Pith. sign in

REVIEW 3 major objections 5 minor 60 references

This paper establishes that the motivic Lie algebra of mixed Tate motives over the integers embeds canonically into the compactly-supported cohomology of GL_g(Z) and of the moduli space of abelian varieties.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 02:08 UTC pith:6PYBRL2E

load-bearing objection The embedding result is real and well-built, but the decisive bridge (I^can = I^RW) is proved only in the third author's companion preprint, so the paper is conditional on that external verification. the 3 major comments →

arxiv 2607.29671 v1 pith:6PYBRL2E submitted 2026-07-31 math.AG math.ATmath.QA

The motivic Lie algebra embeds into the cohomology of the general linear group

Francis Brown , Erik Panzer , Jean-Luc Portner This is my paper
classification math.AG math.ATmath.QA MSC 11F7514G3514T20
keywords motivic Lie algebramixed Tate motivescompactly-supported cohomologygeneral linear groupgraph complextropical Torelli mapcanonical graph integralsmoduli of abelian varieties
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the motivic Lie algebra, the pro-nilpotent Galois group governing mixed Tate motives over Z, appears concretely as a canonical Lie subalgebra inside the unstable compactly-supported cohomology of the locally symmetric spaces for GL_g(Z) and inside the weight-zero compactly-supported cohomology of the moduli space A_g of principally polarized abelian varieties. The route passes through tropical geometry: compactly-supported analogues of the stable cohomology classes pull back along the tropical Torelli map to graph cocycles defined by canonical graph integrals. The paper identifies these integrals with previously studied configuration-space integrals and uses single-valued periods to prove that the resulting cocycles are motivic and generate the motivic Lie algebra. If correct, this makes the motivic Lie algebra a visible, geometrically constructed part of the cohomology of arithmetic groups and moduli spaces, not merely an object from algebraic K-theory and the projective line minus three points.

Core claim

The paper's central claim is Theorem 3: there is a canonical embedding of the free graded Lie algebra on symbols omega_c^5, omega_c^9, ... into the degree-zero graph cohomology H^0(GC_2) tensor R, sending each generator to the canonical graph cocycle C^can_{4k+1} built from graphs with 4k+2 edges and 2k+2 vertices. The paper proves that these classes are motivic: their images in the Ihara Lie algebra lie in and generate the motivic Lie subalgebra g_m tensor R, yielding a canonical isomorphism between the free Lie algebra on the compactly-supported classes and g_m tensor R. Through the tropical Torelli pullback and the Hopf algebra structure on compactly-supported cohomology, the same Lie alg

What carries the argument

The load-bearing objects are the canonical graph integrals I^can_G, defined by integrating tr((Lambda_G^{-1} dLambda_G)^{4k+1}) over the simplex of positive edge-lengths on a graph, and the Rossi-Willwacher configuration-space integrals I^RW_G over positions of vertices in the complex plane. Their equality for every graph, stated as Theorem 10 and proved in a companion paper, turns the canonical cocycles into known deformation-quantization cocycles. The tropical Torelli map pulls compactly-supported Borel-type classes on tropical abelian varieties back to these canonical graph cocycles, and the Lie-theoretic map from graph cohomology to the Ihara Lie algebra transports them into the motivic

Load-bearing premise

The whole chain of theorems depends on the claim, proved in a companion paper rather than here, that two very different ways of integrating over graphs—one over edge-length simplices and one over vertex configurations—always give the same number; if that equality fails, the canonical cocycles are not the known motivic ones and the main theorems no longer follow.

