{"id":"1617a5ff-0427-4cf0-b130-765a23bb1b13","arxiv_id":"2504.19661","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For graded post-Lie structures on free Lie algebras, the coproduct dual to the Grossman-Larson product is given explicitly, yielding new dual Hopf algebra descriptions for the Ihara and ari brackets and a conjectural post-Lie model for the uri bracket.","lead":"This paper gives a general formula for the coproduct dual to the Grossman-Larson product on free Lie algebras, and applies it to the Ihara, ari, and uri brackets from the theory of multiple zeta values. The explicit dual Hopf algebra structures could become a new algebraic tool for studying multiple zeta values and their q-analogues.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The uri-bracket half rests on unproved Conjecture A.7, and even granting it the bm0 restriction is only conjectural; the unconditional core (Theorem 2.4, Ihara/ari) is not affected.","rationale":"The reader's weakest assumption identifies Conjecture A.7 as the crux, and I agree that this is the main load-bearing unproved input for the uri section. My stress-test adds a second, closely related gap: even if Conjecture A.7 were proved, the paper would still not establish the advertised Lie structure on bm0, since Remark 6.12 only conjectures that bm0 is closed under the uri bracket and that the expected bimould isomorphism holds. This does not move the verdict, because the reader already judged the paper CONDITIONAL and because the paper is transparent about both conditions. The central general theorem, Theorem 2.4, is independent of these conjectures and appears well supported by the dualization argument in Theorem 1.26. The Ihara and ari sections are proven unconditionally with explicit formulas. Therefore the appropriate verdict remains CONDITIONAL, with the condition being the proof or independent verification of Conjecture A.7 and, separately, the bm0 closure assertion.","tokens_in":53636,"tokens_out":10633,"duration_ms":115658,"concrete_test":"Verify the threshold shuffle identities of Definition A.5 by exact rational computation for all d1,d2 P Zě0 and all compositions with |σ|+|ϑ| ď 12, and independently check the two post-Lie identities (4)–(5) for the map Źu of Definition 5.5 on generators up to weight 12. If any identity fails, Theorem 5.11 and the uri portion collapse; if all pass, the remaining gap is precisely the unproved closure of bm0 under t·,·u_u.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing soft spot is the dependence of Section 5 on Conjecture A.7. Theorem 5.11 is explicitly conditional, but its proof reduces the post-Lie axioms exactly to the threshold shuffle identities of Definition A.5; no proof of this infinite Bernoulli-number family is supplied, and the numerical checks are not accompanied by code or data. Consequently Theorem 5.14 and the uri part of Section 6 are conditional in a strong sense: if Conjecture A.7 fails, there is no proved post-Lie structure pLie(V), r·,·s, Źu q and no dual Hopf pair for the uri bracket. Moreover, Conjecture A.7 is not the only obstruction to the advertised q-zeta application: Remark 6.12 only expects bm0 to be closed under t·,·u_u and expects the identification with BARIil,swap. Thus even assuming A.7, the statement that the induced bracket restricts to the dual of indecomposables of formal multiple q-zeta values remains conjectural. The unconditional core of the paper — Theorem 2.4 and the Ihara and ari applications — does not rely on this, so the general dual-coproduct contribution stands independently.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for post-Lie structures on free Lie algebras and for the Hopf algebra structures induced on their enveloping algebras via the Grossman-Larson product. Its central result, Theorem 2.4, gives an explicit formula for the coproduct dual to the Grossman-Larson product on the free algebra Q⟨A⟩ in terms of a reduced cotriangle map. This is applied to three families: the Ihara bracket, the ari bracket, and the uri bracket. For Ihara, the authors recover the Goncharov coproduct; for ari, they give explicit formulas for the Grossman-Larson product and its dual coproduct; for uri, they construct a post-Lie structure conditional on Conjecture A.7, a family of Bernoulli-number identities. Section 