{"id":"9da6677e-2d6b-4e9e-ad70-500276a69fc5","arxiv_id":"2411.13830","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A filtered A-infinity category, the Legendrian CI Fukaya category, is defined using moduli spaces of contact instantons with Reeb chord asymptotics.","lead":"This paper constructs a new A-infinity category for contact manifolds, whose objects are Legendrian submanifolds and whose algebraic operations are defined by counting contact instantons. If the construction is completed, it gives contact topology a Fukaya-category-style invariant that could link Legendrian dynamics to homological algebra.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The A-infinity relations in Theorem 16.1 depend on a compactified moduli space of genus-zero contact instantons whose construction is explicitly deferred to a sequel; until that compactness and gluing theory is supplied, Theorem 16.3 is not established.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: the existence of a well-behaved compactified moduli space with the correct virtual dimension, orientations, and gluing laws is not established in the text, and the paper itself defers the genus-zero compactification to a sequel. My reading of the full text confirms this. The algebraic framework is elaborate and the paper is explicit about many of its analytical dependencies, but the central theorem asserting the existence of the filtered A-infinity category is not self-contained. Because the structure maps and their A-infinity relations are defined by counts of these moduli spaces, the gap is not a technicality but a missing proof of the main object. I therefore agree with the reader's rejection. I considered whether a 'CONDITIONAL' or 'UNVERDICTED' verdict might be more appropriate given the framework's plausibility, but the paper presents Theorem 16.3 as a theorem, not as a conditional construction. Since the proof as written does not establish the theorem, rejecting the manuscript in its current form is the correct editorial verdict. The suggested concrete test—checking whether the cited compactness results actually cover the multi-puncture genus-zero case—would settle whether the gap is repairable by existing work or requires genuinely new analysis.","tokens_in":35673,"tokens_out":2567,"duration_ms":26726,"concrete_test":"Check the compactness theorems in [Oh21b] and [OY24] for whether any of them covers finite-energy contact instantons on a genus-zero punctured bordered Riemann surface with multiple boundary punctures, prescribed Reeb chord asymptotics, and a fixed energy bound. Specifically, verify whether they imply that a sequence of such instantons has a subsequence converging to a stable map whose boundary strata are broken configurations of contact instantons. If no such statement exists for disc-type domains with more than two punctures, the proof of Theorem 16.1 is missing a necessary analytic input, and the category of Theorem 16.3 should be regarded as conditional until that compactification and gluing theorem is written.","verdict_should_be":"REJECT","load_bearing_attack":"The central construction counts finite-energy contact instantons on punctured discs and uses those counts to define the structure maps m_k in (16.1). The A-infinity relations (16.2) are then asserted to follow from the boundary structure of the compactified moduli spaces. This is the load-bearing step, and it is not proved in the text. Proposition 10.2 and Theorem 16.1 invoke 'exactly the same proof as [FOOO09a]', but that proof requires a compactification of the moduli spaces M(E;γ;B) whose codimension-one boundary strata are broken configurations. The present paper does not establish this compactification: Section 5.1 states that the energy introduced there 'will be used in our construction of the compactification of moduli space of contact instantons of genus 0 in a sequel.' Lemma 15.1 and Theorem 15.2 cite [Oh21b] for compactness, but the cited results are developed for strip-like domains with small energy, not for the genus-zero, multi-puncture, Legendrian-boundary stable-map compactification with prescribed Reeb chord asymptotics needed here. Equation (15.2) for the virtual dimension is quoted from [OY24], and the coherent orientations in Theorems 14.3 and 14.4 are stated to be provable 'in the same way' as FOOO. Without an actual proof that the boundary of a one-dimensional moduli space is the union of products of lower moduli spaces, the counts in (16.1) are not known to be well-defined, and the A-infinity relation (16.2) has no geometric basis. This is a genuine gap in the central argument of the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to construct a filtered A-infinity category, called the Legendrian contact instanton (CI) Fukaya category, associated