What would settle it

Take a trivalent graph with 18 edges and 10 vertices, compute I^can_G and I^RW_G to high precision, and compare them; any nonzero difference would refute Theorem 10 and with it the paper's central conclusions.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The motivic Lie algebra g_m tensor R is realized as a canonical subspace of H^0(GC_2) tensor R, with explicit graph-cocycle representatives for every generator.
  • The free Lie algebra generated by the compactly-supported classes [omega_c^{4k+1}] embeds primitively into the compactly-supported cohomology of GL_g(Z) and into W_0 H_c^{2g}(A_g;R), so all higher commutators yield additional cohomology classes.
  • The cohomology of A_g contains a copy of the motivic Lie algebra in weight-zero compactly-supported cohomology, independent of conjectures about the Grothendieck-Teichmüller Lie algebra.
  • By Poincaré duality, corresponding classes appear in the homology of SL_g(Z); the paper expects these classes to be permanent in the Quillen spectral sequence and to compute Borel regulators via wheel-graph periods.
  • Canonical graph cocycles are computed explicitly through 14 edges, including a new cocycle C^can_13 supported on 56 graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the equality of integral families survives close scrutiny, the same recipe of pulling compactly-supported stable classes back along tropical Torelli maps is likely to embed the motivic Lie algebra into the cohomology of other arithmetic groups, a possibility the paper explicitly anticipates for the symplectic group.
  • The identity between parametric Feynman integrals and position-space configuration integrals suggests a concrete computational pipeline: single-valued periods and double-copy formulas may express individual generators of g_m as explicit period integrals, not merely as abstract Lie words.
  • A testable consequence of the canonical isomorphism is that each canonical graph cocycle should reduce modulo commutators to a fixed rational multiple of zeta(2k+1) times the corresponding generator of g_m; verifying this for k >= 4 would extend the low-degree consistency checks in the appendix.
  • The paper leaves open whether the embedding into the primitives of the cohomology of A_g is surjective; if it is, the motivic Lie algebra would be exactly the diagonal part of the Hopf algebra of A_g, giving a new and very concrete characterization of g_m.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper claims that the motivic Lie algebra g_m of mixed Tate motives over Z embeds canonically into the compactly-supported cohomology of GL_g(Z), into the weight-zero compactly-supported cohomology of A_g, and into H^0(GC_2)⊗R. The construction passes through the tropical Torelli map: compactly-supported Borel classes pull back to canonical graph cocycles C^can_{4k+1}, given by integrals I^can_G. These are identified with Rossi–Willwacher integrals I^RW_G via a concurrent preprint [Por26], and the associated Lie words are shown, using single-valued periods and [RW14, Thm 1.4], to lie in and generate g_m⊗R. The main theorems are Theorem 1 (embedding into H^2g_c(P_g/GL_g(Z);R)), Theorem 2 (embedding into W_0 H^{2g}_c(A_g;R)), and Theorem 3 (canonical isomorphism L(ω^5_c,ω^9_c,…)⊗R ≅ g_m⊗R). The paper also computes explicit canonical cocycles for graphs with ≤14 edges in Appendix A.

Significance. If the main results are correct, they give a new and remarkably concrete realization of the motivic Lie algebra: the pro-nilpotent Galois group of mixed Tate motives over Z is exhibited inside unstable compactly-supported cohomology of arithmetic groups and inside the graph complex, without reference to the Grothendieck–Teichmüller Lie algebra. The paper is clearly organized, and the low-degree computations and ancillary files provide useful concrete data. The principal weakness is that the central bridge, the equality I^can_G = I^RW_G, is not proved in this paper; it is entirely delegated to the third author's concurrent preprint [Por26]. Consequently the main theorems are conditional on an external result that is not auditable from the present manuscript. I found no internal logical inconsistency in the surrounding argument, but the verification gap is load-bearing.