6 relates these structures to Ecalle's bimoulds, giving an identification for the ari bracket and a conjectural correspondence for the uri bracket. The paper is careful to mark many of the q-zeta statements as conjectural.","tokens_in":53898,"tokens_out":10785,"duration_ms":112651,"significance":"If the main results hold, Theorem 2.4 is a genuinely useful and clean contribution to the theory of dual Hopf algebras attached to post-Lie algebras; the explicit Ihara and ari formulas, together with the recovery of the Goncharov coproduct, are concrete and valuable. The identification of the ari bracket with a post-Lie structure on a free Lie algebra is an instructive bridge between post-Lie theory and Ecalle's bimoulds. The paper is honest about its conditional parts: the uri construction is explicitly contingent on Conjecture A.7, and the relation to formal multiple q-zeta values is described as expected rather than established. However, since the uri section and the advertised q-zeta application depend on unproved identities for which no reproducible numerical evidence is supplied, that part of the paper remains conditional in a strong sense.","major_comments":[{"comment":"Both theorems are conditional on Conjecture A.7, which asserts an infinite family of identities for Bernoulli numbers. The conjecture is not proved, and the numerical checks mentioned in the introduction and Appendix A.2 are not accompanied by code or data specifying the range or the computational method. Since Theorem 5.11 is the only route to the post-Lie structure for the uri bracket, this is load-bearing for the q-zeta application: if Conjecture A.7 fails, Theorem 5.14 and the uri part of Section 6 lose their foundation. Please either prove the conjecture, or include reproducible verification with a precise range, and state clearly in the abstract and introduction which results are unconditional.","section":"§5, Theorems 5.11 and 5.14"},{"comment":"Even assuming Conjecture A.7, the statement that the induced uri bracket restricts to bm0 and that bm0 is isomorphic to BARIil,swap is formulated as an expectation, not as a theorem. The abstract and introduction should not imply that the dual of indecomposables of formal multiple q-zeta values is known to carry this Lie structure; currently the q-zeta conclusion is doubly conjectural, depending both on Conjecture A.7 and on the unproved closure of bm0 under the uri bracket.","section":"§6, Remark 6.12"},{"comment":"The key identification of the ari post-Lie structure with Ecalle's arit bracket is delegated to [Bur23, Proposition 5.70] without reproducing the computation. Since this identification is central to the paper's claim that the ari bracket 'aligns with Ecalle's theory of bimoulds', the authors should either include the proof of that proposition or provide a detailed enough summary that the reader can verify the step without consulting the thesis.","section":"§6, Theorem 6.5"}],"minor_comments":[{"comment":"The formula for ∆a is very hard to parse as typeset, especially the conditions 'ls=H or |ls|≤kjs−1' and the notation 'mpkj1,...,jn+l, pkj1,...,jn'. Please add a short explanation of the indexing conventions or spell out a small example beyond the one already given.","section":"§4, Theorem 4.25"},{"comment":"The conjecture is stated abstractly, but the numerical evidence is only described as 'checked in a lot of cases'. Please specify the weight/depth ranges that were checked and include the PARI/GP code so the checks are reproducible.","section":"Appendix A.2, Conjecture A.7"},{"comment":"There are several typos and minor errors: 'Ecallé' in the first paragraph of Section 6, 'Propositon' in Proposition 4.27, 'unqiue' in the proof of Theorem 1.26, and 'f' missing in the sentence 'we focus on the algebra of formal multiple zeta values' in the introduction. These should be corrected.","section":"Throughout"},{"comment":"The generalization of Lemma 5.8 to n factors is stated as checked only up to degree 15 and not proved. If this formula is used in any of the paper's conclusions, its conjectural status should be flagged; otherwise it would be helpful to note explicitly that it is not needed for the main theorems.","section":"§5, Remark 5.9"}],"recommendation":"major_revision","confidential_remarks":"The unconditional core of the paper is solid and the authors are appropriately careful about the conditional nature of the uri results. My main concern is that the advertised q-zeta application rests on an unproved Bernoulli-number conjecture for which no reproducible verification is provided, and even under that conjecture some key identifications remain conjectural. I would like to see the authors either prove Conjecture A.7, provide full reproducible numerical evidence with a precise range, or clearly mark the uri part as conjectural in the abstract and introduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read on arXiv:2504.19661.