{"id":"fed88d5a-5c07-4272-bb9d-feab250a9cee","arxiv_id":"2412.17253","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs the minimal model of the Nijenhuis associative operad and uses it to produce a cohomology and L-infinity deformation theory for Nijenhuis associative algebras.","lead":"This paper builds a full homotopy theory for Nijenhuis associative algebras, including a minimal operad model, a deformation complex, and an L-infinity algebra whose solutions are exactly the algebra structures. Specialists may read it for a worked example of operadic deformation theory applied to a non-Koszul algebraic structure.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The minimal-model theorem in Theorem 4.4 rests on two delegated computations, ∂²=0 in Proposition 3.1 and the 'verbatim' carry-over of the Wang–Zhou homotopy argument; neither is independently demonstrated in the text.","rationale":"The reader identified the same weakest assumption: the cancellation table in Proposition 3.1 and the carry-over from [43] in Theorem 4.4 are not fully demonstrated. My stress-test reviewed the relevant sections in the manuscript and found no additional independent flaw: the algebraic setup is coherent, the operad NjA is clearly defined, and the claimed relation between Maurer-Cartan elements and Nijenhuis structures in Section 5 is consistent with the explicit formulas. The main theorems are plausible and the paper makes a genuine contribution if the two delegated computations are correct. However, because the proof of ∂²=0 is summarized by a cancellation table without the actual sign and index bookkeeping, and because Theorem 4.4 delegates the core acyclicity argument to a cited paper sharing authors with the present work rather than reproducing it, the central claim is not yet independently verifiable from the text. A targeted computational check on arities 2–4 and a direct verification of the leading-term comparison would settle the matter. Since this is exactly the basis for the reader's CONDITIONAL verdict, my read does not move the verdict: UNCHANGED.","tokens_in":32687,"tokens_out":6676,"duration_ms":66911,"concrete_test":"Run an independent computation, by hand or with a computer algebra system for planar operadic trees, that (i) evaluates ∂²(y_n) and ∂²(x_n) from formulas (9)–(10) for n=2,3,4 with all signs tracked, and checks that the coefficients of every tree monomial cancel as claimed in Proposition 3.1; and (ii) evaluates ∂(m_n) and ∂(P_n) from formulas (11)–(12) for n=3,4, orders the resulting tree monomials under the order Ξ defined in §4.2, and verifies that the leading terms are exactly mn−1 m2 and P_{n-1} m2 P1. If both checks pass, the conditional acceptance of Theorem 4.4 is justified; if either fails, the minimal model theorem and the L∞-algebra of Section 5 are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 4.4, that NjA∞ = Ω(NjA¡) is the minimal model of NjA, and the subsequent controlling L∞-algebra statement of Proposition 5.4 depends on it. The proof has two load-bearing gaps. First, Proposition 3.1 verifies that the cooperad structure is a homotopy cooperad by showing ∂²=0 on the cobar construction. The proof of ∂²(y_n) expands the expression into seven groups I–VII and asserts cancellations such as (I)+(II)=0 and (III,i=1)+(VII,s≠0)=0, but it does not display the signs or the index bijections that make the cancellations happen. Since the differential ∂ is a derivation determined by formulas (9)–(10), one sign error in the Δ_T operations would destroy ∂²=0 and with it the dg operad structure of NjA∞. Second, Theorem 4.4 reduces to a monomial order argument and says: 'Once the leading terms are seen to be the same as the case of Rota-Baxter associative operad, the remaining part of the proof carries verbatim as that of [43, Theorem 3.5].' The claimed identity of leading terms is not shown, and the transfer to the Nijenhuis case is asserted rather than proved. If the leading term of ∂(P_n) under the order Ξ differs from the Rota-Baxter leading term P_{n-1} m2 P1, the induction in [43] does not automatically apply. These are verification gaps, not demonstrated internal contradictions; the construction is plausible and likely correct, but the two