major comments (3)
  1. [§2.4, Theorem 10] The equality I^can_G = I^RW_G is the bridge between the canonical graph cocycles arising from the compactly-supported Borel forms and the Rossi–Willwacher cocycles to which [RW14] applies. Its proof is delegated in full to the concurrent preprint [Por26]. Definition 12, Theorem 19, and therefore Theorems 1–3, 21, and 24 all cross this bridge. Appendix A computes C^can_13 by evaluating I^RW_G with adapted kontsevint code, not by evaluating the canonical integrals of Definition 6, so the computations provide no independent test of the equality. A normalization mismatch (e.g., the factor in footnote 2) or a subtle domain issue in the Schwinger correspondence would invalidate the main conclusions. This is a load-bearing verification gap, not an internal inconsistency.
  2. [§4, Theorem 19] The proof relies on [RW14, Theorem 1.4] (ψ∘Z = Z^σ), the uniqueness in (20), and Corollary 18. The displayed constant in Theorem 19 depends on the normalization of Definition 9 via footnote 2 and on c_{2k} from (23). Since neither the relevant statement of [RW14] nor the comparison of conventions is reproduced, the exact factors in [φ_{2k+1}] = −2(4k+1) binom(4k,2k) ζ(2k+1)[σ_{2k+1}] cannot be checked from the manuscript. It would be helpful to restate the necessary part of [RW14] and to verify the sign and power-of-2 factors explicitly.
  3. [§5.4, Theorem 24] The authors explicitly state that Theorem 24 is not a formal consequence of Theorem 21 and depends on the lift to W_0 H^{2g}_c(A_g) via the inflation sequence of [Bra+24]. The lift is not proved here. If [Bra+24] is published, this is an acceptable citation, but its role should be stated precisely and the relevant statement quoted, so that the proof of Theorem 2 is auditable.
minor comments (5)
  1. [§2.5, Definition 12] The notation τ_{2k+1} in footnote 3 is not introduced in the main text; clarify the relation to φ_{2k+1} and to the normalization of Definition 9.
  2. [§5.2] The graph-complex variant GC'_2 is only referenced. Since the factorization through H^0(GC_2) in (28) is a key step, define GC'_2 and state its relation to GC_2, or give a precise reference to the relevant statement in [Bro25].
  3. [§2.1] The Lie bracket on GC_2 is quoted from [BHP26]. Since the paper uses Lie-algebra morphisms, state which bracket convention is used and check sign conventions; the sign in the bracket formula is important for the claim that λ* and φ are Lie algebra morphisms.
  4. [Theorem 20] The sentence 'It is strongly suspected that these two Hopf algebras are one and the same' is ambiguous. Later the paper uses only the [Bro+24] construction; clarify that all subsequent primitives and Lie-algebra structures are taken with respect to that Hopf algebra.
  5. [Bibliography and Appendix A] The reference '[R W14]' has a spurious space; the code name 'kontsevint' should be formatted consistently as code or linked; the table captions should state that the full list of graphs is in the ancillary files.

Circularity Check

0 steps flagged

No circular reduction found: the derivation relies on external results and a deferred equality in [Por26], but that is a verification gap, not a by-construction circularity.

full rationale

The paper's derivation chain does not exhibit any step in which a prediction or conclusion is equivalent to its inputs by definition, by fitted parameters, or by a self-citation chain. The central bridge is Theorem 10, I^can_G = I^RW_G, which is delegated to the third author's concurrent preprint [Por26]. This is a real verification/completeness gap: the paper's own Appendix A computes the 'canonical' cocycles using the Rossi–Willwacher integrals rather than directly evaluating the canonical integrals of Definition 6, so the paper does not independently test the equality. However, an external or deferred proof is not the same as circularity. The equality is not assumed in the definition of I^can or I^RW, and no equation in the paper reduces to the target embedding. The other load-bearing inputs are established results with independent content: [RW14, Thm 1.4] is by Rossi and Willwacher; [Bro12] proves injectivity of the motivic Lie algebra into the Ihara Lie algebra; [Bro14b] supplies the single-valued period machinery; [Bro25] and [Bro+24] supply the geometric and Hopf-algebra structures. These are self-citations by the first author, but they are prior published theorems with independent proofs, not assumptions of the conclusion. The use of such citations is normal mathematical practice and does not constitute circularity. The paper could be vulnerable if [Por26]'s equality fails or if the normalization conventions differ, but that is a correctness/verification risk, not a circular one. Therefore the honest finding is no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The central claim rests on: (i) the established motivic framework — Tannakian MT(Z), Borel's K-theory computation of the free Lie algebra structure of g_m, the injectivity of ι: g_m → (L(e0,e1),{,}) ([Bro12]), and the single-valued period map sv ∈ U_dR(R) with Z_sv ∘ Z_σ = Z ([Bro14b]); (ii) two external theorems not proved in this paper: [RW14, Thm 1.4] (published, independent) and Theorem 10 of [Por26] (concurrent preprint by the third author, unverified); (iii) the geometry chain from [Bro25], [Bro+24], [Bra+24] (bordifications, tropical Torelli pull-back, Hopf algebra primitivity, inflation sequence), quoted rather than proved. No free parameters are fitted and no entities are invented; the new objects (C^can_{4k+1}, [ω^{4k+1}_c]) are explicitly defined and computed, not postulated.

