Learning Manifold and It\^o Dynamics with Branched Neural Rough Differential Equations

# Learning Manifold and Itô Dynamics with Branched^"" Neural Rough Differential Equations
Source: [https://arxiv.org/html/2606.05272](https://arxiv.org/html/2606.05272)
## Learning Manifold and Itô Dynamics with Branched\{\}^\{\\mkern\-9\.0mu\\hbox to16\.05pt\{\\vbox to17\.16pt\{\\pgfpicture\\makeatletter\\hbox\{\\quad\\lower\-2\.33311pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{ \{\}\{\{\}\}\{\}\{\{\{\}\} \{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\} \}\{\}\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@moveto\{1\.70709pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.70709pt\}\{0\.94281pt\}\{0\.94281pt\}\{1\.70709pt\}\{0\.0pt\}\{1\.70709pt\}\\pgfsys@curveto\{\-0\.94281pt\}\{1\.70709pt\}\{\-1\.70709pt\}\{0\.94281pt\}\{\-1\.70709pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.70709pt\}\{\-0\.94281pt\}\{\-0\.94281pt\}\{\-1\.70709pt\}\{0\.0pt\}\{\-1\.70709pt\}\\pgfsys@curveto\{0\.94281pt\}\{\-1\.70709pt\}\{1\.70709pt\}\{\-0\.94281pt\}\{1\.70709pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\{\}\} \{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\} \}\{\}\\pgfsys@moveto\{\-5\.69046pt\}\{9\.95863pt\}\\pgfsys@moveto\{\-3\.98337pt\}\{9\.95863pt\}\\pgfsys@curveto\{\-3\.98337pt\}\{10\.90144pt\}\{\-4\.74765pt\}\{11\.66573pt\}\{\-5\.69046pt\}\{11\.66573pt\}\\pgfsys@curveto\{\-6\.63327pt\}\{11\.66573pt\}\{\-7\.39755pt\}\{10\.90144pt\}\{\-7\.39755pt\}\{9\.95863pt\}\\pgfsys@curveto\{\-7\.39755pt\}\{9\.01582pt\}\{\-6\.63327pt\}\{8\.25154pt\}\{\-5\.69046pt\}\{8\.25154pt\}\\pgfsys@curveto\{\-4\.74765pt\}\{8\.25154pt\}\{\-3\.98337pt\}\{9\.01582pt\}\{\-3\.98337pt\}\{9\.95863pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-5\.69046pt\}\{9\.95863pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\{\}\} \{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\} \}\{\}\\pgfsys@moveto\{5\.69046pt\}\{9\.95863pt\}\\pgfsys@moveto\{7\.39755pt\}\{9\.95863pt\}\\pgfsys@curveto\{7\.39755pt\}\{10\.90144pt\}\{6\.63327pt\}\{11\.66573pt\}\{5\.69046pt\}\{11\.66573pt\}\\pgfsys@curveto\{4\.74765pt\}\{11\.66573pt\}\{3\.98337pt\}\{10\.90144pt\}\{3\.98337pt\}\{9\.95863pt\}\\pgfsys@curveto\{3\.98337pt\}\{9\.01582pt\}\{4\.74765pt\}\{8\.25154pt\}\{5\.69046pt\}\{8\.25154pt\}\\pgfsys@curveto\{6\.63327pt\}\{8\.25154pt\}\{7\.39755pt\}\{9\.01582pt\}\{7\.39755pt\}\{9\.95863pt\}\\pgfsys@closepath\\pgfsys@moveto\{5\.69046pt\}\{9\.95863pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\} \{\}\{\}\{\}\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@lineto\{\-5\.69046pt\}\{9\.95863pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\} \{\}\{\}\{\}\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@lineto\{5\.69046pt\}\{9\.95863pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.53311pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{$\\scriptstyle\{\}$\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-5\.69046pt\}\{12\.49174pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{$\\scriptstyle\{\}$\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{\{\}\}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\}\{\{ \{\}\{\}\}\}\{ \{\}\{\}\} \{\{\}\{\{\}\}\}\{\{\}\{\}\}\{\}\{\{\}\{\}\} \{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{5\.69046pt\}\{12\.49174pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{$\\scriptstyle\{\}$\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}Neural Rough Differential Equations

