication map and is not injective; the intended statement is that ι is injective and ker(µ) ⊂ Im(ι).","section":"Section 2–3"},{"comment":"In Definition 5.3, “∂− + M c” appears to be a typo. Section 6 has “rater sketchy” for “rather sketchy”. In Section 5, the proof of Lemma 5.13 says “details are left for the reader”; this is fine as a remark, but the diagram in Figure 1 is not enough to verify the claim that the map is a chain map.","section":"Section 5–6"},{"comment":"The manuscript contains the text “profile.{ps,eps,pdf} not found (or no BBox)” instead of an actual figure. This must be fixed before publication.","section":"Section 8, Figure 3"},{"comment":"There are several typos in technical terms: “Checkanov” for “Chekanov” (Section 9), “criitical” for “critical” (Appendix A), “cobnordisms” for “cobordisms”. These do not affect the mathematics but should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"This is a paper with strong ideas and a substantial amount of correct-looking algebra and geometry, but the main theorem is not proven in the present text because the immersed Lagrangian cobordism Floer theory of Section 6 is deferred. The authors mention an alternative route via geometric generation and Legendrian surgery, but they do not carry it out, so the paper’s own argument must stand on Section 6. I would be willing to review a revised version that either supplies the deferred Floer-theoretic details or restricts the main theorem to the cases where those details are already available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper openly says its main theorem is not new and sketches how to derive it from existing generation and surgery results. What is actually new is the route: a generalization of Ekholm–Lekili to arbitrary transverse intersection multiplicities, via an immersed version of the Cthulhu Floer theory, a relative exact triangle, and computations of the cobordism algebra of multiple parallel cores. The authors are upfront that Section 6 is sketchy and details are deferred to future work.\n\nCredit where it is due: the paper is well organized, the alternative derivation in the introduction is credible, and the cap algebra computations in Sections 9–10 plus Appendix C are substantial. The multiple-copy dg algebra relation and the use of SFT compactness for varying boundary conditions are real pieces of work. The authors also clearly state what they have and have not proved, which is rarer than it should be.\n\nThe soft spot is exactly where the stress-test puts it: the load-bearing immersed Floer theory is not actually proved here. Section 6 says the presentation is 'rather sketchy, leaving the details to a future work.' Lemma 7.8's stretched-neck differential matrix (14) is asserted rather than derived; the relevant SFT compactness and gluing for immersed boundary conditions with double points are not written. Lemma 11.5 identifies HF(L0,L1) with Cthulhu homology using Theorem 7.3, so if any of the asserted properties fail, the proof of Theorem 1.2 as presented lacks support. The authors note an alternative route via generation and surgery but do not carry it out, so the paper's own argument stands or falls with Section 6. That said, the failure mode is 'deferred proof,' not circular reasoning or fitted parameters. The central claim is very likely true, and I found no internal contradiction or sign of intellectual dishonesty.\n\nWho this is for: people working on Chekanov–Eliashberg algebras, wrapped Fukaya categories, and Lagrangian cobordisms. A reader willing to accept the sketched Floer theory will get a clear picture of the strategy and useful algebraic machinery. It deserves a serious referee: the gaps are large but explicitly flagged, and the paper prepares the ground for the announced A-infinity functor. I would send it to review, with the expectation that the referee will push for either a full treatment of Section 6 or a statement that the proof relies on the alternative derivation.","headline":"An honest, technically rich paper whose advertised proof leans on a deferred Floer theory; the central claim is likely right but the paper's own argument is not yet complete.","tokens_in":65244,"tokens_out":2022,"would_cite":false,"duration_ms":24340,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D37","53D40","57R17"],"pacs":[],"model":"deepseek-v4-flash","headline":"Closed exact Lagrangians in a Weinstein domain give finite-dimensional modules over the Chekanov–Eliashberg algebra, and Lagrangian Floer cohomology