axioms (7)
  • domain assumption The category MT(Z) of mixed Tate motives over Z is Tannakian over Q, with a graded fiber functor; Lie(U_dR) = g_m is a free graded Lie algebra on one generator σ_{2k+1} in each odd degree (via Borel's K-theory and the injectivity of ι, [Bro12]).
    Background established elsewhere (§3.1.4); the paper cites [Lev98], [DG05], [Bro12]. Not proved in this paper.
  • domain assumption The single-valued period map sv ∈ U_dR(R) exists and satisfies Z_sv ∘ Z_σ = Z with ζ_sv(2k+1)=2ζ(2k+1), ζ_sv(2n)=0.
    From [Bro14b] (§3.3); the paper derives Corollary 18 from it. Prior published work of the first author.
  • domain assumption [RW14, Theorem 1.4]: for the Rossi–Willwacher elements ϕ_{2k+1}, ψ = Pexp_∘(Σ ∫ α_{2k} ϕ_{2k+1}) satisfies ψ ∘ Z = Z_σ.
    External theorem cited without proof (§4, proof of Theorem 19). Not auditable here; the associator convention mapping to x0 = X/(2πi), x1 = Y/(2πi) is given in a footnote.
  • ad hoc to paper I^can_G = I^RW_G for all graphs (Theorem 10, [Por26]).
    The principal new bridge: canonical graph integrals equal Rossi–Willwacher integrals. Proof is deferred to a concurrent preprint by the third author, arXiv:2607.25595; not included in this paper. The entire Theorem 19 depends on it.
  • domain assumption The compactly-supported Borel classes [ω^{4k+1}_c] exist, are nonzero, and pull back along the tropical Torelli map to the canonical cocycles [C^can_{4k+1}]; λ* is a morphism of graded Lie algebras (Prim H^Δ_c → H^0(GC_2)).
    From [Bro25] and [Bro+24] (§5.1–5.2); technical bordification and Hopf-algebra arguments are cited, not reproduced.
  • domain assumption The Hopf algebra structure on ⊕_g W_0 H^{2g}_c(A_g) and the lift of [ω^{4k+1}_c] to its primitives (via the inflation sequence of [Bra+24]).
    Quoted from [Bro+24] and [Bra+24]; §5.4 notes Theorem 24 is not a formal consequence of Theorem 21 alone.
  • domain assumption Computational infrastructure: the Rossi–Willwacher integrals were evaluated by adapting the kontsevint code of [BPP20]; in weights 3, 5, 7 the space of single-valued multiple zeta values is one-dimensional, so I^RW_G = q_G ζ(2k+1).
    Appendix A; code is linked but no parameter files/commit hash; the one-dimensionality of SV MZV spaces in those weights is standard.

pith-pipeline@v1.3.0-daily-deepseek · 25392 in / 30120 out tokens · 287340 ms · 2026-08-03T02:08:24.493798+00:00 · methodology

0 comments
read the original abstract

We show that the motivic Lie algebra of mixed Tate motives over $\mathbb{Z}$ embeds canonically into the unstable compactly-supported cohomology of locally symmetric spaces for $\mathrm{GL}_g(\mathbb{Z})$, and into the weight-zero compactly-supported cohomology of $\mathcal{A}_g$, the moduli space of principally polarized abelian varieties. Our construction passes through tropical geometry and graph complexes: compactly-supported analogues of the Borel classes pull back via the tropical Torelli map to canonical graph cocycles. The latter were recently identified with cocycles studied previously by Rossi and Willwacher. We combine their results with the theory of single-valued periods to conclude that the cocycles map to generators of the motivic Lie algebra. We also compute the canonical graph cocycles for all graphs with $\leq14$ edges.

Figures

Figures reproduced from arXiv: 2607.29671 by Erik Panzer, Francis Brown, Jean-Luc Portner.

