{"id":"dc7c4a9a-02b8-436b-b203-a5a0e9e93829","arxiv_id":"2501.18414","paper_version":1,"verdict":"REJECT","confidence":"LOW","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper transfers crossed modules from triassociative and Leibniz algebras to ternary Leibniz algebras, but omits the supporting computations and contains garbled formulas.","lead":"This paper claims to transfer crossed modules from triassociative and Leibniz algebras to ternary Leibniz algebras, using known translation functors. The central proofs are skipped as 'straightforward', and several key formulas are misprinted, so the constructions cannot be checked from the manuscript.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's action maps (26)-(27) use the wrong operation: expanding T(A⋊B) via Lemma 3.1 gives m'_2 = y⊣D−D⊢y, not y⊥D−D⊢y, so the claimed Peiffer condition fails for the identity crossed module.","rationale":"The reader's rejection is correct, and the specific defect above makes it concrete rather than merely 'proof omitted.' The six maps in Theorem 3.1 are clearly meant to come from the semidirect product: Theorem 3.2 asserts T(A⋊B)=T(A)⋊T(B), and formulas (22)-(25) do match a direct expansion of Lemma 3.1. But the same expansion for m'_2 and m'_3 gives y⊣D−D⊢y, not y⊥D−D⊢y. Therefore the displayed action is not the induced action; in the identity crossed module the Peiffer condition fails. Since the central claim of the paper depends on exactly these formulas, the main construction fails as written. This is not a disagreement with outside consensus: it is an internal inconsistency with Lemma 3.1 and the semidirect product structure that Theorem 3.2 itself asserts. If the authors intended ⊣ in (26) and (27), the theorem may be repairable, but a full verification of Definition 2.18 would still be needed, and Theorem 3.3 separately contains ill-defined, self-referential maps. As submitted, the central argument does not establish its claims.","tokens_in":17107,"tokens_out":26278,"duration_ms":209770,"concrete_test":"Run a normal-form computation in the free triassociative algebra on generators y,c,z using a rewriting system for the triassociative axioms, and compare the normal forms of y⊣(c⊥z−z⊥c) and y⊥(c⊥z−z⊥c); they should be distinct. Then instantiate the crossed module (A,A,id) with the canonical self-action: by Definition 2.19(2), m'_2(y,c,z) must equal [y,c,z]_{T(A)}. Since formula (26) gives y⊥D−D⊢y with D=c⊥z−z⊥c, while the bracket gives y⊣D−D⊢y, the inequality of the two normal forms witnesses failure of the Peiffer condition. An independent check is to expand T(A⋊B) with Lemma 3.1 and compare the first components with formulas (22)-(27); the m'_2 and m'_3 entries will not match.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Even if Lemma 3.1 is accepted, the action displayed in Theorem 3.1 is not the action obtained by applying Lemma 3.1 to the semidirect product, so the central transfer claim is internally inconsistent. Computing the first component of [(y,0),(0,c),(z,0)] in T(A⋊B) from Lemma 3.1 gives y ⊣ (μ⊥_2(c,z) − μ⊥_1(z,c)) − (μ⊥_2(c,z) − μ⊥_1(z,c)) ⊢ y. Equation (26) instead writes m'_2(y,c,z) = y ⊥ (μ⊥_2(c,z) − μ⊥_1(z,c)) − (...) ⊢ y, replacing the left operation ⊣ by the middle operation ⊥; equation (27) has the same defect. Take the crossed module (A,A,id) with the canonical self-action of Example 2.6.2. Condition 2 of Definition 2.19 then forces m'_2(y,c,z) = [y,c,z]_{T(A)} = y⊣(c⊥z−z⊥c) − (c⊥z−z⊥c)⊢y. The displayed m'_2 differs by y⊥(...)−y⊣(...). This difference is not an identity of triassociative algebras: it fails in the free triassociative algebra on y,c,z, because no triassociative axiom identifies x⊣(y⊥z) with x⊥(y⊥z). Hence Theorem 3.1(b), and with it Theorem 3.2, is false as stated; the defect is not merely a missing proof. If the intended symbol in (26)-(27) is ⊣ rather than ⊥, the theorem may be repairable, but the typeset claim is wrong.