delegated computations are exactly what must be checked before Theorem 4.4 can be accepted as proved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an operadic deformation theory for Nijenhuis associative algebras. It defines a homotopy cooperad NjA¡ with generators u_n, v_n, constructs the cobar dg operad NjA∞ = Ω(NjA¡), and claims in Theorem 4.4 that NjA∞ is the minimal model of the operad NjA for Nijenhuis associative algebras. It then uses NjA¡ to produce an L∞-algebra CNjA(V) whose Maurer-Cartan elements are claimed to correspond to Nijenhuis associative structures (Proposition 5.4), and shows that twisting this L∞-algebra by a Maurer-Cartan element recovers the cochain complex defined in Section 2 (Proposition 5.5). The final section relates homotopy relative Rota-Baxter associative algebras of weight 0 to homotopy Nijenhuis associative algebras.","tokens_in":33067,"tokens_out":10164,"duration_ms":100055,"significance":"If the main theorems are fully established, this would be a substantial contribution: an explicit minimal model for a non-Koszul operad, a resulting controlling L∞-algebra for deformations, and a homotopy notion of Nijenhuis associative algebras. The paper is constructive: generators, differentials, monomial orders, and Maurer-Cartan equations are written out in detail, and the proposed relationship with homotopy relative Rota-Baxter algebras is concrete. However, the proof of the minimal-model statement relies on two delegated computations — the ∂²=0 verification in Proposition 3.1 and the 'carries verbatim' transfer of the Wang–Zhou argument in Theorem 4.4 — and these are precisely the load-bearing points. The current version is therefore conditional, but the gaps appear fillable.","major_comments":[{"comment":"The proof of ∂²=0 on the generators y_n is not complete. The expansion of ∂²(y_n) is grouped into seven families I–VII and the text asserts cancellations such as (I)+(II)=0, (III,i=1)+(VII,s≠0)=0, and (IV,i=t,t=p)+(VI)=0, but it does not display the sign factors or the index bijections that realize these cancellations. The formulas (9)–(10) and the definition of Δ_T(v_n) contain several nontrivial signs, including (−1)^t and (−1)^{p(p−1)/2}; a sign error in any of these operations would destroy ∂²=0 and hence the dg cooperad structure. Since NjA∞ is defined as Ω(NjA¡), this verification is essential for Theorem 4.4 and for every subsequent result built on the minimal model. Please supply the complete cancellation argument, or provide a machine-checkable verification of Proposition 3.1.","section":"Section 3, Proposition 3.1"},{"comment":"The proof reduces the quasi-isomorphism statement to showing that the leading terms of ∂(m_n) and ∂(P_n) under the monomial order Ξ are respectively m_{n−1}m_2 and P_{n−1}m_2P_1, and then asserts that 'the remaining part of the proof carries verbatim as that of [43, Theorem 3.5]'. The leading-term comparison is not shown. In particular, ∂(P_n) in equation (12) is a sum over many partitions, grafting positions i_h, k_j, and signs α′; one must verify that every other summand is strictly smaller than P_{n−1}m_2P_1 in the order Ξ. Moreover, the phrase 'carries verbatim' is a substantive claim: the homotopy used in [43] must be compatible with the present differential, with the present filtration, and with the new monomial order. This requires proof rather than assertion, because the minimal model theorem and the L∞-algebra of Section 5 depend on it.","section":"Section 4.2, Theorem 4.4"},{"comment":"Lemma 2.2 and Proposition 2.4 are not proved where they are introduced. The text states that they 'can be verified through direct inspection' but says the proofs will be deduced later from the L∞-algebra constructed in Section 5, with the general coefficient case following from Subsection 2.4. This makes Definition 2.5 — the deformation complex and cohomology of Nijenhuis associative algebras — logically dependent on Theorem 4.4 and Proposition 5.5. The dependency should be made explicit at the point of Definition 2.5, or the authors should provide direct proofs of Lemma 2.2 and Proposition 2.4 in Section 2. As written, a reader cannot check the cochain complex without first accepting the minimal-model