{"id":"97e20779-e187-4363-9627-20bd01ace609","arxiv_id":"2502.04023","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Embedding tensors on 3-Leibniz algebras induce 3-tri-Leibniz algebras, but the paper's deformation-cohomology bijection fails for the zero embedding tensor.","lead":"This paper defines 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras, and shows the two notions generate each other. The structural results are plausible, but the advertised bijection between linear deformations and first cohomology is false.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.6's converse is false: a 1-cocycle need not generate a linear deformation, as the zero embedding tensor shows.","rationale":"The reader's strongest claim is precisely that Theorem 5.6 overstates the relation between 1-cocycles and linear deformations, and the zero embedding tensor gives an immediate counterexample. This concern is load-bearing because Theorem 5.6 is presented as a headline result of Section 5, and the entire cohomological classification of linear deformations rests on the asserted converse. The mathematics is clear: equations (5.1)-(5.3) are obtained by expanding the embedding-tensor condition for T + tT1, and only (5.1) is used to define Z1_T(V, g). There is no argument in the paper showing that (5.1) implies (5.2) or (5.3), and the T = 0 example shows that no such implication can hold. The chosen counterexample is especially clean because for T = 0 equation (5.1) is vacuous, reducing the claim to the assertion that every linear map is an averaging operator, which is false. The concrete test with a specific 3-Lie algebra and a non-averaging projection makes the failure explicit and verifiable by direct substitution. Given that a central theorem is false, the reader's REJECT verdict is justified. The structural results in Sections 2-4 may be salvageable, and the deformation equations themselves are correctly derived, but the key classification theorem is not.","tokens_in":20555,"tokens_out":5060,"duration_ms":51591,"concrete_test":"Take g = span{e1, e2, e3} with the 3-Lie bracket [e1, e2, e3]_g = e1 (and its skew-symmetric permutations), set T = 0, and take V = g with the adjoint representation. Define T1 by T1(e1) = e1, T1(e2) = e2, T1(e3) = e1. (a) Verify that T1 satisfies equation (5.1) because every term contains a factor T u, T v, or T w equal to zero. (b) Evaluate equation (5.3) at (u, v, w) = (e1, e3, e2): the left side is [e1, e1, e2]_g = 0, while the middle right-hand term is T1[T1e1, e3, T1e2]_g = T1[e1, e3, e2]_g = T1(-e1) = -e1. Hence equation (5.3) fails and T + tT1 is not an embedding tensor for t ≠ 0, even though T1 is a 1-cocycle.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The bijection in Theorem 5.6 depends on the sentence before it: 'Conversely, any 1-cocycle T1 gives rise to the linear deformation T + tT1.' This is the load-bearing step, and it is false. Expanding T + tT1 yields equations (5.1)-(5.3); only (5.1) defines a 1-cocycle. Equations (5.2) and (5.3) are further independent constraints. The zero embedding tensor T = 0 on a non-abelian 3-Leibniz algebra g with adjoint representation is a direct counterexample. For T = 0, equation (5.1) is vacuous, so every T1 in Hom(g, g) lies in Z1_T(g, g); but equation (5.3) becomes [T1u, T1v, T1w]_g = T1[T1u, T1v, w]_g = T1[T1u, v, T1w]_g = T1[u, T1v, T1w]_g, which is exactly the condition that T1 be an averaging operator. Hence a generic 1-cocycle does not generate a linear deformation. Thus HH1_T(V, g) is not in bijection with equivalence classes of linear deformations, and Theorem 5.6 collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras, proving structural results that connect embedding tensors with Nijenhuis operators, graph subalgebras, and induced 3-tri-Leibniz algebra structures. Section 4 introduces 3-tri-Leibniz dialgebras and shows that homomorphic embedding tensors induce them. Section 5 defines 1-cocycles and a first cohomology group HH^1_T(V,g) for an embedding tensor T, and claims in Theorem 5.6 that equivalence classes of linear deformations of T are in bijection with HH^1_T(V,g).","tokens_in":20907,"tokens_out":10169,"duration_ms":99596,"significance":"If