###### Abstract

Neural rough differential equations \(NRDEs\) stay accurate under irregular sampling while taking far fewer integration steps than standard neural differential equations, summarising a finely sampled driver by its log\-signature and advancing the hidden state over coarse intervals using the log\-ODE method\. This efficiency rests on the shuffle algebra, the algebraic counterpart of Stratonovich calculus\. This reliance means NRDEs cannot expose the quadratic\-variation terms Itô dynamics require, nor the ordered covariant derivatives that govern Itô flows on connection\-equipped manifolds\. Ameliorating this, we introduce Branched Neural Rough Differential Equations \(B\-NRDEs\), a Hopf\-algebraic framework that recasts the NRDE log\-ODE step as geometric numerical integration on the state\-space manifold, matching the driving algebra to the governing calculus: Grossman–Larson rooted trees for Euclidean Itô dynamics, Munthe–Kaas–Wright planar rooted trees for ordered covariant derivatives on manifolds, and the shuffle algebra in the classical Stratonovich case\. This yields intrinsic coarse\-step dynamics that exactly preserve manifold constraints\. Finally, we introduce a branched signature\-kernel objective to enable Itô\-consistent law matching by making quadratic variation terms visible during training\. On rough Bergomi volatility, sim\-to\-realSO​\(3\)\\mathrm\{SO\}\(3\)forecasting, and SPD covariance dynamics, B\-NRDEs offer a unified, effective approach to stochastic and manifold\-valued dynamics beyond the Euclidean\-Stratonovich setting\.

Dynamical systems, Manifold learning, Differential equations

## 1Introduction

Learning continuous\-time dynamics from time series is a foundational problem in machine learning, with applications ranging from robotics\(Duong and Atanasov,[2021](https://arxiv.org/html/2606.05272#bib.bib56); Cheeet al\.,[2022](https://arxiv.org/html/2606.05272#bib.bib58)\)and molecular dynamics\(Huanget al\.,[2023](https://arxiv.org/html/2606.05272#bib.bib60); Schreineret al\.,[2023](https://arxiv.org/html/2606.05272#bib.bib59)\)to quantitative finance\(Gierjatowiczet al\.,[2020](https://arxiv.org/html/2606.05272#bib.bib62); Issaet al\.,[2023](https://arxiv.org/html/2606.05272#bib.bib46)\)\.Neural controlled differential equations \(NCDEs\)\(Kidgeret al\.,[2020](https://arxiv.org/html/2606.05272#bib.bib8)\)approach this task by parameterising the vector field of a controlled differential equation with a neural network\. To improve scalability and robustness to sampling rates,neural rough differential equations \(NRDEs\)\(Morrillet al\.,[2021b](https://arxiv.org/html/2606.05272#bib.bib3)\)lift the control path to its log\-signature, a series of iterated integrals, and solve the resulting system via the log\-ODE method at coarser discretisations\(Castell and Gaines,[1996](https://arxiv.org/html/2606.05272#bib.bib53)\)\.

![Refer to caption](https://arxiv.org/html/2606.05272v1/x1.png)Figure 1:Illustration of the log\-ODE method forB\-NRDEs\. The control segmentXs,tX\_\{s,t\}is lifted to𝐗s,tℋ∈ℋ\\mathbf\{X\}^\{\\mathcal\{H\}\}\_\{s,t\}\\in\\mathcal\{H\}for the chosen Hopf algebraℋ∈\{ℋ,ℋGL,ℋMKW\}\\mathcal\{H\}\\in\\\{\\operatorname\{\\mathcal\{H\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\\\}, then transformed to a log\-signatureλs,t\\lambda\_\{s,t\}in primitive coordinatesp∈ℬℋprimp\\in\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}\. The tangent vector fields𝒲\\mathcal\{W\}are lifted through the pseudo bialgebra mapF𝒲F\_\{\\mathcal\{W\}\}to matching primitive\-indexed vector fields on the state manifold\. These are combined into the log\-ODE vector fieldLs,tL\_\{s,t\}, whose ODE flow gives the local update fromsstott\.However, geometricNRDEsare driven by geometric signatures whose coordinates satisfy the shuffle product identity\. This is appropriate for Stratonovich integration, which preserves the usual chain rule, but not for Itô integration\. Itô’s product rule generates quadratic\-variation corrections, so products of Itô iterated integrals contain second\-order terms that are not represented by the original tensor coordinates\. This limits the use of geometric signatures in settings where adapted causal Itô modelling assumptions are required\. On manifolds, the obstruction is sharper\. Itô integration is defined relative to a connection, and its expansion involves higher covariant derivatives\. Since these operators do not generally commute, the signature algebra must preserve their ordered, branched composition rather than identify them through shuffle relations\.