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes functorial constructions that send crossed modules of triassociative algebras and crossed modules of Leibniz algebras to crossed modules of ternary Leibniz algebras. The main mechanism is the known bracket [x,y,z] = x ⊣ (y ⊥ z - z ⊥ y) - (y ⊥ z - z ⊥ y) ⊢ x of Lemma 3.1, applied to semidirect products, together with an action transfer stated in Theorems 3.1 and 3.3. The paper also states Rota-Baxter deformations of such crossed modules in Theorems 3.4 and 3.5. The central claims are that the six maps (22)-(27) define an action and a crossed module, that T(A⋊B) = T(A)⋊T(B), and that analogous transfers work for Leibniz algebras and for Rota-Baxter operators.","tokens_in":17498,"tokens_out":8501,"duration_ms":68420,"significance":"If the transfer theorems were correct, the paper would provide a useful bridge between crossed modules of triassociative algebras, Leibniz algebras, and ternary Leibniz algebras, complementing known constructions in the literature. The paper does not contain machine-checked proofs or reproducible code; all central verifications are delegated to 'straightforward' computation. Because several displayed formulas are type-incorrect or internally inconsistent, the significance of the claimed results cannot be assessed from the submitted manuscript.","major_comments":[{"comment":"The displayed maps m'_2 and m'_3 use the middle product in the first term, writing y ⊥ (µ⊥_2(c,z) - µ⊥_1(z,c)). Lemma 3.1, however, defines the ternary bracket with the left product in that position: [x,y,z] = x ⊣ (y ⊥ z - z ⊥ y) - (y ⊥ z - z ⊥ y) ⊢ x. Expanding [(y,0),(0,c),(z,0)] in T(A⋊B) therefore gives y ⊣ (µ⊥_2(c,z)-µ⊥_1(z,c)) - (...) ⊢ y, not y ⊥ (...). For the crossed module (A,A,id) with the canonical self-action, Condition 2 of Definition 2.19 forces m'_2(y,c,z) = [y,c,z]_{T(A)} = y ⊣ (c⊥z-z⊥c) - (c⊥z-z⊥c) ⊢ y. The printed formula differs by y⊥(...) - y⊣(...), which is not an identity in the free triassociative algebra. Thus Theorem 3.1(b) and Theorem 3.2 are false as stated; the defect is not merely a missing proof. If the intended operation in (26)-(27) is ⊣ rather than ⊥, the theorem may be repairable, but the typeset claim is incorrect.","section":"Theorem 3.1, Eqs. (26)-(27)"},{"comment":"The formulas defining the six action maps are type-incorrect. For instance, m2(p1,l,p2) is defined as µ2(p2, m2(l,p2)), which refers to m2 on the right-hand side before m2 is defined and with an inadmissible number of arguments; m3(p1,p2,l) is defined as µ2(p1, m2(p2,l)), again using the yet-undefined ternary map m2 as a binary map; similarly m'_2(l1,p,l2) = µ1(l1, m2(p,l2)) and m'_3(l1,l2,p) = µ1(l1, m1(l2,p)) use binary applications of ternary maps. Consequently the induced action of T(P) on T(L) is not actually defined, and part (2) of the theorem is unsubstantiated.","section":"Theorem 3.3"},{"comment":"The action axioms for ternary Leibniz algebras are presented in a badly corrupted form. Equations (7)-(21) contain unmatched parentheses, missing commas, and variable mismatches, e.g. equation (9) contains '[p1,p3,p4)' with a closing parenthesis instead of a bracket, equation (12) contains '[l1, m1(l2,p2,p3), l2]' where the middle term has the wrong type, and equation (13) contains 'l[p2,p3,p4]' with no comma. Since every subsequent theorem relies on this definition, the paper does not provide a usable and checkable notion of action.","section":"Definition 2.18"},{"comment":"The main results are all asserted with 'the proof is long but straightforward' or 'it comes from direct computation' without any verification of the action axioms or the crossed-module conditions. These verifications are the core content of the paper, not routine decoration. In view of the concrete type errors in the displayed formulas, such delegation is insufficient and prevents the reader from distinguishing a repairable typo from an incorrect construction.","section":"Proofs of Theorems 3.1, 3.3, 3.4, 3.5"}],"minor_comments":[{"comment":"The first line states 'Let (A, ⊣, ⊥, ⊣) be an triassociative algebra'; the fourth operation should be ⊢.","section":"Definition 2.3"},{"comment":"In the proof, the same symbols x,y,z are used both for elements of A and for their classes in A/I, and the final equality omits the class notation; the displayed chain is not a well-formed proof as written.","section":"Proposition 2.1"},{"comment":"The second and third multiplication formulas contain 'bA ⊥B b2' and 'bA ⊢B b2', which appear to be typographical errors for 'b1 ⊥B b2' and 'b1 ⊢B b2'.","section":"Proposition 2.2"},{"comment":"The example refers to a 'morphism of triassociative dialgebras'; the terminology should be 'triassociative algebras'.","section":"Example 2.4"},{"comment":"The notation mR_2 is used in the definitions of m2, m3, m'_2, and m'_3, but mR_2 is never defined; presumably a definition or a reference to µR_2 is intended.","section":"Theorem 3.4"},{"comment":"There are numerous typographical and grammatical errors, including 'satisfyied', 'propreties', 'connexion', 'communt fact', 'echanging', and 'associate trialgebra'; these should be corrected in any revision.