is the derived Hom of these modules.","keywords":["Chekanov–Eliashberg algebra","Weinstein manifold","exact Lagrangian submanifold","Lagrangian Floer cohomology","dg-modules","Lagrangian cobordism","Cthulhu homology","augmentations"],"falsifier":"Take any closed exact Lagrangian sphere L in a Weinstein domain and compute the self-Floer group HF(L,L) ≅ H*(L). The paper predicts H*Rhom_AS(VL,VL) ≅ H*(L); computing V_L from the recipe and checking that the derived endomorphism homology has the Poincaré polynomial of L would confirm or refute the theorem.","tokens_in":64371,"feed_emoji":"🔗","tokens_out":11980,"duration_ms":118074,"temperature":0.7,"pith_summary":"This paper bridges two ways of organising symplectic information for a Weinstein manifold: the Floer-theoretic invariants of closed exact Lagrangians and the Chekanov–Eliashberg (CE) differential graded algebra of the Legendrian spheres along which critical handles are attached. Its main theorem associates to each such Lagrangian L a finite-dimensional dg-module V_L over the CE algebra, with the dimension and Euler characteristic of each idempotent part dictated by the intersections of L with the corresponding cocore. For two Lagrangians, it proves that Lagrangian Floer cohomology HF(L0,L1) is isomorphic to the derived Hom H*Rhom(V_L0,V_L1). If correct, this is the object-level and morphism-level first half of a fully faithful embedding of the compact Fukaya category into the derived category of finite-dimensional CE-modules. The payoff is that a priori intricate holomorphic-curve invariants of Lagrangians become explicit finite-dimensional algebraic objects built from the handle attachment.","feed_headline":"Lagrangian Floer cohomology becomes derived Hom of one algebra","feed_subtitle":"Closed exact Lagrangians become finite-dimensional modules over the Chekanov–Eliashberg algebra of the attaching link.","key_machinery":"Mechanism: deform L to an immersed exact Lagrangian C∪Σ, with C a standard cap of parallel perturbed critical cores and Σ an immersed filling of its Legendrian boundary. The cap algebra D_C has generators the self-intersections of the parallel cores and the Reeb chords of the boundary link; gradient flow trees compute its differential. The filling yields an augmentation of D_C. Three algebraic operations — minimal morsification, expansion, and omission of idempotents — convert that augmentation into a finite-dimensional dg-module V_L over the Chekanov–Eliashberg algebra A_S. A neck-stretching relative exact triangle compares Floer complexes of two caps, and the short-resolution bimodule of A","core_discovery":"Central claim: to each closed exact Lagrangian L meeting all critical cocores transversely, associate a finite-dimensional dg-module V_L over the Chekanov–Eliashberg algebra A_S of the attaching link S. Its σ-part has dimension |L∩D_σ| and Euler characteristic L•D_σ; for two Lagrangians, HF*(L0,L1) ≅ H*Rhom_{A_S}(V_L0,V_L1). The module is produced by deforming L to an immersed Lagrangian of parallel perturbed cores plus an immersed filling, whose augmentation of the cap algebra is converted by algebraic operations on idempotent dg-algebras into a module over A_S. The isomorphism follows by neck-stretching and a relative exact triangle identifying the resulting complex with the short resoluti","pith_inferences":["The recipe suggests concrete computations: in a domain where A_S is known, HF(L0,L1) can be computed from the intersection numbers |L_i∩D_σ| alone, without building full holomorphic-curve moduli spaces in the ambient manifold.","Not pursued here: the same scheme may extend to non-exact or monotone Lagrangians if a curved version of the CE algebra is used; the paper works only over exact settings.","The relative exact triangle for concatenations could be iterated along a handle decomposition, giving a Mayer–Vietoris-style algorithm for Floer groups in multi-handle Weinstein domains."],"forward_implications":["Lagrangian Floer cohomology in a Weinstein domain becomes an algebraic computation from the attaching-link data and the cocore intersection pattern.","Finite-dimensionality plus the dimension and Euler-characteristic formulas constrain the Floer groups and Euler characteristics of closed exact Lagrangians.","Under the homological-smallness hypothesis on A_S, each closed connected Maslov-zero exact Lagrangian has primitive homology class and intersects every cocore in at most one point, up to sign.","With the announced sequel, the compact Fukaya category embeds