\n\nThe paper does two things. First, it proves a genuinely new general formula for the graded dual coproduct of a Grossman-Larson product on a free Lie algebra: for a post-Lie structure on Lie(A), the dual coproduct of a word is a sum over shuffles/concatenations with the reduced triangle map. I checked the logic in Theorem 2.4 and Proposition 1.26, and the proof is coherent, with explicit examples taking up real space. That alone is a solid contribution to the post-Hopf algebra literature. Second, it applies this framework to three brackets: the Ihara bracket, the ari bracket, and the uri bracket. The Ihara section gives a generalized magmatic construction and a clean formula for the dual coproduct that recovers Goncharov's coproduct without iterated integrals. The ari section is unconditional: explicit formulas for the Grossman-Larson product and its dual coproduct, with a verification that the post-Lie axioms hold. These parts stand on their own.\n\nThe soft spots are all in the uri half, and the authors flag most of them. Theorem 5.11 is explicitly conditional on Conjecture A.7, a family of Bernoulli-number identities that is checked numerically but not proved and without code or data. If that conjecture fails, the post-Lie structure for uri collapses; the paper then loses the q-MZV application. Even granting A.7, Remark 6.12 only expects bm0 to be closed under the bracket and expects the identification with BARIil,swap; no proof is given. The bimould identification for ari in Theorem 6.5 is also delegated to a calculation in the first author's thesis, so a referee cannot fully verify it from the text. None of this threatens the unconditional core, but it does mean the uri part is an invitation to a conjecture rather than a theorem.\n\nWho should read this? People working on post-Lie algebras and Hopf structures in MZV theory, and anyone trying to get a handle on Ecalle's bimould brackets via free Lie algebras. The paper is honestly written, with the conditional status of the uri section made explicit. It deserves a serious referee: the unconditional results are interesting and proved with care, and the conjectural part is clearly framed as such. I would engage with it and probably cite it for the dual-coproduct formula and the ari formulas. It is not a candidate for desk rejection, but it should be sent for review with a note that the uri section is conditional and the thesis-dependent calculation should be checked.","headline":"Solid unconditional core (general dual-coproduct formula plus Ihara/ari constructions) with a clearly flagged conjectural uri-bracket half; deserves peer review, not desk rejection.","tokens_in":54430,"tokens_out":2192,"would_cite":true,"duration_ms":22313,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B01","17B70","16T05","11M32","11B68"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that every graded post-Lie structure on a free Lie algebra determines an explicit dual coproduct for its Grossman-Larson product, and that the Ihara, ari, and uri brackets fit this pattern, yielding a conditional post-Lie…","keywords":["post-Lie algebras","free Lie algebras","Grossman-Larson product","dual coproduct","multiple zeta values","multiple q-zeta values","Bernoulli numbers","bimoulds"],"falsifier":"Compute the Bernoulli numbers $B_1(m,n)$ and test the threshold shuffle identities of Definition A.5 for parameter tuples $(d_1,d_2,\\sigma,\\tau,\\vartheta)$ beyond those checked numerically; a single failure would disprove Conjecture A.7 and remove the post-Lie axiom from $(\\mathrm{Lie}(V),\\triangleright_u)$, while a proof for all tuples would complete Theorem 5.11. Alternatively, search for a word $w$ whose $\\Delta_a(w)$ right tensor factor