the main results were correct, the paper would provide a uniform higher-arity analogue of known embedding-tensor constructions and a deformation classification. Some structural observations, such as the graph and Nijenhuis characterizations in Theorems 3.8 and 3.9 and the explicit examples in Section 2 and 3, are natural and potentially useful. However, the central deformation-theoretic claim, Theorem 5.6, is false: the stated cohomology group ignores the nonlinear equations (5.2)-(5.3) that a genuine deformation generator must satisfy. Since this theorem is the paper's principal new contribution, the paper cannot be accepted in its current form.","major_comments":[{"comment":"The bijection stated in Theorem 5.6 is false. Take T=0 on a non-abelian 3-Leibniz algebra g with the adjoint representation. Then Eq. (5.1) is identically zero, so every T1 in Hom(g,g) is a 1-cocycle. Moreover, the equivalence relation in Definition 5.2 is trivial because delta(a,b)u = Tρl(a,b,u) - [a,b,Tu]_g = 0 when T=0, so HH^1_T(g,g)=Hom(g,g). But Tt=tT1 is an embedding tensor only if Eq. (5.3) holds, which for the adjoint representation is exactly the averaging-operator condition (3.2): [T1u,T1v,T1w]_g = T1[T1u,T1v,w]_g = T1[T1u,v,T1w]_g = T1[u,T1v,T1w]_g. A generic linear map is not an averaging operator, so a generic 1-cocycle does not generate a linear deformation. Thus the sentence immediately before Theorem 5.6, 'Conversely, any 1-cocycle T1 gives rise to the linear deformation T+tT1', is false, and the claimed bijection between HH^1_T(V,g) and equivalence classes of linear deformations collapses.","section":"Section 5 (Theorem 5.6, Definition 5.2)"},{"comment":"The proof of Theorem 3.12 asserts that the subspace I_g spanned by the differences of the three brackets is an ideal of g, and then asserts without proof that the maps ρl, ρm, ρr are well-defined on the quotient and form a representation. These facts are load-bearing for the claim that every 3-tri-Leibniz algebra is induced by an embedding tensor. The well-definedness of ρm and ρr with respect to representatives of the quotient requires checking membership in I_g for each slot separately, and the ideal property itself is not immediate from axioms (2.7)-(2.11). The theorem may be true, but as written its proof is incomplete and should be supplied explicitly.","section":"Section 3 (Theorem 3.12)"}],"minor_comments":[{"comment":"The name 'Nejinhuis operator' is a typo and should read 'Nijenhuis operator'.","section":"Definition 3.7"},{"comment":"The phrase 'the set of theses 1-cocycles' should read 'the set of these 1-cocycles'.","section":"Definition 5.2"},{"comment":"The text 'a homomorphism from from ~Tt to Tt' contains a duplicated 'from'.","section":"Section 5, before Theorem 5.6"},{"comment":"Writing each brace-enclosed triple of equations as three separate labeled equations would improve readability and avoid ambiguity about which right-hand side corresponds to which expression.","section":"Equations (5.1) and (5.2)"},{"comment":"The proof displays only the first-order terms and writes 'mod t^2'; since a linear deformation of a 3-tri-Leibniz algebra requires the deformed brackets to satisfy all axioms for all t, the argument should explicitly use that T_t is an embedding tensor for all t to conclude that the displayed ω define a genuine deformation.","section":"Proposition 5.3"},{"comment":"The proof establishes the embedding-tensor identity for |t| sufficiently small and then concludes that Tt is an embedding tensor for all t; this inference should be justified by noting that Eqs. (5.1)-(5.3) are polynomial identities in t, so validity on an open interval implies validity for all t.","section":"Theorem 5.8"},{"comment":"The notation (x,x) for elements of g3Leib ⊕ g is confusing because x denotes both an element of g and its class in g3Leib; using different symbols for the two components would clarify the formula.","section":"Remark 3.13"}],"recommendation":"reject","confidential_remarks":"The counterexample in my first major comment is elementary and decisive: the paper's main deformation-theoretic theorem is false as stated. The missing constraints (5.2) and (5.3) are independent of the cocycle condition, so a correct statement would require either a restricted notion of integrable cocycle or a genuinely different cohomology theory. This is not a local fix. The structural parts of the paper may be salvageable after the proof of Theorem 3.12 is completed, but the present version should not be published with Theorem 5.6 in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The structural sections are a competent template-following write-up, but the deformation section is not just shaky—it falls apart at Theorem 5.6. The claimed bijection between equivalence classes of linear deformations and the first cohomology is false, and the counterexample is immediate.\n\nWhat is new: the definitions of 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras, plus the mutual generation theorem (embedding tensor yields a 3-tri-Leibniz algebra; conversely every 3-tri-Leibniz algebra arises from an embedding tensor via a quotient). The hemisemidirect product, graph characterization, and Nijenhuis-operator version are all transported sensibly from the 3-Lie and Lie triple system papers. The dialgebra construction is formal but probably correct. I checked the proof of Proposition 2.15 and it goes through; Theorem 3.9 is a direct computation.\n\nNow the problem. Section 5 defines a 1-cocycle by equation (5.1), then asserts \"conversely, any 1-cocycle gives rise to the linear deformation T + tT1.\" That is wrong. Expanding Tt = T + tT1 gives three conditions; only (5.1) is the cocycle condition. Equations (5.2) and (5.3) are extra constraints. Take T = 0 on a non-abelian 3-Leibniz algebra with the adjoint representation. Then (5.1) is vacuous, so every linear map is a 1-cocycle, but Tt = tT1 is an embedding tensor only when T1 satisfies (5.3), which is precisely the averaging-operator condition. A generic 1-cocycle is not an averaging operator. So HH1 does not classify linear deformations. The theorem is load-bearing and false.\n\nThere is also a small unproved step in Theorem 3.12: the quotient ideal Ig is asserted to be an ideal, and the induced representation maps are asserted to be well-defined, but neither is verified. That is fixable; Theorem 5.6 is not.\n\nThe paper is for readers who want the dictionary between averaging operators and tri-Leibniz structures; those parts could be salvageable. But the deformation classification advertised in the abstract is invalid, and the internal contradiction between the definition of 1-cocycle and the claimed converse is visible from the paper's own equations. I would desk reject with an invitation to resubmit after fixing or removing the deformation section. Not worth referee time as is.","headline":"The structural dictionary between embedding tensors and 3-tri-Leibniz algebras is largely fine, but the deformation-cohomology bijection in Theorem 5.6 is false, and the zero embedding tensor kills the headline result.","tokens_in":21431,"tokens_out":3566,"would_cite":false,"duration_ms":35067,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17A01","17A60","17A32","17A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces embedding tensors on 3-Leibniz algebras and shows they are in two-way correspondence with 3-tri-Leibniz algebras, then classifies linear deformations of such tensors through a first cohomology group.","keywords":["3-Leibniz algebra","embedding tensor","3-tri-Leibniz algebra","averaging operator","linear deformation","first cohomology","3-tri-Leibniz dialgebra","Nijenhuis element"],"falsifier":"Take a non-abelian 3-Leibniz algebra (for instance $R^{4}$ with the vector-product bracket of Example 2.5) and set T = 0 with the adjoint representation. Then every linear map T1 is a 1-cocycle, but T + tT1 is an embedding tensor only when T1 is an averaging operator; a T1 that is not an averaging operator therefore has a cohomology class but no associated linear deformation, so the claimed bijection would fail.","tokens_in":20321,"feed_emoji":"📐","tokens_out":8953,"duration_ms":73756,"temperature":0.7,"pith_summary":"The paper introduces embedding tensors—linear maps from a representation space into a 3-Leibniz algebra that intertwine