Figure 1
Figure 1. Figure 1: The graph complex differential (edge contractions). 1.4. Acknowledgments. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe programme (grant agreement No. 101167287). Erik Panzer is funded as a Royal Society University Research Fellow through grant URF\R\251041. For the purpose of Open Access, the authors have applied a CC BY public copyrig… view at source ↗
Figure 2
Figure 2. Figure 2: The 3 types of Lie trees obtained from the wheel with 5 spokes graph. 2.4. Equality of the cocycles. In [Por26] the third-named author proved Theorem 10. The canonical and Rossi–Willwacher integrals are equal, I can G = I RW G , for all graphs. Therefore, the corresponding graph cocycles are identical: C can = C RW . 2.5. From graphs to words. Willwacher constructed in [Wil15] a morphism of graded Lie alge… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

60 extracted references · 8 canonical work pages

  1. [1]

    , TITLE =

    Lee, R. , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1978 , NUMBER =

    1978

  2. [2]

    and Hu, S

    Brown, F. and Hu, S. and Panzer, E. , fjournal =. Unstable cohomology of. Adv. Math. , volume =. 2026 , doi =. 2406.12734 , archivePrefix=

    Pith/arXiv arXiv 2026

  3. [3]

    , TITLE =

    Garland, H. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1971 , PAGES =

    1971

  4. [4]

    and Serre, J.-P

    Borel, A. and Serre, J.-P. , TITLE =. Comment. Math. Helv. , FJOURNAL =. 1973 , PAGES =

    1973

  5. [5]

    and Bruce, J

    Brandt, M. and Bruce, J. and Chan, M. and Melo, M. and Moreland, G. and Wolfe, C. , title =. 2024 , archivePrefix =. doi:10.2140/gt.2024.28.497 , journal =. 2012.02892 , primaryClass =

    Pith/arXiv arXiv 2024

  6. [6]

    Kontsevich, Maxim , TITLE =. Ast\'. 2019 , PAGES =

    2019

  7. [7]

    , title =

    Portner, J.-L. , title =. 2026 , month = jul, archivePrefix =. 2607.25595 , primaryClass =

    Pith/arXiv arXiv 2026

  8. [8]

    and Kiselev, A

    Buring, R. and Kiselev, A. V. and Rutten, N. J. , title =. Physics of Particles and Nuclei , year = 2018, month = sep, volume =. doi:10.1134/S1063779618050118 , archivePrefix =. 1712.05259 , primaryClass =

    Pith/arXiv arXiv 2018

  9. [9]

    , title =

    Kontsevich, M. , title =. Deformation theory and symplectic geometry. Proceedings of the Ascona meeting, Switzerland, June 17--21, 1996 , isbn =. 1997 , publisher =

    1996

  10. [10]

    On Motives Associated to Graph Polynomials , Author =. Commun. Math. Phys. , Year =. arXiv , Doi =:math/0510011 , ISSN =

    Pith/arXiv arXiv

  11. [11]

    Inventiones Mathematicae , Year =

    Logarithms and deformation quantization , Author =. Inventiones Mathematicae , Year =. arXiv , Doi =:1401.3200 , ISSN =

    Pith/arXiv arXiv

  12. [12]

    , Journal =

    Brown, F. , Journal =. Multiple zeta values and periods of moduli spaces. 2009 , Month = jun, Number =

    2009

  13. [13]

    and Dupont, C

    Brown, F. and Dupont, C. , title =. Commun. Math. Phys. , year =. doi:10.1007/s00220-021-03969-4 , archiveprefix =. 1910.01107 , primaryclass =

    Pith/arXiv arXiv 1910

  14. [14]

    and Dupont, C

    Brown, F. and Dupont, C. , title =. J. Reine Angew. Math. doi:doi:10.1515/crelle-2020-0042 , year =

    work page doi:10.1515/crelle-2020-0042 2020

  15. [15]

    Multiple zeta values in deformation quantization , Author =. Invent. math. , fjournal =. doi:10.1007/s00222-020-00970-x , Year =

    doi:10.1007/s00222-020-00970-x

  16. [16]

    and Kiselev, A

    Buring, R. and Kiselev, A. V. , title =. J. Phys.: Conf. Ser. , publisher =. doi:10.1088/1742-6596/1194/1/012017 , archivePrefix =. 1811.07878 , primaryClass =

    Pith/arXiv arXiv

  17. [17]

    and Kiselev, A

    Buring, R. and Kiselev, A. V. and Rutten, N. , title =. Journal of Physics: Conference Series , publisher =. doi:10.1088/1742-6596/965/1/012010 , archivePrefix =. 1710.02405 , primaryClass =

    Pith/arXiv arXiv

  18. [18]

    and Kiselev, A

    Buring, R. and Kiselev, A. V. and Rutten, N. J. , title =. Journal of Nonlinear Mathematical Physics , volume =. 2017 , publisher =. doi:10.1080/14029251.2017.1418060 , archivePrefix =. 1710.00658 , primaryClass =

    arXiv 2017

  19. [19]