to any tame contact manifold. Objects are oriented Legendrian submanifolds equipped with a bounding cochain and a relative spin structure; morphism spaces are the chain complexes CI(R,R') generated by iso-speed Reeb chords; and the structure maps m_k are defined in (16.1) by counts of finite-energy contact instantons on punctured disc-type domains with prescribed Reeb chord asymptotics. The construction follows the FOOO blueprint: graded bridged Legendrian links provide gradings via a polygonal Maslov index, relative spin structures give coherent orientations, and abstract indices encode the gluing rules. The paper also defines a Legendrian CI DGA and its augmentations in Part 1, and then moves to the categorical construction in Part 2, culminating in Theorem 16.3.","tokens_in":35951,"tokens_out":3489,"duration_ms":34603,"significance":"If the construction were fully established, it would provide a Fukaya-type category generated by Legendrian submanifolds on every tame contact manifold, with morphisms given by Reeb chords and higher operations by contact instantons. This would be a substantial new structure unifying and extending contact instanton cohomology, Legendrian contact homology, and symplectic Fukaya categories. The paper is genuinely useful for its algebraic and combinatorial scaffolding: the abstract index formalism (Section 12), the polygonal Maslov index (Section 13), and the orientation package (Section 14) are carefully developed, and the paper is explicit about what it delegates to other sources. However, the central existence theorem is conditional on analytic compactness and gluing results that are not proved or cited from existing literature; the paper itself states that the genus-zero compactification is deferred to a sequel. As written, the category is not known to exist, and the main theorem is therefore not established.","major_comments":[{"comment":"Theorem 16.3, the main result, asserts the existence of a filtered A-infinity category whose structure maps are the operators in (16.1). These maps are counts of moduli spaces of contact instantons on punctured discs, and the A-infinity relation (16.2) is asserted to follow from the boundary structure of the compactified moduli spaces. However, the genus-zero compactification is not constructed in this paper. Section 5.1 states that the energy introduced there 'will be used in our construction of the compactification of moduli space of contact instantons of genus 0 in a sequel.' Until that compactification — including the codimension-one strata and gluing laws — is supplied, the counts in (16.1) are not known to be well-defined and the relation (16.2) has no geometric basis. This is a load-bearing gap in the central claim.","section":"§5.1, §16"},{"comment":"The proof of the A-infinity relation is delegated: Proposition 10.2 says that the compactification from [Oh21b] and 'exactly the same proof as [FOOO09a]' prove that m-circumflex composed with m-circumflex vanishes, and Theorem 16.1 says the same 'in the same way as the symplectic analog was proved in [FOOO10]'. The FOOO proof relies on a gluing analysis for pseudoholomorphic polygons with Lagrangian boundary conditions that produces the boundary of the one-dimensional moduli space as a union of fiber products of lower-dimensional moduli spaces. No such gluing theorem for contact instantons with Reeb chord asymptotics, disc bubbles, and sphere bubbles is proved or cited from an existing source. Lemma 15.1 cites [Oh21b] for compactness, but the paper itself describes [Oh21b] as the small-energy, no-bubbling case. This is not a harmless reference to a standard result; it is the missing analytical core of the construction.","section":"§10, Proposition 10.2 and §16, Theorem 16.1"},{"comment":"The virtual dimension formula (15.2), quoted from [OY24, Section 11.3], and the index formula Theorem 7.9 are used to identify the dimension-zero moduli spaces whose counts define m_k. Even granting these index formulas, the well-definedness of the counts requires that the relevant moduli spaces be compact, oriented, and cut out transversely for the full multi-puncture, genus-zero configuration space with prescribed Reeb chord asymptotics. The paper does not establish these properties; it refers to [Oha] and [Oh21b] for transversality and compactness, but those references do not cover the stable-map compactification with the boundary marked points and evaluation maps used in (16.1). Theorems 14.3 and 14.4, which supply coherent orientations, are likewise stated with proofs 'in the same way' as FOOO, without presenting the contact-specific argument needed for signs in the counts over Z.","section":"§15, Eq. (15.2) and §7, Theorem 7.9"},{"comment":"The gapped condition and the filtration used in the