construction.","section":"Sections 2.2–2.4"},{"comment":"The proof that twisting CNjA(A) by the Maurer-Cartan element corresponding to (m,P) yields the complex sC*NjA(A) is central, but several identifications are merely asserted. For example, the text states that '−[ν,sf]_G corresponds to −(−1)^{n+1}δ^n_Alg(~sf) under the fixed isomorphism (19)' and that certain combinations 'correspond to Φ^n(~sf)' and 'correspond to (−1)^n δ^{n−1}_NjO(ĝ)' without showing the intermediate sign computations. Since the cochain complex of Section 2 was defined before the L∞-algebra, an independent and fully displayed sign verification is necessary; a sign error in these correspondences would change the stated cohomology theory. Please expand the computation of l^α_1 on both components.","section":"Section 5.2, Proposition 5.5"}],"minor_comments":[{"comment":"The text contains many typographical artifacts, including 'n /greaterorequalslant0' in Section 2, 'Provinence' in the affiliation, and 'Iheoret. Phys.' in reference [24]; please proofread the manuscript carefully.","section":"Throughout"},{"comment":"The seven-group expansion of ∂²(y_n) is typeset with very large brace diagrams that are difficult to read. A structured cancellation table with explicit signs and index ranges for each group would greatly improve readability and verifiability.","section":"Section 3, Proposition 3.1"},{"comment":"In the proof of Theorem 4.4, the phrase 'It can be easily seen that the differential ∂ satisfies Conditions (i) and (ii) in Definition 4.2' is too terse. In particular, the filtration M(i) required by Condition (ii) is not specified; although a degree filtration is plausible, it should be stated explicitly.","section":"Section 4.2"},{"comment":"The corollary is described as a direct consequence of Theorem 6.7, but Theorem 6.7 gives a structure-wise construction on A⊕M for each pair (A,M); it does not by itself define a natural operad morphism from NjA∞ to the color-forgotten operad RBArel∞. Please clarify the intended naturality or state a weaker conclusion.","section":"Section 6, Corollary 6.8"},{"comment":"Proposition 5.4 is stated to follow from [5, Proposition 3.9]; this is plausible, but the paper should explicitly check that the Maurer-Cartan equation in CNjA(V) is equivalent to equations (17)–(18) with the stated signs, since this equivalence underpins the identification of homotopy Nijenhuis algebras with Maurer-Cartan elements.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"This is a technically demanding paper whose main ideas are promising and which fits well in the journal. My recommendation is driven by the need to fill the two delegated computations: the cancellation proof in Proposition 3.1 and the transfer argument in Theorem 4.4. The authors have previously proved the analogous Rota-Baxter statement, so I expect the gaps to be fixable, but the current version is not yet self-contained. I would encourage the authors to include the full computations, possibly in an appendix or as an ancillary computer script, and to state explicitly the logical dependencies between Sections 2, 4, and 5."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is a genuine extension of the Wang–Zhou Rota-Baxter program to Nijenhuis associative algebras, and the main constructions—the homotopy cooperad NjA¡, the cobar minimal model NjA∞, and the controlling L∞-algebra—are new. The paper is the first operadic treatment of Nijenhuis associative algebras, and since the operad is not Koszul (Remark 1.7 is explicit about this), the minimal model is not coming from standard Koszul duality. That makes it a meaningful test case for the non-Koszul technology.\n\nWhat the paper does well: it gives a clean cochain complex via a mapping cone in Section 2, then later recovers that complex from the twisted L∞-algebra in Proposition 5.5. That is a satisfying way to justify the cohomology. The authors are also honest about what they are doing: Theorem 4.4 states that, once the leading terms match, the homotopy argument carries verbatim from [43], and Proposition 3.1 lists the cancellations I–VII rather than pretending every sign is displayed.