{"id":"3643f2db-d973-4969-a4a9-a2834db4521b","arxiv_id":"2501.11158","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Pre-Calabi-Yau extensions of A∞ algebras induce actions of a newly constructed dg properad of oriented ribbon graphs on cyclic Hochschild cohomology, whose cohomology matches compactly supported cohomology of moduli spaces of curves.","lead":"This paper introduces a new algebraic object built from directed ribbon graphs, the oriented gravity properad, and shows it acts on the cyclic cohomology of pre-Calabi-Yau algebras. If correct, it provides a new combinatorial model for the cohomology of moduli spaces of algebraic curves, linking noncommutative algebra to geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central proof is carried out in an interpolating properad whose differential δ = δ• + D_γ is only asserted to square to zero; Theorem 1.3 is conditional on that unshown identity.","rationale":"The reader's conditional verdict is appropriate. The main theorem is a cohomology computation whose only new ingredient is the interpolating properad ORGraphs_{d,d+1}, and both quasi-isomorphism propositions depend on the same asserted differential. The omitted verification of δ•^2 = 0 and Lemma 4.1.1 is therefore genuinely load-bearing rather than a cosmetic gap. The rest of the comparison with RGraphs^d is plausible and the intended spectral sequence strategy is standard, but as written the proof is incomplete. I do not see an independent internal contradiction in the statement of Theorem 1.3 itself; the concern is about the proof's foundation. Since the reader already assigned CONDITIONAL with low confidence, the stress-test does not move the verdict.","tokens_in":16690,"tokens_out":16599,"duration_ms":173003,"concrete_test":"Independently expand δ^2 on the generators of ORGraphs_{d,d+1}(m,n) for small m,n (e.g. m+n ≤ 4) using the explicit formulas for δ• and D_γ: verify δ•^2 = 0 and Lemma 4.1.1, then verify the full identity δ^2 = δ•^2 + δ•D_γ + D_γδ• + D_{γ∘_1γ} = 0 on the same generators. If the identity fails, Theorem 1.3 has no interpolating dg properad. If it passes, the remaining check is to compute the homology of gr ker π1 including properadic composites that no longer contain an unlabelled target; a non-acyclic contribution there would invalidate Proposition 4.3 as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.1 defines the dg properad (ORGraphs_{d,d+1}, δ) with δ = δ• + D_γ. The square-zero property δ•^2 = 0 is dismissed as 'a straightforward but tedious calculation' (p. 11), and Lemma 4.1.1, δ•γ + γ∘_1γ = 0, is stated with proof deferred to [MWW]. This is load-bearing: δ^2 = 0 is equivalent to δ•^2 = 0 together with the cross-identities δ•D_γ + D_γδ• = 0 and D_{γ∘_1γ} = 0, the last being exactly Lemma 4.1.1. If any sign or coefficient in the colour-changing term δv^{◦→•}, in the k-edge chain γ, or in the γ-composition is wrong, the interpolating object is not even a complex; then Propositions 4.3 and 4.4 cannot be quasi-isomorphisms of dg properads and Main Theorem 1.3 has no proof. The paper supplies no computation for this check, and [MWW] is an unpublished preprint (arXiv:2411.19657). A secondary gap is that the acyclicity proof of Proposition 4.3 asserts that every graph in the associated graded complex has a vertex of type (ii); properadic composites can replace an unlabelled target by a graph with no unlabelled targets, so this step also needs independent justification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a dg properad ORGraphs^d of oriented ribbon graphs and studies its actions and cohomology. Theorem 1.1 asserts that any degree-d pre-Calabi-Yau extension of an A-infinity structure induces an action of ORGraphs^d on the dual cyclic Hochschild complex. Proposition 3.5.1 gives a morphism from the minimal resolution Holie^d to ORGraphs^{d+1}, yielding Corollary 1.2. The main theorem (Theorem 1.3) computes H^•(ORGraphs^{d+1}(m,n)) as H^•(RGraphs^d(m,n)), which by [C, M1] equals the compactly supported cohomology of moduli spaces M_{g,m+n}. The proof uses an interpolating properad ORGraphs_{d,d+1} with two differentials and two quotient maps to ORGraphs^{d+1} and RGraphs^d.","tokens_in":17131,"tokens_out":3426,"duration_ms":31661,"significance":"If Theorem 1.3 is correct, the paper provides a new combinatorial model for the totality of compactly supported cohomology groups of moduli spaces of algebraic curves, and connects pre-Calabi-Yau structures to cyclic cohomology and the gravity operad. The constructions are explicit and the paper is well-organized, with several detailed proofs such as Lemma 3.2.1 and Proposition 3.4.1. However, the central proof of Theorem 1.3 rests on several asserted but unverified computations and on citations to an unpublished preprint, so the result is presently conditional. The potential significance is high: it would extend the properadic approach to moduli spaces and offer a new setting for actions on cyclic Hochschild complexes.","major_comments":[{"comment":"The