construction of CI(R,R') depend on the assertion in Lemma 15.1 that the set Γ_RR' is a (Γ,Γ')-set, which the lemma derives from the compactness theorem of [Oh21b]. Since the full compactified moduli spaces — including those with bubbles and with arbitrary numbers of positive and negative punctures — are not established in this paper, the gappedness of the structure maps is also not established. The abstract index framework of Section 12 is coherent on its own, but its applicability to the geometric counts in (16.1) is conditional on the same missing compactness and gluing input.","section":"§9, Definition 9.6 and §15, Lemma 15.1"}],"minor_comments":[{"comment":"There are several typos and small errors that should be corrected: 'Legencrianl' in Theorem 15.2, 'obatin' in the conclusion, 'dimension' for 'dimension' in Theorem 15.3, and the missing closing parenthesis in the expression 'C(R,R';ℓ;K[q,q^{-1})' at the start of Section 15.","section":"Throughout"},{"comment":"The statement that the newly defined energy will be used in a sequel for the genus-zero compactification should be moved to a prominent place in the introduction, since it is a central limitation of the present paper's main theorem.","section":"§5.1"},{"comment":"Remark 14.6 says that the moduli parameters of marked points and the automorphism group action are handled as in [FOOO09b, Section 8.3], but no explanation is given for why the contact case, with its Reeb chord asymptotics and the PSL(2,R) quotient on the disc, follows the same argument. A brief justification or a precise statement of the adapted result would help.","section":"§14, Remark 14.6"},{"comment":"The definition of the LCI-DGA differential in (11.1) implicitly assumes that the dimension-zero moduli spaces are compact, oriented, and carry a count over the chosen coefficient ring. Since the compactification is deferred to a sequel, the definition should be presented as conditional, or the missing analytic hypothesis should be stated explicitly.","section":"§11, Definition 11.2"}],"recommendation":"reject","confidential_remarks":"The paper relies heavily on the author's own prior preprints and works in preparation, including [KO], [Oha], [Ohb], and [AFO+], for the analytic foundations. For a paper whose central theorem is the existence of a category, it would be essential that the compactness and gluing results appear either in the same paper or in published, verifiable sources. As it stands, Theorem 16.3 is a conditional statement, and the missing analysis is not a local fix but a substantial separate development. This is the basis for my recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a program paper with a genuinely new categorical construction, but the central theorem is not proved in the text. The A-infinity relations depend on a compactified moduli space of genus-zero contact instantons whose construction is explicitly postponed to a sequel. If you read it as a complete proof of Theorem 16.3, it falls short. If you read it as a research announcement with the algebra laid out carefully and the analysis delegated to upcoming papers, it is coherent and honest.\n\nWhat is new: the idea of defining an A-infinity category whose objects are Legendrians, whose morphisms are Reeb chord complexes, and whose higher operations count contact instantons, is novel. The algebraic scaffolding—graded bridges, polygonal Maslov indices, abstract index systems, coherent orientations—is developed in real detail and is largely self-consistent. The DGA and augmentation story falls out naturally, and the paper is upfront about what is assumed. That transparency counts for something.\n\nThe soft spot is load-bearing. Section 5.1 states that the energy introduced there will be used for the genus-zero compactification in a sequel. Proposition 10.2 proves the A-infinity relation by saying \"exactly the same proof as FOOO\", and Theorem 16.1 does the same. But the analytic setting here is contact instantons with Reeb chord asymptotics, not pseudoholomorphic curves in a symplectic manifold. The virtual dimension formula (15.2) is quoted from a companion paper, and the orientations in Theorems 14.3 and 14.4 are \"proved in the same way\". None of this is obviously wrong, but the counts in (16.1) are not shown to be well-defined. If the compactification or gluing analysis fails, the category is not known to exist. The paper also relies on several in-preparation preprints, which makes independent verification difficult.\n\nProportionately: this is not a case of a lazy author hand-waving. The analytic foundations may well exist in the cited preprints. But the present manuscript is not self-contained, and the gap sits exactly at the point where the A-infinity relations are supposed to come from.