the bracket with the representation—and studies the algebraic structures they create. Its central structural result is a two-way correspondence: an embedding tensor endows the representation space with a 3-tri-Leibniz algebra, and conversely every 3-tri-Leibniz algebra arises as the induced structure on a representation of a 3-Leibniz algebra. The authors also show that any 3-tri-Leibniz algebra embeds into an averaging 3-Leibniz algebra, and that homomorphic embedding tensors produce 3-tri-Leibniz dialgebras. Finally, following Gerstenhaber's deformation method, they define 1-cocycles and a first cohomology for embedding tensors and claim a bijection between equivalence classes of linear deformations and that cohomology.","feed_headline":"Embedding tensors build every 3-tri-Leibniz algebra","feed_subtitle":"The correspondence links embedding tensors to 3-tri-Leibniz algebras, and a first cohomology classifies their linear deformations.","key_machinery":"The central object is the embedding tensor T: V → g, a linear map from a representation (V; ρl, ρm, ρr) of a 3-Leibniz algebra (g, [·,·,·]) satisfying [Tu,Tv,Tw] = Tρl(Tu,Tv,w) = Tρm(Tu,v,Tw) = Tρr(u,Tv,Tw). The derived object is the 3-tri-Leibniz algebra, a space with three trilinear brackets [·,·,·]⊣, [·,·,·]⊥, [·,·,·]⊢ satisfying five identities; the induced brackets on V are [u,v,w]⊢ = ρl(Tu,Tv,w), [u,v,w]⊥ = ρm(Tu,v,Tw), and [u,v,w]⊣ = ρr(u,Tv,Tw). The correspondence is carried by the hemisemidirect product g ⋉_{l,m,r} V: T is an embedding tensor exactly when its graph is a subalgebra of this 3-tri-Leibniz algebra, equivalently when the map (x,u) ↦ (Tu,0) is a Nijenhuis operator. For deformations, the mechanism is the cochain complex whose 1-cocycles are maps T1 satisfying Eq. (5.1), with equivalence generated by δ(a,b)u = Tρl(a,b,u) − [a,b,Tu].","core_discovery":"The paper's central claim is that embedding tensors on 3-Leibniz algebras are governed by 3-tri-Leibniz algebras: the representation space of an embedding tensor carries a canonical 3-tri-Leibniz structure, and conversely every 3-tri-Leibniz algebra is induced by an embedding tensor on a 3-Leibniz algebra with respect to a representation. A further claim is that homomorphic embedding tensors induce 3-tri-Leibniz dialgebras, and that linear deformations of an embedding tensor can be studied through a first cohomology group, with Theorem 5.6 asserting a bijection between equivalence classes of linear deformations and HH1_T(V,g).","pith_inferences":["A reader's check of Theorem 5.6: the converse claim that any 1-cocycle gives a linear deformation needs the higher-order equations (5.2)–(5.3). With T=0 on a non-abelian 3-Leibniz algebra in its adjoint representation, every T1 is a 1-cocycle, but T+tT1 is an embedding tensor only if T1 is an averaging operator, so the bijection would need extra conditions.","By analogy with embedding tensors on Lie algebras and on 3-Lie algebras, one expects an L∞-algebra whose Maurer–Cartan elements are exactly embedding tensors; the paper's cohomology would then be the cohomology of that structure, and the full deformation equations (5.1)–(5.3) would be the Maurer–Cartan equation rather than just a linear cocycle condition.","The dialgebra section proves one direction (a homomorphic embedding tensor gives a 3-tri-Leibniz dialgebra); the natural converse question—whether every 3-tri-Leibniz dialgebra arises this way—is not addressed and would be a direct test of how sharp the construction is."],"forward_implications":["Every 3-tri-Leibniz algebra is realized from an embedding tensor, so structural results for 3-tri-Leibniz algebras and for embedding tensors transfer across the correspondence.","Any 3-tri-Leibniz algebra embeds into an averaging 3-Leibniz algebra, meaning its three brackets can be realised as the image of a single averaging operator on an ambient algebra.","Homomorphic embedding tensors turn the representation space into a 3-tri-Leibniz dialgebra, connecting crossed modules of 3-Leibniz algebras to dialgebra structures.","If Theorem 5.6's bijection holds, equivalence classes of linear deformations of an embedding tensor are completely described by the first