To address these limitations, we introduceB\-NRDEs, a unified framework for learning dynamics using the log\-ODE method under Itô integration and over manifolds\. By lifting the control path to a Hopf algebra ofrooted trees— theGrossman–Larsonalgebra for Euclidean Itô calculus or theMunthe–Kaas–Wrightalgebra for manifolds, we derive a generalisedNRDEthat strictly respects the geometric and causal constraints of the domain\. See[Table1](https://arxiv.org/html/2606.05272#S2.T1)for comparisons of Hopf algebras and[Figure1](https://arxiv.org/html/2606.05272#S1.F1)for an illustration of the proposed framework\. Ourmain contributionsare as follows:

1. 1\.We model path signatures in theGrossman–LarsonHopf algebraℋGL\\mathcal\{H\}\_\{\\text\{GL\}\}, whose rooted\-tree basis represents Itô\-type iterated integrals\. We then introduce the first neural dynamics model utilising theMunthe–Kaas–WrightHopf algebraℋMKW\\mathcal\{H\}\_\{\\text\{MKW\}\}of planar rooted trees to strictly enforce manifold constraints for Itô\-type dynamical systems\.
2. 2\.We unify the above approaches viapseudo bialgebra maps\(Kern and Lyons,[2023](https://arxiv.org/html/2606.05272#bib.bib6)\), which convert Hopf algebraic structures to learned vector fields and differential operators on the target manifold\. This yields a general log\-ODE formulation for Stratonovich and Itô dynamics on connection\-equipped manifolds\. In this work, we instantiate the numerical solver on homogeneous spaces, with tangent vectors represented in frame or Lie\-algebra coordinates and integrated through a group action\.
3. 3\.We enable Itô\-consistent neural dynamics training both in Euclidean space and on manifolds by developing a branched signature\-kernel objective\.
4. 4\.We release[Roughrax](https://github.com/luke-a-thompson/Roughrax), a JAX package for branched rough paths, and autodifferentiable numerical solution forrough differential equations \(RDEs\)over manifolds\.

## 2Preliminaries

In this section, we first review the standard formulation of Neural CDEs and RDEs to establish the Euclidean baseline \([Section2\.1](https://arxiv.org/html/2606.05272#S2.SS1)\)\. We then discuss the machinery required to extend the Log\-ODE framework to non\-Euclidean and Itô settings\. We first introduce rough path Hopf algebras \([Section2\.2](https://arxiv.org/html/2606.05272#S2.SS2)\), then present pseudo bialgebra maps that send algebra elements to differential operators onℳ\\mathcal\{M\}\([Section2\.3](https://arxiv.org/html/2606.05272#S2.SS3)\)\.

### 2\.1Neural Controlled and Rough Differential Equations

Let\(\(t0,x0\),…,\(tn,xn\)\)\\big\(\(t\_\{0\},x\_\{0\}\),\\dots,\(t\_\{n\},x\_\{n\}\)\\big\)be a potentially irregular time series with observationsxi∈ℝdx\_\{i\}\\in\\mathbb\{R\}^\{d\}\. We construct a continuous interpolationX:\[t0,tn\]→ℝdX\\colon\[t\_\{0\},t\_\{n\}\]\\to\\mathbb\{R\}^\{d\}such thatXti=xiX\_\{t\_\{i\}\}=x\_\{i\}\.

Neural Controlled Differential Equations\.ANCDE\(Kidgeret al\.,[2020](https://arxiv.org/html/2606.05272#bib.bib8)\)evolves a hidden stateht∈ℝuh\_\{t\}\\in\\mathbb\{R\}^\{u\}driven by the pathXX\. Letξϕ:ℝd→ℝu\\xi\_\{\\phi\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{u\},ℓψ:ℝu→ℝw\\ell\_\{\\psi\}:\\mathbb\{R\}^\{u\}\\to\\mathbb\{R\}^\{w\}be an encoder and output network respectively, andgθ:ℝu→ℝu×dg\_\{\\theta\}:\\mathbb\{R\}^\{u\}\\to\\mathbb\{R\}^\{u\\times d\}be a vector field parametrisation\. The forward pass is defined by the Riemann\-Stieltjes integral:

ht=ht0\+∫t0tgθ​\(hs\)​dXs,\\displaystyle h\_\{t\}=h\_\{t\_\{0\}\}\+\\int\_\{t\_\{0\}\}^\{t\}g\_\{\\theta\}\(h\_\{s\}\)\\,\\mathrm\{d\}X\_\{s\},\(1\)where initial hidden stateht0=ξϕ​\(Xt0\)h\_\{t\_\{0\}\}=\\xi\_\{\\phi\}\(X\_\{t\_\{0\}\}\), and outputyt=ℓψ​\(ht\)y\_\{t\}=\\ell\_\{\\psi\}\(h\_\{t\}\),gθ​\(hs\)​d​Xsg\_\{\\theta\}\(h\_\{s\}\)\\,\\mathrm\{d\}X\_\{s\}denotes matrix\-vector multiplication\. This formulation allows the latent state to react continuously to incoming data\.