    Algebra Number Theory , FJOURNAL =

    Chan, Melody , TITLE =. Algebra Number Theory , FJOURNAL =. 2012 , NUMBER =. doi:10.2140/ant.2012.6.1133 , URL =

    doi:10.2140/ant.2012.6.1133 2012

  20. [20]

    , title =

    Willwacher, T. , title =

  21. [21]

    , Booktitle =

    Brown, F. , Booktitle =. Motivic periods and. 2014 , pages =

    2014

  22. [22]

    and Vogtmann, K

    Conant, J. and Vogtmann, K. , Title =. Algebr. Geom. Topol. , ISSN =. 2003 , DOI =

    2003

  23. [23]

    and Melo, M

    Brannetti, S. and Melo, M. and Viviani, F. , Title =. Adv. Math. , ISSN =. 2011 , DOI =

    2011

  24. [24]

    , title =

    Willwacher, T. , title =. doi:10.48550/arXiv.2503.17131 , archivePrefix =. 2503.17131 , primaryClass =

    Pith/arXiv arXiv doi:10.48550/arxiv.2503.17131

  25. [25]

    , TITLE =

    Willwacher, T. , TITLE =. Invent. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00222-014-0528-x , Archiveprefix =. 1009.1654 , Primaryclass =

    Pith/arXiv arXiv 2015

  26. [26]

    and Galatius, S

    Chan, M. and Galatius, S. and Payne, S. , Title =. J. Am. Math. Soc. , ISSN =. 2021 , Publisher =. doi:10.1090/jams/965 , MSC2010 =

    work page doi:10.1090/jams/965 2021

  27. [27]

    Duke Math

    Caporaso, Lucia and Viviani, Filippo , TITLE =. Duke Math. J. , FJOURNAL =. 2010 , NUMBER =

    2010

  28. [28]

    , TITLE =

    Deligne, P. , TITLE =. Inst. Hautes \'. 1971 , PAGES =. doi:10.1007/BF02684692 , ISSN =

    doi:10.1007/bf02684692 1971

  29. [29]

    , TITLE =

    Matsushima, Y. , TITLE =. Osaka Math. J. , FJOURNAL =. 1962 , PAGES =. doi:10.18910/5711 , ISSN =

    work page doi:10.18910/5711 1962

  30. [30]

    and Miller, J

    Ash, A. and Miller, J. and Patzt, P. , title =

  31. [31]

    and Calegari, F

    Boxer, G. and Calegari, F. and Gee, T. , title =. J. Am. Math. Soc. , issn =. 2025 , doi =

    2025

  32. [32]

    and Chan, M

    Brown, F. and Chan, M. and Galatius, S. and Payne, S. , title =

  33. [33]

    , title =

    Brown, F. , title =. Invent. Math. , year = 2025, pages =. doi:10.1007/s00222-025-01335-y , archivePrefix =. 2309.12753 , primaryClass =

    Pith/arXiv arXiv 2025

  34. [34]

    , TITLE =

    Brown, F. , TITLE =. SIGMA Symmetry Integrability Geom. Methods Appl. , FJOURNAL =. 2021 , PAGES =. doi:10.3842/SIGMA.2021.103 , archivePrefix =. 2101.04419 , primaryClass =

    Pith/arXiv arXiv 2021

  35. [35]

    and Schnetz, O

    Brown, F. and Schnetz, O. , title =. Geom. Topol. , volume =. doi:10.2140/gt.2025.29.4389 , archivePrefix =. 2402.06757 , primaryClass =

    arXiv 2025

  36. [36]

    , title =

    Magnus, W. , title =. Commun. Pure Appl. Math. , issn =. 1954 , doi =

    1954

  37. [37]

    , title =

    Chen, K.-T. , title =. Ann. Math. (2) , volume =. 1957 , doi =

    1957

  38. [38]

    Rossi, C. A. and Willwacher, T. , title =. 2014 , adsurl =

    2014

  39. [39]

    , Journal =

    Brown, F. , Journal =. Mixed. 2012 , Number =. doi:10.4007/annals.2012.175.2.10 , Archiveprefix =. 1102.1312 , Primaryclass =

    Pith/arXiv arXiv 2012

  40. [41]