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"This manuscript appears to be an early draft. The central constructions are not merely under-proved; at least one main theorem is false as stated and another is ill-typed. I would not encourage resubmission unless the formulas are corrected and full verifications are supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The long story short: the paper's main transfer theorem is stated with a type error in the action formulas, so as it stands it doesn't prove what it claims. The underlying idea is a natural extension of Casas's T functor, but the execution is too careless to be useful.\n\nWhat's genuinely new: the explicit six action maps (22)-(27) and the transfer statements for crossed modules, which I don't think appear in the cited literature. The paper also collects known constructions and examples in one place. If someone repairs the formulas and supplies the missing verifications, the transfer result would be a reasonable small contribution to the crossed-module program.\n\nThe soft spots are substantial. The big one: equations (26)-(27) use the middle product ⊥ on the left, while Lemma 3.1 forces the left product ⊣. This is not a cosmetic typo—checking the identity crossed module against the Peiffer condition fails exactly because of this mismatch. So Theorem 3.1(b) is false as stated. Theorem 3.3 has its own issue: the maps m2, m'2, and m'3 are defined self-referentially, and the defining equations don't type-check. Definition 2.18's list of action axioms is also full of broken equations, which makes it hard to even parse what an action is supposed to be. And all the central proofs are dismissed as 'long but straightforward.' No comparison is made with the Leibniz n-algebra crossed module results in [36], which is a gap given the T functor already appears there. The self-citation is not the problem; the problem is that the load-bearing formulas don't check out.\n\nThis paper is for someone working in the niche of crossed modules for non-associative algebras who might mine it for a repairable construction. It shouldn't have been submitted in this shape. A serious revision with full computations, or at least a formal verification of the transferred action axioms, could make it publishable.\n\nI would not send this to peer review as is. If the authors fix the formulas and actually prove the action axioms, then a short paper in a specialized journal would be reasonable. Right now, the internal contradictions put it below the bar.","headline":"Salvageable idea, but the central transfer formulas have a load-bearing type error and the proofs are missing, so the paper as submitted should be rejected.","tokens_in":17957,"tokens_out":2434,"would_cite":false,"duration_ms":22728,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17A40","18G45","17A32"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that every crossed module of triassociative algebras, and of Leibniz algebras, transfers to a crossed module of ternary Leibniz algebras by explicit action maps.","keywords":["Leibniz algebras","ternary Leibniz algebras","triassociative algebras","crossed modules","actions","Rota-Baxter operators","triassociative crossed modules","Leibniz crossed modules"],"falsifier":"Take the two-dimensional triassociative algebra from Example 2.3, with basis $\\{e_1,e_2\\}$, all three products equal, $e_1 \\dashv e_1 = e_2 \\dashv e_1 = 0$, $e_1 \\dashv e_2 = a e_1$, $e_2 \\dashv e_2 = e_2$, and compute the bracket $[x,y,z] = x \\dashv (y \\perp z - z \\perp y) - (y \\perp z - z \\perp y) \\vdash x$ on all triples of basis elements. If any