Neural Rough Differential Equations\.While highly expressive,NCDEsare expensive for long sequences\.NRDEs\(Morrillet al\.,[2021b](https://arxiv.org/html/2606.05272#bib.bib3)\)reduce this cost by applying the log\-ODE method: over each windowIj=\[tj,tj\+1\]I\_\{j\}=\[t\_\{j\},t\_\{j\+1\}\], the pathXXis summarised by its log\-signatureλj\\lambda\_\{j\}\. This log\-signature determines an autonomous vector field, denotedg~θ,λj\\tilde\{g\}\_\{\\theta,\\lambda\_\{j\}\}, obtained by applying the log\-ODE lift to the base neural vector field\. The hidden state is then advanced by theordinary differential equation \(ODE\)

ht=htj\+∫tjtg~θ,λj​\(hs\)​ds,t∈Ij\.h\_\{t\}=h\_\{t\_\{j\}\}\+\\int\_\{t\_\{j\}\}^\{t\}\\tilde\{g\}\_\{\\theta,\\lambda\_\{j\}\}\(h\_\{s\}\)\\,\\mathrm\{d\}s,\\qquad t\\in I\_\{j\}\.\(2\)Here,λj\\lambda\_\{j\}sets the coefficients of the ODE solved onIjI\_\{j\}, rather than serving as a new path differential, yielding a higher\-order approximation that permits larger step sizes\.

### 2\.2Rough Path Hopf Algebras

Hopf algebras organise iterated\-integral combinatorics and the product identities needed for computation\.[Table1](https://arxiv.org/html/2606.05272#S2.T1)summarises the regimes supported by each Hopf algebra\.

###### Definition 2\.1\(Hopf algebra and primitives\)\.

Letℋ=\(H,⋆,η,Δ,ε,S\)\\mathcal\{H\}=\(H,\\star,\\eta,\\Delta,\\varepsilon,S\)be a Hopf algebra on a vector spaceHHwith product⋆:H⊗H→H\\star:H\\otimes H\\to Hand coproductΔ:H→H⊗H\\Delta:H\\to H\\otimes H; we denote a basis ofHHbyℬℋ\\mathcal\{B\}\_\{\\mathcal\{H\}\}, and writePrim​\(ℋ\)≔\{p∈H:Δ​p=p⊗1\+1⊗p\}\\mathrm\{Prim\}\(\\mathcal\{H\}\)\\coloneqq\\\{p\\in H:\\Delta p=p\\otimes 1\+1\\otimes p\\\}with a chosen primitive basisℬℋprim⊂Prim​\(ℋ\)\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}\\subset\\mathrm\{Prim\}\(\\mathcal\{H\}\)\. We suppressη\\eta,ε\\varepsilonandSSbelow, since only⋆\\starandΔ\\Deltaare immediately relevant to our constructions\.

Table 1:Summary of the integration regimes encoded by each Hopf algebra and their first application to neural differential equations\.##### Stratonovich \(geometric\) signatures andℋ\\mathcal\{H\}\.

For add\-dimensional control pathX=\(X\(1\),…,X\(d\)\)X=\(X^\{\(1\)\},\\dots,X^\{\(d\)\}\), the truncated geometric signature collects its iterated Stratonovich integrals in word coordinates indexed byei1⊗⋯⊗eime\_\{i\_\{1\}\}\\otimes\\cdots\\otimes e\_\{i\_\{m\}\}\. We use the standard tensor\-algebra convention in which signatures are group\-like, and log\-signatures are computed using the tensor product; the shuffle product appears dually as the product identity satisfied by signature coordinate functions\(Kidgeret al\.,[2020](https://arxiv.org/html/2606.05272#bib.bib8); Morrillet al\.,[2021b](https://arxiv.org/html/2606.05272#bib.bib3)\)\. The defining feature of this setting is the Stratonovich product rule\. At level two, we have

Xs,t\(i\)​Xs,t\(j\)=∫stXs,u\(i\)∘dXu\(j\)\+∫stXs,u\(j\)∘dXu\(i\)\.X\_\{s,t\}^\{\(i\)\}X\_\{s,t\}^\{\(j\)\}=\\int\_\{s\}^\{t\}X\_\{s,u\}^\{\(i\)\}\\circ\\mathrm\{d\}X\_\{u\}^\{\(j\)\}\+\\int\_\{s\}^\{t\}X\_\{s,u\}^\{\(j\)\}\\circ\\mathrm\{d\}X\_\{u\}^\{\(i\)\}\.\(3\)Algebraically, \([3](https://arxiv.org/html/2606.05272#S2.E3)\) is encoded byei​ej=ei⊗ej\+ej⊗eie\_\{i\}\\shuffle e\_\{j\}=e\_\{i\}\\otimes e\_\{j\}\+e\_\{j\}\\otimes e\_\{i\}\. Thusℋ\\mathcal\{H\}naturally represents geometric \(Stratonovich\) rough paths: iterated integrals are indexed by tensor coordinates and obey shuffle identities\. Since Stratonovich integration is coordinate\-free, the geometric signature remains appropriate for Stratonovich dynamics on manifolds\.