cohomologically fully faithfully into the derived category of finite-dimensional A_S-modules."],"supporting_citations":[{"why":"Provides the Lagrangian-cobordism Floer theory (Cthulhu homology) whose immersed version and relative exact triangle carry the proof.","marker":"[6]"},{"why":"Establishes the single-point-intersection case and supplies the identification of the Cthulhu complex with wrapped Floer homology used to discard the filling term.","marker":"[24]"},{"why":"Gives the vanishing of wrapped Floer homology in subcritical Weinstein domains used in Lemma 11.5.","marker":"[7]"},{"why":"Supplies gradient flow trees as the tool for computing the cap algebra differential via rigid holomorphic discs.","marker":"[15]"},{"why":"Extends gradient flow-tree computations to non-compact cobordisms through the morsification construction needed in Appendix B.","marker":"[23]"},{"why":"Provides the short resolution of the diagonal bimodule used to compute derived Hom of dg-modules.","marker":"[31]"},{"why":"Supplies the regularity of anchored holomorphic discs underlying the curve counts used throughout.","marker":"[25]"},{"why":"Is the earlier correspondence between augmentations and higher-dimensional representations that the construction of V_L from a filling generalises.","marker":"[35]"}],"fun_headline_variants":["Exact Lagrangians become modules over CE algebra","Floer cohomology as derived Hom of CE modules","Closed exact Lagrangians yield CE algebra modules","Floer theory via modules over Chekanov-Eliashberg","Finite-dimensional CE modules from exact Lagrangians"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof leans on a promised Floer theory for immersed exact Lagrangian cobordisms that is only sketched here and deferred to a later paper; if any of its stated properties fail, the main isomorphism lacks a proof.","fun_headline_variants_meta":{"raw":{"variants":["Exact Lagrangians become modules over CE algebra","Floer cohomology as derived Hom of CE modules","Closed exact Lagrangians yield CE algebra modules","Floer theory via modules over Chekanov-Eliashberg","Finite-dimensional CE modules from exact Lagrangians"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000657,"raw_usage":{"total_tokens":2835,"prompt_tokens":724,"completion_tokens":2111,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":2030}},"tokens_in":468,"tokens_out":2111,"duration_ms":15250,"temperature":1.0,"reasoning_tokens":2030,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:39:59.043681+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any closed exact Lagrangian sphere L in a Weinstein domain and compute the self-Floer group HF(L,L) ≅ H*(L). The paper predicts H*Rhom_AS(VL,VL) ≅ H*(L); computing V_L from the recipe and checking that the derived endomorphism homology has the Poincaré polynomial of L would confirm or refute the theorem.","supporting_citations":[{"cited_title":"Chantraine, G","cited_arxiv_id":null,"evidence_quote":"Provides the Lagrangian-cobordism Floer theory (Cthulhu homology) whose immersed version and relative exact triangle carry the proof."},{"cited_title":"Ekholm and Y","cited_arxiv_id":null,"evidence_quote":"Establishes the single-point-intersection case and supplies the identification of the Cthulhu complex with wrapped Floer homology used to discard the filling term."},{"cited_title":"Geometric generation of the wrapped Fukaya category of Weinstein mani- folds and sectors","cited_arxiv_id":null,"evidence_quote":"Gives the vanishing of wrapped Floer homology in subcritical Weinstein domains used in Lemma 11.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies gradient flow trees as the tool for computing the cap algebra differential via rigid holomorphic discs."},{"cited_title":"Ekholm, K","cited_arxiv_id":null,"evidence_quote":"Extends gradient flow-tree computations to non-compact cobordisms through the morsification construction needed in Appendix B."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the short resolution of the diagonal bimodule used to compute derived Hom of dg-modules."},{"cited_title":"Holomorphic curves for Legendrian surgery","cited_arxiv_id":"1906.07228","evidence_quote":"Supplies the regularity of anchored holomorphic discs underlying the curve counts used throughout."},{"cited_title":"Ng and D","cited_arxiv_id":null,"evidence_quote":"Is the earlier correspondence between augmentations and higher-dimensional representations that the construction of V_L from a filling generalises."}],"review_version":1}