of the 32 instances of the identity $[[x,y,z],t,u] = [x,y,[z,t,u]] + [x,[y,t,u],z] + [[x,t,u],y,z]$ fails, Lemma 3.1 is false and Theorem 3.1 collapses; if all pass, the load-bearing bracket is confirmed on this example.","tokens_in":16927,"feed_emoji":"🔁","tokens_out":7366,"duration_ms":58490,"temperature":0.7,"pith_summary":"This paper aims to extend the notion of crossed module—a model for homotopy 2-types—from triassociative and Leibniz algebras to ternary Leibniz algebras. It proves that the transfer functor $T$, defined on a triassociative algebra by the ternary bracket $[x,y,z] = x \\dashv (y \\perp z - z \\perp y) - (y \\perp z - z \\perp y) \\vdash x$, and on a Leibniz algebra by $[x,[y,z]]$, carries actions and crossed modules along with the algebras themselves. The central result gives six explicit bilinear maps (22)–(27) that turn an action of $B$ on $A$ into an action of $T(B)$ on $T(A)$, so that $(T(A),T(B),\\phi)$ is a crossed module of ternary Leibniz algebras whenever $(A,B,\\phi)$ is a crossed module of triassociative algebras. The same transfer works for crossed modules of Leibniz algebras. If the paper is right, this gives a systematic way to build higher-arity Leibniz structures from classical ones, with consequences for homotopy invariants and deformation theory.","feed_headline":"Crossed modules survive the leap to ternary Leibniz algebras","feed_subtitle":"A transfer functor sends triassociative and Leibniz crossed modules to crossed modules of ternary Leibniz algebras.","key_machinery":"The central object is the functor $T$ between algebraic categories. On a triassociative algebra $(A,\\dashv,\\perp,\\vdash)$ it puts the ternary bracket $[x,y,z] = x \\dashv (y \\perp z - z \\perp y) - (y \\perp z - z \\perp y) \\vdash x$; on a Leibniz algebra $(L,[-,-])$ it puts $\\{x,y,z\\} = [x,[y,z]]$. The work of this functor is to push the crossed-module action through: the six maps $m_1,\\dots,m'_3$ in (22)–(27) are defined so that each ternary Leibniz action axiom becomes a consequence of the six triassociative action equalities from Definition 2.7.","core_discovery":"On the paper's own terms, the discovery is a pair of transfer theorems. Theorem 3.1 states that any crossed module of triassociative algebras yields, through the bracket $[x,y,z] = x \\dashv (y \\perp z - z \\perp y) - (y \\perp z - z \\perp y) \\vdash x$ and the six maps (22)–(27), a crossed module of ternary Leibniz algebras, with morphisms of triassociative crossed modules inducing morphisms of the ternary ones. Theorem 3.3 states the analogous statement for crossed modules of Leibniz algebras via the bracket $\\{x,y,z\\} = [x,[y,z]]$. The construction is explicit: every action map for the source category is repackaged, using the middle product $\\perp$ and the two-sided actions, into the six ternary action maps required by Definition 2.18.","pith_inferences":["Because $T$ is functorial on crossed modules, it plausibly induces a functor between the associated 2-categories of crossed modules; this would make $T$ a map of homotopy-theoretic models, though the paper does not spell that out.","The two alternative brackets in Remark 3.1 suggest that the transfer is not unique; one could test whether those brackets also lift actions and crossed modules, and whether the resulting ternary crossed modules are isomorphic.","A computational sanity check of the ternary Leibniz identity for (22)–(27) on a small example, such as the two-dimensional algebra in Example 2.3, would give independent evidence that no hidden case distinction is missing.","Averaging operators and Nijenhuis operators, which the paper uses to build new triassociative algebras, might similarly lift to operators on the induced ternary Leibniz crossed modules; the paper does not claim this."],"forward_implications":["Any crossed module of triassociative algebras, for instance an ideal inclusion, becomes a crossed module of ternary Leibniz algebras once the six maps (22)–(27) are installed.","Any crossed module of Leibniz algebras becomes a ternary Leibniz crossed module without changing the underlying map $\\phi$, only reinterpreting the bracket as $[x,[y,z]]$.","The transfer is functorial: a morphism of crossed modules in the source category