##### Itô \(branched\) signatures andℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}\.

Under Itô integration, the product rule includes a quadratic\-variation term:

Xs,t\(i\)​Xs,t\(j\)=∫stXs,u\(i\)​dXu\(j\)\+∫st\\displaystyle X\_\{s,t\}^\{\(i\)\}X\_\{s,t\}^\{\(j\)\}=\\int\_\{s\}^\{t\}X\_\{s,u\}^\{\(i\)\}\\mathrm\{d\}X\_\{u\}^\{\(j\)\}\+\\int\_\{s\}^\{t\}Xs,u\(j\)​d​Xu\(i\)\\displaystyle X\_\{s,u\}^\{\(j\)\}\\mathrm\{d\}X\_\{u\}^\{\(i\)\}\(4\)\+⟨X\(i\),X\(j\)⟩s,t\.\\displaystyle\+\\langle X^\{\(i\)\},X^\{\(j\)\}\\rangle\_\{s,t\}\.Over the originaldd\-dimensional word coordinates, the correction term⟨X\(i\),X\(j\)⟩\\langle X^\{\(i\)\},X^\{\(j\)\}\\rangleis not exposed as an independent level\-two driver coordinate\. It can be represented only indirectly, for example, by augmenting the path through a lead–lag lift\. Hence, the unaugmented shuffle representation is not the natural signature space for Itô modelling\. We therefore work in the Hopf algebra ofGrossman and Larson \([1989](https://arxiv.org/html/2606.05272#bib.bib24)\),ℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}, on rooted trees, which underpins branched rough paths\(Gubinelli,[2010](https://arxiv.org/html/2606.05272#bib.bib4)\)\. Here, trees index the additional non\-geometric iterated\-integral coordinates, and in particular provide a \(symmetric\) second\-order coordinate corresponding to the quadratic variation in \([4](https://arxiv.org/html/2606.05272#S2.E4)\)\.

##### Itô \(branched\) signatures over manifolds andℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\.

For dynamics on a manifoldℳ\\mathcal\{M\}with connection∇\\nabla, the local flow involves*ordered*compositions of covariant derivatives: in general∇U∇V≠∇V∇U\\nabla\_\{U\}\\nabla\_\{V\}\\neq\\nabla\_\{V\}\\nabla\_\{U\}, and the induced curvature terms depend on the order in which directions are applied\. This order\-sensitivity is captured by theMunthe\-Kaas and Wright \([2006](https://arxiv.org/html/2606.05272#bib.bib23)\)Hopf algebra,ℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}, defined over*planar*rooted trees which fix a left\-to\-right child order at each node to index ordered iterated covariant derivatives\. In contrast, non\-planar rooted trees symmetrise child order and collapse the order\-sensitive flow terms\. See[AppendixA](https://arxiv.org/html/2606.05272#A1)for further algebraic details\. We show a visual representation of the Stratonovich and Itô conventions in[Figure2](https://arxiv.org/html/2606.05272#S2.F2)\.

![Refer to caption](https://arxiv.org/html/2606.05272v1/x2.png)\(a\)Itô convention\.
![Refer to caption](https://arxiv.org/html/2606.05272v1/x3.png)\(b\)Stratonovich convention\.

Figure 2:Riemann–Stieltjes approximations under different evaluation conventions\. Redrawn fromBellingeriet al\.\([2024](https://arxiv.org/html/2606.05272#bib.bib33)\)\.

### 2\.3Vector Fields and Pseudo Bialgebra Maps

Pseudo\-bialgebra maps\(Kern and Lyons,[2023](https://arxiv.org/html/2606.05272#bib.bib6)\)identify how signature coordinates act on functions through vector fields\. Letℳ\\mathcal\{M\}be a smooth manifold, letΓ​\(T​ℳ\)\\Gamma\(T\\mathcal\{M\}\)denote its smooth vector fields, and letDiff​\(ℳ\)\\mathrm\{Diff\}\(\\mathcal\{M\}\)be the algebra of linear differential operators onC∞​\(ℳ\)C^\{\\infty\}\(\\mathcal\{M\}\)\. Given a connection∇\\nablawhen covariant derivatives are required, a pseudo\-bialgebra map sends each word or tree basis element of the chosen Hopf algebra to the corresponding differential operator onℳ\\mathcal\{M\}\. Thus, the signature expansion and the induced flow expansion are evaluated in the same word\- or tree\-indexed coordinates\.

###### Definition 2\.2\(pseudo bialgebra map\(Kern and Lyons,[2023](https://arxiv.org/html/2606.05272#bib.bib6), Def\. 4\.1\)\)\.