differential δ = δ• + D_γ on ORGraphs_{d,d+1} is the backbone of the proof of Theorem 1.3. The identity δ•^2 = 0 is dismissed as 'a straightforward but tedious calculation', and Lemma 4.1.1, which gives the cross-identity δ•γ + γ ∘_1 γ = 0, is stated with proof deferred to the unpublished preprint [MWW]. These identities are load-bearing: without them, (ORGraphs_{d,d+1}, δ) is not a dg properad, and Propositions 4.3 and 4.4 cannot be quasi-isomorphisms of dg properads. The manuscript should supply these computations or give a complete, publicly accessible proof.","section":"§4.1, displayed equation after Lemma 4.1.1"},{"comment":"The acyclicity argument for ker π1 considers the associated graded complex gr ker π1^marked and claims that 'Any generating graph Γ has least one vertex of type (ii)'. This claim is used to conclude that each tensor factor CΓ has an acyclic factor. However, properadic composites of graphs can replace an unlabelled target by a graph with no unlabelled targets, so it is not clear that every graph occurring in the associated graded has a vertex of type (ii). This step needs an independent justification, as it is essential for the quasi-isomorphism ORGraphs_{d,d+1} → ORGraphs^{d+1}.","section":"§4.3, Proof of Proposition 4.3"},{"comment":"The morphism Holie^d → ORGraphs^{d+1} is the basis for Corollary 1.2, but its proof is omitted entirely, described only as 'a straightforward calculation'. Since the map is given by an explicit sum with signs, verifying that it commutes with the differentials is a non-trivial check. The reader cannot verify this claim from the text, and the result is used in the main narrative. The author should provide the calculation or a detailed reference.","section":"§3.5, Proposition 3.5.1"},{"comment":"The proofs of Propositions 4.3 and 4.4 defer crucial acyclicity claims to [MWW] (an unpublished preprint) and to [M3]. Specifically, the proof of Proposition 4.3 says 'cf. §4.1 in [MWW]' and Proposition 4.4 says 'cf. Lemma 6.2.3 in [M3]', while Lemma 4.1.1 is deferred to [MWW]. This makes the proof of Theorem 1.3 not self-contained and unverifiable in its present form. The author should either prove these claims in the paper or state precisely which published arguments imply them.","section":"§4.3 and §4.4"}],"minor_comments":[{"comment":"The notation is inconsistent: the text uses both RGraphsor_d and RGraphs^or_d for the same object, and similarly ORGraphs^d versus ORGraphsd in different sections. A unified notation would improve readability.","section":"§3.2"},{"comment":"The statement 'It is easy to check that ORGraphsd is a dg sub-properad' would benefit from a short verification, since closure under the differential and compositions is not obvious given the quotient by the ideal of black targets.","section":"§3.4"},{"comment":"In the infinite-dimensional case, the action is described only 'along the same lines' with substitution of outputs into inputs. A more explicit description of the operation on generators would help the reader verify the claims.","section":"§3.2.1, Proof of Lemma 3.2.1"},{"comment":"There are numerous typographical errors and formatting artifacts in the arXiv text, such as 'RG raphsor', missing spaces, and garbled diagrams. These should be corrected in the final version.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the paper is clearly written in its overall structure, but the proof currently depends on several omitted computations and on [MWW], an unpublished preprint. I recommend asking the author to provide the missing calculations—especially δ•^2 = 0, Lemma 4.1.1, and the acyclicity arguments in Propositions 4.3 and 4.4—or to point to a complete published source. Without these, the central result cannot be verified by the reader."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper constructs ORGraphs^d, a directed ribbon graph properad, proves it acts on the dual cyclic Hochschild complex of any pre-CY algebra, and identifies its cohomology with compactly supported cohomology of M_{g,m+n}. If the main theorem holds, this is a genuinely new combinatorial model and a clean cyclic analogue of [KTV]. The action theorem (1.1) is proven in reasonable detail; the Holie morphism is plausible; the gravity-algebra corollary is a nice bonus.\n\nThe soft spot is exactly where the reader and stress-test put it: Section 4. The proof of Theorem 1.3 goes through an interpolating properad ORGraphs_{d,d+1} with differential δ = δ• + D_γ. Two identities are load-bearing: δ•^2 = 0 and δ•γ + γ∘_1γ = 0. Both are dismissed as \"straightforward\" and the second is deferred to [MWW], an unpublished preprint. Since δ^2 = 0 splits exactly into those identities, the main theorem is conditional on computations the paper does not contain. That is a real incompleteness, not a manufactured one.