violates the conjectural k-level filtration of Conjecture 4.29.","tokens_in":53406,"feed_emoji":"🧮","tokens_out":6647,"duration_ms":65576,"temperature":0.7,"pith_summary":"The paper works out the Hopf-algebra consequences of putting a post-Lie structure on a free Lie algebra. It gives what the authors describe as the first general description of the coproduct dual to the Grossman-Larson product, expressed through word decompositions and a reduced triangle map. Applying this machinery to three brackets, it recovers the Goncharov coproduct for multiple zeta values and constructs a candidate post-Lie algebra whose associated Lie bracket should control formal multiple q-zeta values. The q-zeta part is conditional on an unproved family of Bernoulli-number identities, the threshold shuffle identities; the depth-graded version, the ari bracket, is unconditional.","feed_headline":"New post-Lie structures reach multiple q-zeta values","feed_subtitle":"Explicit dual coproducts recover known MZV constructions and, modulo a Bernoulli-number conjecture, build the uri bracket.","key_machinery":"The load-bearing object is a post-Lie algebra $(\\mathrm{Lie}(A), [\\cdot,\\cdot], \\triangleright)$: a free Lie algebra with its usual bracket and a bilinear product $\\triangleright$ that acts as a derivation in the second argument and satisfies the post-Lie identity that makes $x\\triangleright y - y\\triangleright x + [x,y]$ a second Lie bracket. The central computational tool is the reduced triangle map $\\Phi^{\\mathrm{irr}}$, which records only the part of the dual of $\\triangleright$ whose right factor is a single generator; the universal formula for the dual coproduct inserts $\\Phi^{\\mathrm{irr}}$ at every split of a word. For the uri bracket, the additional combinatorial ingredient is the family of threshold shuffle identities, conditions on Bernoulli numbers that are needed to verify the post-Lie axiom for the uri triangle map.","core_discovery":"For any graded post-Lie algebra on a free Lie algebra $(\\mathrm{Lie}(A), [\\cdot,\\cdot], \\triangleright)$, the coproduct $\\Delta_f$ dual to the Grossman-Larson product $f$ is given on a word $A$ by summing over all decompositions $A=A_1\\cdots A_n$ of $(A_1\\otimes 1) \\;\\bar{\\shuffle}\\; \\Phi^{\\mathrm{irr}}(A_2) \\;\\bar{\\shuffle}\\; \\cdots \\;\\bar{\\shuffle}\\; \\Phi^{\\mathrm{irr}}(A_n)$, where $\\Phi^{\\mathrm{irr}}$ is the reduced triangle map dual to $\\triangleright$ on single-letter right factors and $\\bar{\\shuffle}$ is shuffle on the left tensor factor times concatenation on the right. The paper shows that for the Ihara bracket on $\\mathrm{Lie}(x_0,x_1)$ this formula reproduces the Goncharov coproduct, and for the ari bracket it yields explicit and effective formulas for the Grossman-Larson product and its dual coproduct. For the uri bracket, assuming the Bernoulli numbers satisfy the threshold shuffle identities, the triple $(\\mathrm{Lie}(V),[\\cdot,\\cdot],\\triangleright_u)$ is a post-Lie algebra whose induced Lie bracket is expected to restrict to the dual of indecomposables of formal multiple q-zeta values.","pith_inferences":["If the conjectured one-parameter family of threshold shuffle identities holds, the uri construction deforms as a parameter $t$ varies, suggesting a whole family of post-Lie structures interpolating between the depth-graded ari model and the full uri model; the paper does not explore this deformation.","The reduced-triangle coproduct formula is likely to be a general template for computing dual coproducts in other post-Lie settings, such as Lie-Butcher theory or regularity structures, where explicit dual coproducts are often hard to obtain.","The conjectural k-level filtration on $\\Delta_a$ can be tested computationally on words of increasing length; a proof would sharpen the analogy with the filtration arguments used for classical multiple zeta values and might point toward an explicit generating set for formal multiple q-zeta values.","If the expected isomorphism between $\\mathfrak{bm}_0$ with the uri bracket and alternil bimoulds with the uri bracket is established, it would independently show that $\\mathfrak{bm}_0$ is a Lie algebra, since the bimould operation is known to preserve the relevant space in low depths."],"forward_implications":["Every graded post-Lie structure