A linear mapF:ℋ→Diff​\(ℳ\)F\\colon\\mathcal\{H\}\\to\\mathrm\{Diff\}\(\\mathcal\{M\}\)is a pseudo bialgebra map if

F​\(τ⋆σ\)=F​\(τ\)∘F​\(σ\)F\(\\tau\\star\\sigma\)=F\(\\tau\)\\circ F\(\\sigma\)and, for allτ∈ℋ\\tau\\in\\mathcal\{H\}andϕ,ψ∈C∞​\(ℳ\)\\phi,\\psi\\in C^\{\\infty\}\(\\mathcal\{M\}\),m∘\(F⊗F\)​\(Δ​τ\)​\(ϕ⊗ψ\)=F​\(τ\)​\(ϕ​ψ\),m\\circ\(F\\otimes F\)\(\\Delta\\tau\)\(\\phi\\otimes\\psi\)=F\(\\tau\)\(\\phi\\psi\),wherem​\(a⊗b\)=a​bm\(a\\otimes b\)=abdenotes pointwise multiplication of functions\.

Given a collection of smooth vector fields𝒲=\(W\(1\),…,W\(d\)\)\\mathcal\{W\}=\(W^\{\(1\)\},\\dots,W^\{\(d\)\}\)onℳ\\mathcal\{M\}, the elementary differential associated with a rooted tree is defined recursively\. For the single\-node treeF𝒲​\(∙i\)=W\(i\)F\_\{\\mathcal\{W\}\}\(\\bullet\_\{i\}\)=W^\{\(i\)\}\. For a rooted treeτ=\[τ1​…​τk\]i\\tau=\[\\tau\_\{1\}\\dots\\tau\_\{k\}\]\_\{i\}, whose root has colouri∈\{1,…,d\}i\\in\\\{1,\\dots,d\\\}andτ1,…,τk\\tau\_\{1\},\\dots,\\tau\_\{k\}are its immediate child nodes, set

F𝒲​\(\[τ1,…,τk\]i\)=∇F𝒲​\(τ1\),…,F𝒲​\(τk\)kW\(i\)\.F\_\{\\mathcal\{W\}\}\(\[\\tau\_\{1\},\\dots,\\tau\_\{k\}\]\_\{i\}\)=\\nabla^\{k\}\_\{F\_\{\\mathcal\{W\}\}\(\\tau\_\{1\}\),\\dots,F\_\{\\mathcal\{W\}\}\(\\tau\_\{k\}\)\}W^\{\(i\)\}\.\(5\)
Intuitively, we apply thekk\-th covariant derivative to the base vector fieldW\(i\)W^\{\(i\)\}, and ask how much it moves along the directions of vector fields specified by the subtreesτ1,…,τk\\tau\_\{1\},\\dots,\\tau\_\{k\}\. Crucially, ifℋ=ℋMKW\\mathcal\{H\}=\\mathcal\{H\}\_\{\\text\{MKW\}\}, the order of arguments in the covariant derivative∇k\\nabla^\{k\}corresponds to the planar order of subtrees, and the chain rule for covariant derivatives\.

## 3Methodology

We defineB\-NRDEby extending log\-NCDE\(Walkeret al\.,[2024](https://arxiv.org/html/2606.05272#bib.bib1)\)to non\-Euclidean geometries\. The model learns an algebra\-valued vector field and integrates the dynamics using the log\-ODE method\.[Figures1](https://arxiv.org/html/2606.05272#S1.F1)and[1](https://arxiv.org/html/2606.05272#alg1)summarise the resulting piecewise log\-ODE formulation\.

Algorithm 1B\-NRDEInput:Hopf algebra

ℋ∈\{ℋ,ℋGL,ℋMKW\}\\mathcal\{H\}\\in\\\{\\operatorname\{\\mathcal\{H\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\\\}, truncation depth

NN, partition

\{\[tk,tk\+1\]\}k=0M−1\\\{\[t\_\{k\},t\_\{k\+1\}\]\\\}\_\{k=0\}^\{M\-1\}, path segments

Xtk,tk\+1X\_\{t\_\{k\},t\_\{k\+1\}\}, vector\-field lift

FWF\_\{W\}, and initial condition

Y0∈ℳY\_\{0\}\\in\\mathcal\{M\}
Precompute:build the depth\-

NN,

dd\-decorated basis of

ℋ\\mathcal\{H\}, primitive basis

𝒫←ℬℋprim\\mathcal\{P\}\\leftarrow\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}, signature routines, Hopf logarithm, and primitive\-field evaluation schedules