cohomology HH1_T(V,g), and Nijenhuis elements describe exactly the trivial deformations."],"supporting_citations":[{"why":"defines n-Leibniz algebras, giving the 3-Leibniz identity the whole paper is built on.","marker":"[1]"},{"why":"supplies the representation theory (three actions ρl, ρm, ρr) that defines embedding tensors.","marker":"[5]"},{"why":"motivates embedding tensors from gauged supergravity, the origin of the notion.","marker":"[7]"},{"why":"identifies embedding tensors with averaging operators, the adjoint-representation case studied here.","marker":"[8]"},{"why":"the deformation-and-cohomology treatment of embedding tensors on 3-Lie algebras that the present paper extends to 3-Leibniz algebras.","marker":"[13]"},{"why":"Gerstenhaber's deformation theory, whose linear-deformation framework Section 5 follows.","marker":"[15]"},{"why":"introduces dialgebras, the template for the 3-tri-Leibniz dialgebra definition.","marker":"[16]"},{"why":"introduces triLeibniz algebras, used as a starting point for 3-tri-Leibniz dialgebras.","marker":"[17]"}],"fun_headline_variants":["Embedding tensors on 3-Leibniz algebras yield 3-tri-Leibniz algebras","3-tri-Leibniz algebras arise from embedding tensors on 3-Leibniz","Homomorphic embedding tensors induce 3-tri-Leibniz dialgebras","First cohomology classifies linear deformations of embedding tensors","Embedding tensors on 3-Leibniz algebras biject with 3-tri-Leibniz algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The deformation classification rests on the unstated assumption that every 1-cocycle automatically satisfies the higher-order equations that make T + tT1 an embedding tensor for every t.","fun_headline_variants_meta":{"raw":{"variants":["Embedding tensors on 3-Leibniz algebras yield 3-tri-Leibniz algebras","3-tri-Leibniz algebras arise from embedding tensors on 3-Leibniz","Homomorphic embedding tensors induce 3-tri-Leibniz dialgebras","First cohomology classifies linear deformations of embedding tensors","Embedding tensors on 3-Leibniz algebras biject with 3-tri-Leibniz algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001759,"raw_usage":{"total_tokens":6881,"prompt_tokens":820,"completion_tokens":6061,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":436,"completion_tokens_details":{"reasoning_tokens":5951}},"tokens_in":436,"tokens_out":6061,"duration_ms":39169,"temperature":1.0,"reasoning_tokens":5951,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T23:50:00.354402+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a non-abelian 3-Leibniz algebra (for instance $R^{4}$ with the vector-product bracket of Example 2.5) and set T = 0 with the adjoint representation. Then every linear map T1 is a 1-cocycle, but T + tT1 is an embedding tensor only when T1 is an averaging operator; a T1 that is not an averaging operator therefore has a cohomology class but no associated linear deformation, so the claimed bijection would fail.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines n-Leibniz algebras, giving the 3-Leibniz identity the whole paper is built on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the representation theory (three actions ρl, ρm, ρr) that defines embedding tensors."},{"cited_title":"Nicolai, H","cited_arxiv_id":null,"evidence_quote":"motivates embedding tensors from gauged supergravity, the origin of the notion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"identifies embedding tensors with averaging operators, the adjoint-representation case studied here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the deformation-and-cohomology treatment of embedding tensors on 3-Lie algebras that the present paper extends to 3-Leibniz algebras."},{"cited_title":"Gerstenhaber","cited_arxiv_id":null,"evidence_quote":"Gerstenhaber's deformation theory, whose linear-deformation framework Section 5 follows."},{"cited_title":"Dialgebras and related opera ds","cited_arxiv_id":null,"evidence_quote":"introduces dialgebras, the template for the 3-tri-Leibniz dialgebra definition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces triLeibniz algebras, used as a starting point for 3-tri-Leibniz dialgebras."}],"review_version":1}