\n\nThe soft spots are exactly the two the stress-test note names, and they are real. Proposition 3.1 needs ∂²=0 to hold for the homotopy cooperad structure; the proof asserts cancellations like (I)+(II)=0 without showing the sign conventions or the index bijections. Since a single sign error would destroy the dg structure, this is load-bearing. Similarly, Theorem 4.4 delegates the quasi-isomorphism to [43, Theorem 3.5] after saying the leading terms are the same. The leading terms are displayed, but the identification with the Rota-Baxter case is asserted, not demonstrated, and the induction in [43] is long. Self-citation is not the issue; the issue is that the argument is not reproduced or even sketched. These are verification gaps rather than demonstrated contradictions, and the structure is coherent, so I would bet the cancellations do work out. But \"likely correct\" is not the same as \"proved.\"\n\nOne thing in the paper's favor: Proposition 2.6 is proved directly, so the cochain complex is well-defined without needing the L∞-algebra; Lemmas 2.2 and Proposition 2.4 are deferred but later justified. No circularity.\n\nThis paper is for people working on operadic deformation theory of operated algebras and possibly on Nijenhuis geometry via the announced sequel. It deserves a serious referee, not a desk reject, and I would take the referee report.\n\nRecommendation: send to peer review. Ask the authors to expand Proposition 3.1 with the actual ∂² computation, and to write out the carry-over in Theorem 4.4 instead of one sentence. Those are fixable revisions, and the core novelty is worth the effort.","headline":"A real first: minimal model and L∞-deformation theory for Nijenhuis associative algebras, but the proof leans on two delegated computations that a referee should ask to see in full.","tokens_in":33618,"tokens_out":2572,"would_cite":true,"duration_ms":26784,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E40","16S80","17B38","18M60","18M65","18M70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs homotopy Nijenhuis associative algebras and proves that the differential graded operad governing them is the minimal model of the operad for ordinary Nijenhuis associative algebras.","keywords":["homotopy Nijenhuis associative algebra","minimal model","Koszul dual homotopy cooperad","Nijenhuis associative algebra","deformation complex","cohomology","L∞-algebra","operad"],"falsifier":"Work out $\\partial^2=0$ directly on the generator $y_4$ using Formulas (9)–(10); the seven groups of terms labelled (I)–(VII) must cancel exactly, and any leftover monomial would destroy the cooperad structure and with it the minimal model. Alternatively, compute $H^1(\\mathrm{NjA}_\\infty,\\partial)$ from Formulas (11)–(12): a nonzero class would contradict the quasi-isomorphism to $\\mathrm{NjA}$.","tokens_in":32478,"feed_emoji":"🧮","tokens_out":20837,"duration_ms":157559,"temperature":0.7,"pith_summary":"Nijenhuis associative algebras are associative algebras equipped with an endomorphism $P$ satisfying the Nijenhuis identity, a structure that appears in quantum bi-Hamiltonian systems and as the associative analogue of the classical Nijenhuis tensor. The paper's goal is to give such algebras a full deformation and homotopy theory. It constructs a differential graded operad $\\mathrm{NjA}_\\infty$ whose algebras are homotopy Nijenhuis associative algebras, and proves that $\\mathrm{NjA}_\\infty$ is the minimal model of the operad $\\mathrm{NjA}$ of Nijenhuis associative algebras. From this it extracts an $L_\\infty$-algebra whose Maurer-Cartan elements are exactly Nijenhuis associative algebra structures, and whose twisting recovers the cohomology and deformation complex defined in Section 2. Deformations of the associative product and of the Nijenhuis operator can therefore be studied simultaneously through a single controlling algebra.","feed_headline":"Proved: minimal model for homotopy Nijenhuis algebras","feed_subtitle":"A single L∞-algebra controls simultaneous deformations of the product and the Nijenhuis operator.","key_machinery":"The load-bearing object is the Koszul dual homotopy cooperad $\\mathrm{NjA}^{¡}$, the Hadamard product $S(\\mathrm{NjA}^{¡}) \\otimes_H S^{-1}$; its underlying generators are $e_n$ and $o_n$, with degrees $n-1$ and $n$, and its cooperadic structure is encoded by the type-(I) and type-(II) trees of Section 3. The differential graded operad $\\mathrm{NjA}_\\infty$ is the cobar construction $\\Omega(\\mathrm{NjA}^{¡})$, freely generated by $m_n = s^{-1} e_n$ and $P_n = s^{-1} o_n$, with differential given by Formulas (11)–(12). The differential on the $m_n$ is the standard A-infinity identity; the differential on the $P_n$ is the higher Nijenhuis relation. The proof of Theorem 4.4 introduces a monomial order on tree monomials, with weights $\\phi(m_n)=n-1$ and $\\phi(P_n)=2n-1$, so that the leading terms of $\\partial(m_n)$ and $\\partial(P_n)$ coincide with the leading terms in the Rota-Baxter case; the remaining cancellations are then shown to follow by the same argument as in that case. The convolution homotopy operad $\\mathrm{Hom}(\\mathrm{NjA}^{¡},\\mathrm{End}(V))$ gives the controlling $L_\\infty$-algebra $C_{\\mathrm{NjA}}(V)$.","core_discovery":"A Nijenhuis operator on an associative algebra $(A,m)$ is a linear endomorphism $P$ satisfying $m \\circ (P \\otimes P) = P \\circ (m \\circ (P \\otimes \\mathrm{Id}) + m \\circ (\\mathrm{Id} \\otimes P) - P \\circ m)$. The paper defines homotopy Nijenhuis associative algebras by replacing $m$ and $P$ with families of higher operations $m_n$ and $P_n$, obeying the A-infinity identity for the products and a higher Nijenhuis identity for the operators. The central structural result, Theorem 4.4, is that the differential graded operad $\\mathrm{NjA}_\\infty = \\Omega(\\mathrm{NjA}^{¡})$ is the minimal model of $\\mathrm{NjA}$, so that homotopy Nijenhuis algebras are the up-to-homotopy version of Nijenhuis associative algebras. Proposition 5.4 then realizes a Nijenhuis associative algebra structure on a space $V$ as a Maurer-Cartan element in the $L_\\infty$-algebra $C_{\\mathrm{NjA}}(V)=\\mathrm{Hom}(\\mathrm{NjA}^{¡},\\mathrm{End}(V))^{\\prod}$, which in low degrees combines the classical bracket on associative-algebra cochains with higher brackets that mix cochains and Nijenhuis-operator cochains. Twisting this $L_\\infty$-algebra by a Maurer-Cartan element reproduces, up to suspension, the cochain complex of Nijenhuis associative algebras defined in Section 2, justifying that cohomology as the deformation complex. The paper also shows that homotopy relative Rota-Baxter operators of weight $0$ on $A$ with respect to a bimodule $M$ correspond exactly to homotopy Nijenhuis associative algebra structures on the semi-direct product $A \\oplus M$.","pith_inferences":["If the minimal model is as small as claimed, the same monomial-order strategy should apply to Nijenhuis Lie algebras and to Nijenhuis operators on bimodules, since the proof only needs the leading terms to match; the announced sequel has a concrete template to test.","The $L_\\infty$-algebra $C_{\\mathrm{NjA}}(V)$ should support an explicit obstruction theory for extending formal deformations, with degree-2 classes of the twisted complex serving as obstructions; this is not spelled out in the paper.","Because the Nijenhuis-operator differential contains an extra term $-P_M \\circ \\delta$ beyond the ordinary associative-algebra differential, computing explicit examples of the long exact cohomology sequence would clarify what that extra term measures empirically."],"forward_implications":["Every Nijenhuis associative algebra carries a cochain complex, built as a mapping cone of a chain map from associative-algebra cochains to Nijenhuis-operator cochains, whose cohomology controls simultaneous deformations of the product and the Nijenhuis operator.","Maurer-Cartan elements of the controlling $L_\\infty$-algebra are in bijection with Nijenhuis associative algebra structures, placing these deformation problems in the standard Maurer-Cartan framework.","Homotopy Nijenhuis associative algebras reduce on homology to ordinary Nijenhuis associative algebras: $m_1$ is a differential, $m_2$ induces an associative product, and $P_1$ induces a Nijenhuis operator on homology, with $P_2$ measuring the homotopy failure of the Nijenhuis relation.","Weight-zero relative Rota-Baxter operators, including their homotopy versions, fit into Nijenhuis theory: a homotopy relative Rota-Baxter operator