Cache offline:for each segment

XkX\_\{k\}, compute

λk=logℋ⁡\(SigℋN⁡\(Xk\)\)=∑p∈𝒫λkp​p\.\\lambda\_\{k\}=\\log\_\{\\mathcal\{H\}\}\\\!\\left\(\\operatorname\{Sig\}\_\{\\mathcal\{H\}\}^\{N\}\(X\_\{k\}\)\\right\)=\\sum\_\{p\\in\\mathcal\{P\}\}\\lambda\_\{k\}^\{p\}p\.
for

k=0k=0to

M−1M\-1do

Lk​\(Y\)←∑p∈𝒫λkp​FW​\(p\)​\(Y\)L\_\{k\}\(Y\)\\leftarrow\\sum\_\{p\\in\\mathcal\{P\}\}\\lambda\_\{k\}^\{p\}F\_\{W\}\(p\)\(Y\)⊳\\trianglerightlifted field

Z˙τ=Lk​\(Zτ\),Z0=Yk;Yk\+1←Z1\\dot\{Z\}\_\{\\tau\}\\\!=\\\!L\_\{k\}\(Z\_\{\\tau\}\),\\,Z\_\{0\}\\\!=\\\!Y\_\{k\};\\,Y\_\{k\+1\}\\leftarrow Z\_\{1\}⊳\\trianglerightlog\-ODE step

endfor

Output:trajectory samples

\{Yk\}k=0M\\\{Y\_\{k\}\\\}\_\{k=0\}^\{M\}

### 3\.1Signature Primitivisation

Given a path segmentXs,tX\_\{s,t\}, itsℋ\\mathcal\{H\}\-signature is the group\-like element𝕏s,tℋ∈ℋ\\mathbb\{X\}^\{\\mathcal\{H\}\}\_\{s,t\}\\in\\mathcal\{H\}whose coordinates encode the corresponding iterated integrals, meaning thatΔ​𝕏s,tℋ=𝕏s,tℋ⊗𝕏s,tℋ\\Delta\\mathbb\{X\}^\{\\mathcal\{H\}\}\_\{s,t\}=\\mathbb\{X\}^\{\\mathcal\{H\}\}\_\{s,t\}\\otimes\\mathbb\{X\}^\{\\mathcal\{H\}\}\_\{s,t\}\. The log\-ODE method requires expressing the signature in primitive elements via the Hopf logarithm:

logℋ⁡\(g\)≔∑n≥1\(−1\)n−1n​\(g−1\)⋆n,\\log\_\{\\mathcal\{H\}\}\(g\)\\coloneqq\\sum\_\{n\\geq 1\}\\frac\{\(\-1\)^\{n\-1\}\}\{n\}\(g\-1\)^\{\\star n\},\(6\)for a group\-like elementggwhere11is the unit and⋆\\staris the product inℋ\\mathcal\{H\}\.logℋ\\log\_\{\\mathcal\{H\}\}yields a primitive element, which we term thelogℋ\\log\_\{\\mathcal\{H\}\}\-signature\.

The product⋆\\starof the working Hopf algebra fixes the meaning of \([6](https://arxiv.org/html/2606.05272#S3.E6)\)\. Throughout,ℋ\\mathcal\{H\}denotes the graded dual in which enhanced paths are group\-like and logarithms primitive, with coordinate identities such as the shuffle product rule stated dually\. Forℋ\\mathcal\{H\},⋆\\staris tensor concatenation, whose dual coordinate product is the shuffle; forℋGL\\mathcal\{H\}\_\{\\mathrm\{GL\}\}andℋMKW\\mathcal\{H\}\_\{\\mathrm\{MKW\}\},⋆\\staris the tree grafting product \([SectionsA\.4\.3](https://arxiv.org/html/2606.05272#A1.SS4.SSS3)and[A\.4\.5](https://arxiv.org/html/2606.05272#A1.SS4.SSS5)\)\. A single Hopf logarithm thus yields the segment log\-signatures\{λj\}j=0T−1\\\{\\lambda\_\{j\}\\\}\_\{j=0\}^\{T\-1\}used by B\-NRDE across all three regimes\.

### 3\.2Neural Vector Fields

The learnable component of aB\-NRDEis a collection of atomic driving vector fields, not the full primitive\-indexed log\-ODE field\. Forz∈ℳz\\in\\mathcal\{M\}, we parameterise

Wθ​\(z\)=\(Wθ\(1\)​\(z\),…,Wθ\(d\)​\(z\)\),Wθ\(i\)​\(z\)∈Tz​ℳ\.W\_\{\\theta\}\(z\)=\(W\_\{\\theta\}^\{\(1\)\}\(z\),\\dots,W\_\{\\theta\}^\{\(d\)\}\(z\)\),\\quad W\_\{\\theta\}^\{\(i\)\}\(z\)\\in T\_\{z\}\\mathcal\{M\}\.Thus𝒲θ=\(Wθ\(1\),…,Wθ\(d\)\)\\mathcal\{W\}\_\{\\theta\}=\(W\_\{\\theta\}^\{\(1\)\},\\dots,W\_\{\\theta\}^\{\(d\)\}\)contains one vector field per driver channel\. B\-NRDE learns only these atomic channel fields\. The primitive\-indexed fields used by the log\-ODE are then generated deterministically fromWθW\_\{\\theta\}by the vector\-field lifts described in the sequel\.