\n\nI would also flag Proposition 4.3's acyclicity argument. The proof decomposes the associated graded complex into factors C_v and says every graph has a vertex of type (ii) because there are no closed directed paths. That is true for the generators, but after properadic composition the property can fail; the argument needs a justification it does not give. The same style of appeal to [M3] and [MWW] appears in 4.4. None of this makes me think the theorem is false—the strategy is coherent and the cited results are the right tools—but as written the central proof is a series of deferred checks.\n\nOn circularity: I disagree with any charge that the argument is circular. The comparison is to independently known cohomology of RGraphs^d, and self-citations are used appropriately. The citation pattern is normal for this subfield.\n\nWho this is for: people working on operadic models of moduli spaces, pre-Calabi-Yau algebras, and graph complexes. A specialist referee can probably check Section 4 in a few days; the gaps are tedious-computation-shaped, not conceptual. I would send it to review, and ask the referee to verify δ•^2 = 0 and Lemma 4.1.1 explicitly. I would not cite Theorem 1.3 as established until that check is public.","headline":"A plausible and genuinely new oriented ribbon graph properad, but the proof of the main cohomology theorem is carried by omitted computations and an unpublished preprint; worth refereeing, not yet citable as established.","tokens_in":17575,"tokens_out":2604,"would_cite":false,"duration_ms":23839,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E40","14H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The oriented gravity properad is a new combinatorial model for the cohomology of moduli spaces of algebraic curves.","keywords":["pre-Calabi-Yau algebras","oriented ribbon graphs","properads","moduli spaces of algebraic curves","cyclic Hochschild cohomology","gravity operad","A-infinity algebras","graph complexes"],"falsifier":"A direct computation of δ•^2 on a small generator of ORGraphs_{d,d+1}, for instance the graph with one labelled boundary, one labelled white vertex, and one unlabelled black vertex, would either confirm or refute the omitted square-zero calculation; if a nonzero value appears, Propositions 4.3 and 4.4 cannot both hold, and the main theorem fails.","tokens_in":16515,"feed_emoji":"🔗","tokens_out":5897,"duration_ms":53870,"temperature":0.7,"pith_summary":"This paper proves that a differential graded properad built from oriented ribbon graphs, called the oriented gravity properad, has cohomology equal to that of the ordinary ribbon graph properad, and therefore computes the compactly supported cohomology of moduli spaces of algebraic curves with marked points. It also proves that any pre-Calabi-Yau extension of an A-infinity algebra structure makes the dual cyclic Hochschild complex into a module over this properad, and that the minimal resolution of the Lie operad acts on it. The result matters because it connects noncommutative deformation theory to the geometry of curve moduli and gives a purely combinatorial, graph-theoretic replacement for those moduli cohomology spaces.","feed_headline":"Oriented ribbon graphs encode curve moduli cohomology","feed_subtitle":"A new graph properad gives pre-Calabi-Yau algebras control over cyclic Hochschild cohomology.","key_machinery":"The central object is the dg properad ORGraphs^d (and its unshifted predecessor RGraphs^d): a properad of connected ribbon graphs with directed edges, labelled boundaries, labelled white vertices, and unlabelled black vertices, with vertices carrying cohomological degree d and edges degree 1−d. The differential is the standard splitting-of-vertices operation, with certain target vertices killed by passing to an ideal. The argument that carries the main theorem is the interpolating properad ORGraphs_{d,d+1}, which allows both unlabelled white and black vertices; Proposition 4.3 shows the quotient killing unlabelled white vertices is quasi-isomorphic to $ORGraphs^{{d+1}}$, and Proposition 4.4 shows the quotient killing essential black vertices is quasi-isomorphic to RGraphs^d. These two quasi-isomorphisms, proved by filtrations and the acyclicity of elementary two-term complexes, identify the cohomology of the oriented properad with that of the classical ribbon graph complex.","core_discovery":"Main Theorem 1.3 states that for any integers d, m ≥ 1, n ≥ 1, the cohomology of the oriented ribbon graph properad $ORGraphs^{{d+1}}$(m,n) is isomorphic to H•(RGraphs^d(m,n)), which in turn equals the product of compactly supported cohomology groups ∏_{g≥0, 2g+m+n≥3} $H_c^{{•-m+d(2g-2+m+n)}}$(M_{g,m+n}). In words, the new properad of directed ribbon graphs with no closed directed cycles computes the same compactly supported cohomology of moduli spaces M_{g,m+n} as the ordinary ribbon graph complex, up to a degree shift. Theorem 1.1 shows that a degree d pre-Calabi-Yau extension π of an A∞-structure μ on a graded vector space A induces a natural action of ORGraphs^d on the dual cyclic Hochschild complex Cyc•(A,K), and Proposition 3.5.1 supplies a nontrivial morphism from the minimal resolution Holie^d into $ORGraphs^{{d+1}}$, making Cyc•(A,K) a Holie^d-algebra by Corollary 1.2. The paper also proves that the higher Hochschild cohomology of any pre-CY algebra is naturally an algebra over the gravity operad.","pith_inferences":["Editorial inference: The interpolation argument suggests a general pattern—changing vertex colour and degree in a graph complex can be bridged by a two-colour complex whose quasi-isomorphisms reduce the computation to known complexes; the same scheme may apply to other graph-complex comparisons.","Editorial inference: Because the paper works with coinvariants under cyclic groups, an analogous construction for invariants or for non-cyclic Hochschild complexes might produce oriented properads acting on other natural cohomology theories.","Editorial inference: The explicit verification of the square-zero differential on the interpolating properad is the most direct spot to test the proof; a small computer algebra check on low genus graphs could confirm or refute the asserted acyclicity before relying on the unpublished claims."],"forward_implications":["The cohomology of ORGraphs^{d+1}(m,n) is known explicitly as the product of compactly supported cohomology groups of moduli spaces M_{g,m+n} with the stated degree shifts.","Every pre-Calabi-Yau extension of an A∞ structure gives a representation of ORGraphs^d on the dual cyclic Hochschild complex, so cyclic cohomology carries these graph operations.","Degree (d+1) pre-Calabi-Yau extensions make the dual cyclic Hochschild complex into a Holie^d-algebra, extending the usual Lie structure to all higher operations.","The higher Hochschild cohomology of any pre-CY algebra is naturally an algebra over the gravity operad, with the binary operation coming from the standard Lie bracket.","The genus-zero part of the cohomology recovers Getzler's gravity operad, so the construction is a full properadic extension of that classical structure."],"supporting_citations":[{"why":"Introduces the properad RGrad of ribbon graphs and its representation on cyclic words, which is the base construction.","marker":"[MW]"},{"why":"Supplies the twisting endofunctor used to build the dg properads twRGrad, RGraphs^d, and RGraphsor^d.","marker":"[Wi1]"},{"why":"Computes the cohomology of RGraphs^d(m,n) and introduces the gravity properad; the main theorem reduces to this computation.","marker":"[M1]"},{"why":"Provides the construction of moduli spaces of nodal disks used to identify ribbon graph cohomology with compactly supported cohomology of M_{g,m+n}.","marker":"[C]"},{"why":"The degree-shifted ribbon graph complex with marked boundaries is the classical Penner complex, giving the geometric identification for n = 0.","marker":"[P]"},{"why":"Gives the standard reference for the ribbon graph complex and its cohomology identification.","marker":"[K]"},{"why":"Supplies acyclicity claims used in the spectral sequence arguments of Propositions 4.3 and 4.4.","marker":"[M3]"},{"why":"An unpublished preprint that contains key acyclicity claims on which the interpolating properad proof relies.","marker":"[MWW]"},{"why":"Establishes the isomorphism of the genus-zero ribbon tree complex with the gravity operad, used in Lemma 3.3.1.","marker":"[Wa]"},{"why":"Introduces the gravity operad that the genus-zero part is identified with.","marker":"[G]"}],"fun_headline_variants":["Oriented ribbon graphs recover moduli cohomology","New properad encodes curve moduli cohomology","Gravity operad acts on higher Hochschild cohomology","Pre-CY algebras yield cyclic cohomology via ribbon graphs","Ribbon graph properad computes moduli space cohomology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result rests on a chain of technical checks that are asserted here and partly deferred to earlier papers (one unpublished): the differential on the interpolating graph complex must square to zero, and certain subcomplexes must have no cohomology.","fun_headline_variants_meta":{"raw":{"variants":["Oriented ribbon graphs recover moduli cohomology","New properad encodes curve moduli cohomology","Gravity operad acts on higher Hochschild cohomology","Pre-CY algebras yield cyclic cohomology via ribbon graphs","Ribbon graph properad computes moduli space cohomology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00068,"raw_usage":{"total_tokens":3104,"prompt_tokens":975,"completion_tokens":2129,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":2046}},"tokens_in":591,"tokens_out":2129,"duration_ms":17604,"temperature":1.0,"reasoning_tokens":2046,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:34:39.334777+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct computation of δ•^2 on a small generator of ORGraphs_{d,d+1}, for instance the graph with one labelled boundary, one labelled white vertex, and one unlabelled black vertex, would either confirm or refute the omitted square-zero calculation; if a nonzero value appears, Propositions 4.3 and 4.4 cannot both hold, and the main theorem fails.","supporting_citations":[],"review_version":1}