is automatically a morphism between the induced ternary crossed modules.","The semi-direct product respects the transfer: $T(A \\rtimes B) = T(A) \\rtimes T(B)$, so extensions built from actions survive the passage to ternary structures.","Rota-Baxter deformations of Leibniz crossed modules also induce ternary Leibniz crossed modules, giving a second family of constructions."],"supporting_citations":[{"why":"Supplies Lemma 3.1: the bracket $x \\dashv (y \\perp z - z \\perp y) - (y \\perp z - z \\perp y) \\vdash x$ that turns a triassociative algebra into a ternary Leibniz algebra, the base of the whole transfer.","marker":"[32]"},{"why":"Provides the notion and basic properties of crossed modules for Leibniz, associative and diassociative algebras that the paper transfers to ternary Leibniz algebras.","marker":"[16]"},{"why":"Supplies crossed modules for Leibniz $n$-algebras and the semi-direct product bracket used in Definition 3.1 and Theorem 3.2.","marker":"[36]"},{"why":"Supplies the Rota-Baxter formalism for ternary Leibniz algebras and the lemmas used in Theorem 3.4 to deform crossed modules.","marker":"[27]"}],"fun_headline_variants":["Ternary Leibniz algebras gain crossed modules via transfers","Explicit transfer: triassociative crossed modules to ternary Leibniz","New theorems: crossed modules in ternary Leibniz from known ones","Crossed modules transfer across algebraic structures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction depends on one borrowed formula: the bracket $[x,y,z] = x \\dashv (y \\perp z - z \\perp y) - (y \\perp z - z \\perp y) \\vdash x$ must satisfy the ternary Leibniz identity inside every triassociative algebra, and if it fails, the induced maps (22)–(27) need not define an action or a crossed module.","fun_headline_variants_meta":{"raw":{"variants":["Ternary Leibniz algebras gain crossed modules via transfers","Explicit transfer: triassociative crossed modules to ternary Leibniz","New theorems: crossed modules in ternary Leibniz from known ones","Crossed modules transfer across algebraic structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00067,"raw_usage":{"total_tokens":2951,"prompt_tokens":740,"completion_tokens":2211,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":356,"completion_tokens_details":{"reasoning_tokens":2150}},"tokens_in":356,"tokens_out":2211,"duration_ms":16066,"temperature":1.0,"reasoning_tokens":2150,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T23:34:44.177788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the two-dimensional triassociative algebra from Example 2.3, with basis $\\{e_1,e_2\\}$, all three products equal, $e_1 \\dashv e_1 = e_2 \\dashv e_1 = 0$, $e_1 \\dashv e_2 = a e_1$, $e_2 \\dashv e_2 = e_2$, and compute the bracket $[x,y,z] = x \\dashv (y \\perp z - z \\perp y) - (y \\perp z - z \\perp y) \\vdash x$ on all triples of basis elements. If any of the 32 instances of the identity $[[x,y,z],t,u] = [x,y,[z,t,u]] + [x,[y,t,u],z] + [[x,t,u],y,z]$ fails, Lemma 3.1 is false and Theorem 3.1 collapses; if all pass, the load-bearing bracket is confirmed on this example.","supporting_citations":[{"cited_title":"M., Trialgebras and Leibniz 3-algebras, Bol","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.1: the bracket $x \\dashv (y \\perp z - z \\perp y) - (y \\perp z - z \\perp y) \\vdash x$ that turns a triassociative algebra into a ternary Leibniz algebra, the base of the whole transfer."},{"cited_title":"More on crossed modules of Lie, Leibniz, associative and diassociative algebras","cited_arxiv_id":"1508.01147","evidence_quote":"Provides the notion and basic properties of crossed modules for Leibniz, associative and diassociative algebras that the paper transfers to ternary Leibniz algebras."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies crossed modules for Leibniz $n$-algebras and the semi-direct product bracket used in Definition 3.1 and Theorem 3.2."},{"cited_title":"(2023), Ternary Leibniz color algebras and beyond , In: Hounkonou, M.N., Mitrovic, M., Abbas, M., Khan, M","cited_arxiv_id":null,"evidence_quote":"Supplies the Rota-Baxter formalism for ternary Leibniz algebras and the lemmas used in Theorem 3.4 to deform crossed modules."}],"review_version":1}