on $(A,M)$ is equivalent to a homotopy Nijenhuis structure on the semi-direct product $A \\oplus M$.","Because the Nijenhuis relation is cubic, the operad $\\mathrm{NjA}$ is not Koszul; the paper thereby provides a non-Koszul example where a minimal model and a complete homotopy deformation theory are still obtained."],"supporting_citations":[{"why":"Supplies the monomial-order method and the minimal-model argument whose remaining cancellations are carried over verbatim once the leading terms match.","marker":"[43]"},{"why":"Provides the convolution homotopy operad construction and the general machinery that turns a homotopy cooperad into an $L_\\infty$-algebra, used to define $C_{\\mathrm{NjA}}(V)$.","marker":"[5]"},{"why":"Establishes the classical correspondence between relative Rota-Baxter operators of weight zero and Nijenhuis operators, which Section 6 lifts to the homotopy level.","marker":"[6]"},{"why":"Gives the definition of a minimal model of a dg operad and the uniqueness statement used in Section 4.","marker":"[10]"},{"why":"Provides the A-infinity identity that Equation (13) reproduces for the higher products $m_n$.","marker":"[38]"}],"fun_headline_variants":["Minimal model achieved for homotopy Nijenhuis algebras","One L-infinity algebra controls all Nijenhuis deformations","Homotopy Nijenhuis algebras: minimal model and deformation complex","Nijenhuis associative algebras: homotopy via L-infinity","Rota-Baxter link yields homotopy Nijenhuis structures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on a lengthy sign-cancellation computation that is only summarized as a table of seven term groups, and on the assumption that an argument from the Rota-Baxter case transfers unchanged once the leading monomials are matched; if either point fails, the minimal model theorem and the $L_\\infty$-algebra built from it collapse.","fun_headline_variants_meta":{"raw":{"variants":["Minimal model achieved for homotopy Nijenhuis algebras","One L-infinity algebra controls all Nijenhuis deformations","Homotopy Nijenhuis algebras: minimal model and deformation complex","Nijenhuis associative algebras: homotopy via L-infinity","Rota-Baxter link yields homotopy Nijenhuis structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000506,"raw_usage":{"total_tokens":2571,"prompt_tokens":1149,"completion_tokens":1422,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":765,"completion_tokens_details":{"reasoning_tokens":1332}},"tokens_in":765,"tokens_out":1422,"duration_ms":10579,"temperature":1.0,"reasoning_tokens":1332,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:39:43.498517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work out $\\partial^2=0$ directly on the generator $y_4$ using Formulas (9)–(10); the seven groups of terms labelled (I)–(VII) must cancel exactly, and any leftover monomial would destroy the cooperad structure and with it the minimal model. Alternatively, compute $H^1(\\mathrm{NjA}_\\infty,\\partial)$ from Formulas (11)–(12): a nonzero class would contradict the quasi-isomorphism to $\\mathrm{NjA}$.","supporting_citations":[{"cited_title":"Wang and G","cited_arxiv_id":null,"evidence_quote":"Supplies the monomial-order method and the minimal-model argument whose remaining cancellations are carried over verbatim once the leading terms match."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the convolution homotopy operad construction and the general machinery that turns a homotopy cooperad into an $L_\\infty$-algebra, used to define $C_{\\mathrm{NjA}}(V)$."},{"cited_title":"Das, Deformations of associative Rota-Baxter operators , J","cited_arxiv_id":null,"evidence_quote":"Establishes the classical correspondence between relative Rota-Baxter operators of weight zero and Nijenhuis operators, which Section 6 lifts to the homotopy level."},{"cited_title":"Drummond-Cole and B","cited_arxiv_id":null,"evidence_quote":"Gives the definition of a minimal model of a dg operad and the uniqueness statement used in Section 4."},{"cited_title":"Stashe ff, Homotopy associativity of H-spaces I , Proc","cited_arxiv_id":null,"evidence_quote":"Provides the A-infinity identity that Equation (13) reproduces for the higher products $m_n$."}],"review_version":1}