\n\nWho is this for? Specialists in contact instantons and Fukaya categories who are willing to read the companion papers. A graduate student would not be able to fill in the missing pieces. The referee should be someone comfortable with both FOOO-style algebra and contact instanton analysis.\n\nRecommendation: send to peer review. A serious editor should not desk-reject a construction this new and this plausible. Ask the referees to check that the deferred analytics actually exist, and have the author state clearly which parts are proved here and which are promised elsewhere. If the gaps close, this could be a lasting framework.","headline":"A plausible but unfinished framework for Legendrian Fukaya categories; the analytic core is deferred to sequels, so the main theorem is not proved as written, but the paper deserves a serious referee.","tokens_in":36511,"tokens_out":2604,"would_cite":false,"duration_ms":25529,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D42","58J32"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims every tame contact manifold carries a filtered A-infinity category whose objects are Legendrian submanifolds, whose morphisms are Reeb chords, and whose structure maps count contact instantons.","keywords":["contact instantons","Legendrian submanifolds","Fukaya category","A-infinity category","Reeb chords","contact homology","bounding cochains","Maslov index"],"falsifier":"Enumerate all zero- and one-dimensional moduli spaces of contact instantons for two Legendrian unknots in the standard contact 3-sphere and check whether the $A_\\infty$ relation $m_1\\circ m_1=0$ (or its curved analogue) holds; a boundary component that is not a broken concatenation of lower moduli spaces, or any moduli space whose dimension differs from $\\mu(E,\\vec\\gamma;B)+n+k-2$, would disprove the construction.","tokens_in":35419,"feed_emoji":"","tokens_out":10201,"duration_ms":126576,"temperature":0.7,"pith_summary":"This paper aims to build a Fukaya-type category for contact manifolds, replacing pseudoholomorphic curves with contact instantons. The central claim is that every tame contact manifold carries a filtered $A_\\infty$ category whose objects are Legendrian submanifolds equipped with a bounding cochain and a relative spin structure, whose morphism spaces are generated by Reeb chords (flow lines of the Reeb vector field joining one Legendrian to another), and whose higher structure maps count finite-energy contact instantons on punctured discs with prescribed Reeb chord asymptotics. The $A_\\infty$ relations are meant to follow from the boundary structure of the one-dimensional compactified moduli spaces, in direct analogy with the symplectic Fukaya category. If the construction is correct, contact manifolds acquire an algebraic invariant of the same flavor as Lagrangian Floer theory, with Reeb chords playing the role of Lagrangian intersections.","feed_headline":"Contact manifolds get a Fukaya category built from Reeb chords","feed_subtitle":"If the construction holds, every tame contact manifold carries this Reeb-chord algebraic invariant.","key_machinery":"The load-bearing objects are finite-energy contact instantons: smooth maps $u$ from a punctured bordered Riemann surface into the contact manifold satisfying the contact Cauchy-Riemann equation $\\bar\\partial_\\pi u = 0$ together with $d(u^*\\lambda\\circ j)=0$, with boundary on a Legendrian link and Reeb chord asymptotics at the punctures. The grading is carried by the polygonal Maslov index $\\mu(E,\\vec\\gamma;B)$, defined as the Maslov index of a loop of Lagrangian subspaces in the contact distribution $\\xi$, which enters the virtual dimension formula (15.2). The algebraic packaging is the filtered $A_\\infty$ framework: a bar complex, bounding cochains solving the Maurer-Cartan equation, and coherent orientations induced by relative spin structures on the Legendrian submanifolds, with gluing rules that mirror the standard Fukaya category construction.","core_discovery":"The paper's central claim is that the whole filtered $A_\\infty$ machinery of Lagrangian Floer theory—curved $A_\\infty$ algebras, bounding cochains, energy filtrations, abstract indices, and coherent orientations—can be transplanted to the contact setting by replacing pseudoholomorphic discs with contact instantons and Lagrangian intersections with iso-speed Reeb chords. The morphism complex $CI(R,R')$ is freely generated by Reeb chords from $R$ to $R'$, graded by a polygonal Maslov index $\\mu(E,\\vec\\gamma;B)$, and the structure maps $m_k$ count zero-dimensional moduli spaces $M_{k+1}(E;\\vec\\gamma;B)$ of contact instantons, with virtual dimension given by $\\dim M(E,\\vec\\gamma;B) = \\mu(E,\\vec\\gamma;B) + n + k - 2$. Theorem 16.3 states that, after strictification using