    , TITLE =

    Waterhouse, William C. , TITLE =. 1979 , PAGES =

    1979

  41. [42]

    , TITLE =

    Borel, A. , TITLE =. Ann. Sci. \'. 1974 , PAGES =. doi:10.24033/asens.1269 , ISSN =

    work page doi:10.24033/asens.1269 1974

  42. [43]

    Cycles, transfers, and motivic homology theories , SERIES =

    Voevodsky, Vladimir , TITLE =. Cycles, transfers, and motivic homology theories , SERIES =. 2000 , ISBN =

    2000

  43. [44]

    Hanamura, Masaki , TITLE =. Invent. Math. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/s002220000091 , URL =

    work page doi:10.1007/s002220000091 2000

  44. [45]

    , TITLE =

    Levine, M. , TITLE =. 1998 , PAGES =. doi:10.1090/surv/057 , URL =

    work page doi:10.1090/surv/057 1998

  45. [46]

    , TITLE =

    Deligne, P. , TITLE =. The. 1990 , ISBN =

    1990

  46. [47]

    1972 , PAGES =

    Saavedra Rivano, Neantro , TITLE =. 1972 , PAGES =

    1972

  47. [48]

    , TITLE =

    Levine, M. , TITLE =. Algebraic. 1993 , ISBN =

    1993

  48. [49]

    , TITLE =

    Chen, K.-T. , TITLE =. Bull. Amer. Math. Soc. , FJOURNAL =. 1977 , NUMBER =

    1977

  49. [50]

    Drinfeld, V. G. , TITLE =. Algebra i Analiz , FJOURNAL =. 1990 , NUMBER =

    1990

  50. [51]

    , TITLE =

    Enriquez, B. , TITLE =. Math. Res. Lett. , FJOURNAL =. 2006 , NUMBER =. doi:10.4310/MRL.2006.v13.n2.a5 , URL =

    work page doi:10.4310/mrl.2006.v13.n2.a5 2006

  51. [52]

    and Deligne, P

    Beilinson, A. and Deligne, P. , TITLE =. Motives (. 1994 , MRCLASS =

    1994

  52. [53]

    and Schlotterer, O

    Broedel, J. and Schlotterer, O. and Stieberger, S. and Terasoma, T. , journal =. All order ' -expansion of superstring trees from the. 2014 , month =. doi:10.1103/PhysRevD.89.066014 , url =

    work page doi:10.1103/physrevd.89.066014 2014

  53. [54]

    Forum Math

    Single-valued periods and multiple zeta values , Author =. Forum Math. Sigma , Year =. arXiv , Doi =:1309.5309 , ISSN =

    Pith/arXiv arXiv

  54. [55]

    , TITLE =

    Brown, F. , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2017 , NUMBER =. doi:10.4310/CNTP.2017.v11.n3.a2 , URL =

    doi:10.4310/cntp.2017.v11.n3.a2 2017

  55. [56]

    , TITLE =

    Deligne, P. , TITLE =. Galois groups over. 1989 , MRCLASS =. doi:10.1007/978-1-4613-9649-9_3 , URL =

    doi:10.1007/978-1-4613-9649-9_3 1989

  56. [57]

    and Goncharov, A

    Deligne, P. and Goncharov, A. B. , TITLE =. Ann. Sci. \'. 2005 , NUMBER =. doi:10.1016/j.ansens.2004.11.001 , Archiveprefix =. math/0302267 , Primaryclass =

    Pith/arXiv arXiv 2005

  57. [58]

    Goncharov, A. B. and Manin, Yu. I. , TITLE =. Compos. Math. , FJOURNAL =. 2004 , NUMBER =. doi:10.1112/S0010437X03000125 , URL =

    doi:10.1112/s0010437x03000125 2004

  58. [59]

    and M\"uller-Stach, S

    Huber, A. and M\"uller-Stach, S. , TITLE =. 2017 , PAGES =

    2017

  59. [60]

    , TITLE =

    Schnetz, O. , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2014 , NUMBER =. doi:10.4310/CNTP.2014.v8.n4.a1 , URL =

    doi:10.4310/cntp.2014.v8.n4.a1 2014

  60. [61]

    , title=

    Stieberger, S. , title=. Journal of Physics A: Mathematical and Theoretical , volume=