on a free Lie algebra yields a dual pair of graded Hopf algebras $(\\mathbb{Q}\\langle A\\rangle, \\shuffle, \\Delta_f) \\leftrightarrow (\\mathbb{Q}\\langle A\\rangle, f, \\Delta_{\\shuffle})$, giving a uniform framework for Goncharov-type coproducts.","For the Ihara bracket, the dual coproduct equals the Goncharov coproduct, so that construction is recovered without formal iterated integrals.","The ari-bracket Hopf pair is unconditional: explicit formulas for $f_a$ and $\\Delta_a$ give a concrete depth-graded model for formal multiple q-zeta values modulo formal quasi-modular forms.","If the threshold shuffle identities hold, the uri bracket produces a dual Hopf pair whose depth-graded version is exactly the ari-bracket Hopf pair, connecting the two models by a natural isomorphism.","The right tensor factor of $\\Delta_a$ preserves the subspace generated by $v_1,v_2,v_3$, a property that the paper views as a step toward a q-analogue of the theorem that multiple zeta values are spanned by indices in $\\{2,3\\}$."],"supporting_citations":[{"why":"Provides the recursive definition of the Grossman-Larson product and the theorem that $f$ makes $U(\\mathfrak{g})$ a Hopf algebra, which the paper dualizes.","marker":"[EFLMK15]"},{"why":"Defines the Goncharov coproduct that the Ihara-bracket specialization of the dual coproduct reproduces.","marker":"[Gon05]"},{"why":"Introduces the double shuffle Lie algebra $\\mathfrak{dm}_0$ and proves it is a Lie subalgebra for the Ihara bracket, the classical setting the paper extends.","marker":"[Rac00]"},{"why":"Supplies the theorem that multiple zeta values with indices in $\\{2,3\\}$ span the algebra, motivating the filtration properties studied for the ari bracket.","marker":"[Bro12]"},{"why":"Defines the space $\\mathfrak{bm}_0$, relates noncommutative polynomial models to alternil bimoulds, and supplies the translation statements used for the uri bracket.","marker":"[Bur23]"},{"why":"Defines the ari and uri operations on bimoulds that the paper identifies with its post-Lie-induced brackets.","marker":"[Eca11]"},{"why":"Proposed the uri bracket as a conjectural Lie structure for multiple q-zeta values, the target of the paper's uri construction.","marker":"[Küh19]"},{"why":"Establishes the surjective algebra morphism from the shuffle algebra on $V$ to $Z_q$ that underlies the formal multiple q-zeta setup.","marker":"[BB23]"}],"fun_headline_variants":["Dual coproduct formula for post-Lie algebras","Post-Lie coproduct reproduces Goncharov","Explicit post-Lie coproduct for MZV and q-MZV","New dual coproduct formula targets q-zeta","Post-Lie structures yield explicit dual coproducts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire uri-bracket construction rests on Conjecture A.7: the Bernoulli numbers must satisfy the threshold shuffle identities, a family of identities checked numerically in many cases but not proved; if they fail, the uri post-Lie algebra and its dual Hopf pair collapse, leaving only the unconditional depth-graded ari results.","fun_headline_variants_meta":{"raw":{"variants":["Dual coproduct formula for post-Lie algebras","Post-Lie coproduct reproduces Goncharov","Explicit post-Lie coproduct for MZV and q-MZV","New dual coproduct formula targets q-zeta","Post-Lie structures yield explicit dual coproducts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001293,"raw_usage":{"total_tokens":5309,"prompt_tokens":1003,"completion_tokens":4306,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":619,"completion_tokens_details":{"reasoning_tokens":4222}},"tokens_in":619,"tokens_out":4306,"duration_ms":28428,"temperature":1.0,"reasoning_tokens":4222,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:47:37.496146+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Bernoulli numbers $B_1(m,n)$ and test the threshold shuffle identities of Definition A.5 for parameter tuples $(d_1,d_2,\\sigma,\\tau,\\vartheta)$ beyond those checked numerically; a single failure would disprove Conjecture A.7 and remove the post-Lie axiom from $(\\mathrm{Lie}(V),\\triangleright_u)$, while a proof for all tuples would complete Theorem 5.11. Alternatively, search for a word $w$ whose $\\Delta_a(w)$ right tensor factor violates the conjectural k-level filtration of Conjecture 4.29.","supporting_citations":[],"review_version":1}