bounding cochains satisfying the Maurer-Cartan equation, this produces a filtered $A_\\infty$ category whose homology is claimed to be invariant under Hamiltonian isotopy.","pith_inferences":["The paper leaves implicit that, once the deferred compactification is supplied, this category should serve as the common home for previously studied invariants: the planned sequel explicitly ties it to Rabinowitz Fukaya categories and Lagrangian cobordism Floer theory on Liouville manifolds.","Because the action of a Reeb chord is single-valued, the energy filtration here is simpler than in the symplectic case; this suggests the category could support quantitative invariants of contact dynamics, such as spectral invariants, without Novikov ring complications.","A testable extension is to compute the category explicitly for the standard contact 3-sphere with Legendrian unknots, where Reeb chord counts are concrete and the $A_\\infty$ relations can be checked by hand."],"forward_implications":["Every tame contact manifold carries a filtered $A_\\infty$ category generated by its Legendrian submanifolds, with Reeb chords as morphisms and contact instantons as the higher operations.","Dualizing the $A_\\infty$ algebra yields a Legendrian contact instanton DGA (LCI-DGA) whose differential counts contact instantons, and whose augmentations define linearized contact instanton contact homology.","Bounding cochains on the objects deform the structure maps, and the strictified category's homology groups are claimed to be invariant under Hamiltonian isotopy of the Legendrian submanifolds.","For the 1-jet bundle of a closed manifold with the zero section as the Legendrian, the construction recovers the de Rham cohomology of the base manifold."],"supporting_citations":[{"why":"Supplies the compactness theorem for moduli spaces of contact instantons and the tame contact manifold framework that the whole construction relies on.","marker":"[Oh21b]"},{"why":"Provides the index formula and a priori estimates for contact instantons with Legendrian boundary condition, used for the grading and the virtual dimension formula.","marker":"[OY24]"},{"why":"The proof of the curved $A_\\infty$ relations and bounding cochain machinery is imported from this filtered Fukaya algebra construction.","marker":"[FOOO09a]"},{"why":"Supplies the abstract index formalism and the cocycle/orientation arguments used for the polygonal Maslov index and coherent orientations.","marker":"[FOOO10]"},{"why":"Gives the energy, bubbling, and Fredholm theory for contact Cauchy-Riemann maps that underlies the off-shell energy estimates in Section 5.","marker":"[Oh23]"},{"why":"Contributes the Fredholm theory and generic transversality for bordered contact instantons, including the generic nondegeneracy of Reeb chords.","marker":"[Oha]"}],"fun_headline_variants":["Reeb chords build a Fukaya category via contact instantons","Contact instantons give Legendrian Fukaya category","Tame contact manifolds get an A_infinity category from Reeb chords","Contact instantons yield A_infinity category for Legendrian links","From Reeb chords to a Fukaya category on contact manifolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The existence of a well-behaved compactified moduli space of finite-energy contact instantons on punctured discs, with prescribed Reeb chord asymptotics, the virtual dimension formula, coherent orientations, and the gluing laws needed for the $A_\\infty$ relations, is assumed rather than proved here, with the compactification explicitly deferred to a sequel.","fun_headline_variants_meta":{"raw":{"variants":["Reeb chords build a Fukaya category via contact instantons","Contact instantons give Legendrian Fukaya category","Tame contact manifolds get an A_infinity category from Reeb chords","Contact instantons yield A_infinity category for Legendrian links","From Reeb chords to a Fukaya category on contact manifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001275,"raw_usage":{"total_tokens":5207,"prompt_tokens":931,"completion_tokens":4276,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":4187}},"tokens_in":547,"tokens_out":4276,"duration_ms":26576,"temperature":1.0,"reasoning_tokens":4187,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:50:16.739129+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all zero- and one-dimensional moduli spaces of contact instantons for two Legendrian unknots in the standard contact 3-sphere and check whether the $A_\\infty$ relation $m_1\\circ m_1=0$ (or its curved analogue) holds; a boundary component that is not a broken concatenation of lower moduli spaces, or any moduli space whose dimension differs from $\\mu(E,\\vec\\gamma;B)+n+k-2$, would disprove the construction.","supporting_citations":[],"review_version":1}