The display above isrepresentation\-free: the method only requires that the network parameterisation determines an element ofTz​ℳT\_\{z\}\\mathcal\{M\}for each statezzand driver channel\. In our experiments, we use a homogeneous space realisation, in which the network outputs frame coordinates and the numerical flow is applied through the corresponding group action\. This enforces the manifold constraint at every solver substep, avoiding the ambient\-space projection or retraction errors that arise when integrating extrinsically\.

### 3\.3Vector Field Lifts

We implement the pseudo bialgebra map as a*vector field lift*FW:ℬℋprim→Γ​\(T​M\)F\_\{W\}\\colon\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}\\to\\Gamma\(TM\)mapping primitive basis elements of the chosen Hopf algebra to vector fields on the state space\. This follows from the coproduct Leibniz property of[Definition2\.2](https://arxiv.org/html/2606.05272#S2.Thmtheorem2), which ensuresF𝒲​\(p\)∈Γ​\(T​M\)F\_\{\\mathcal\{W\}\}\(p\)\\in\\Gamma\(TM\)forp∈ℬℋprimp\\in\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}\(Kern and Lyons,[2023](https://arxiv.org/html/2606.05272#bib.bib6), Prop\. 4\.3\); see[PropositionB\.5](https://arxiv.org/html/2606.05272#A2.Thmtheorem5)\. In all cases, the implementation separates basis and combinatorial precomputation fromauto\-differentiation\-based evaluation\. The atomic fields are those defined in[Section3\.2](https://arxiv.org/html/2606.05272#S3.SS2), and we writeW≔𝒲θ,FW≔F𝒲θW\\coloneqq\\mathcal\{W\}\_\{\\theta\},\\ F\_\{W\}\\coloneqq F\_\{\\mathcal\{W\}\_\{\\theta\}\}for notational simplicity\.

Primitive bases and combinatorics\.*Words*\(ℋ\\operatorname\{\\mathcal\{H\}\}\): We use the Lyndon basis of the free Lie algebra, indexed by Lyndon wordsℓ=\(i1,…,im\)\\ell=\(i\_\{1\},\\dots,i\_\{m\}\)over\{1,…,d\}\\\{1,\\dots,d\\\}with\|ℓ\|≤N\|\\ell\|\\leq N; see[DefinitionA\.2](https://arxiv.org/html/2606.05272#A1.Thmtheorem2)\. We enumerate these words by degree using Duval’s generator\(Duval,[1988](https://arxiv.org/html/2606.05272#bib.bib34)\)\. For each non\-letter Lyndon word, we store its standard factorisation

ℓ=u​v,v​the longest Lyndon suffix,\\ell=uv,\\qquad v\\text\{ the longest Lyndon suffix\},\(7\)as the pair of indices corresponding touuandvv\.

*Trees*\(ℋGL,ℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\): primitives are represented by decorated rooted trees, stored recursively as\(τ1,…,τk,c\),\(\\tau\_\{1\},\\ldots,\\tau\_\{k\},c\),wherec∈\{1,…,d\}c\\in\\\{1,\\dots,d\\\}is the root colour andτ1,…,τk\\tau\_\{1\},\\ldots,\\tau\_\{k\}are the root subtrees\. We precompute the indices of these subtrees, giving the recursive evaluation schedule shared by both tree algebras\. ForℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}, unordered rooted\-tree shapes are enumerated inBeyer and Hedetniemi \([1980](https://arxiv.org/html/2606.05272#bib.bib26)\)lexicographic order, with colourings identified under tree symmetries\. ForℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}, ordered rooted\-tree shapes are generated recursively from ordered compositions of the remaining node count; the left\-to\-right order of children is part of the basis element, and all colourings are distinct\.

Shared differential\-operator scheme\.The lifted fields are defined intrinsically from vector fields onℳ\\mathcal\{M\}\. Forℋ\\operatorname\{\\mathcal\{H\}\}, Lyndon primitives are evaluated using the ordinary Lie bracket of vector fields, while forℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}andℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}, tree primitives are evaluated using iterated covariant derivatives with respect to the chosen connection∇\\nabla\. In the numerical implementation, these intrinsic operations are evaluated by forward\-mode automatic differentiation in the chosen coordinate or frame representation\.

Evaluation: shuffle primitives \(bracket recursion\)\.Given atomic channel fieldsW\(1\),…,W\(d\)W^\{\(1\)\},\\dots,W^\{\(d\)\}, we set

FW​\(ℓ\)=\{W\(i\),ℓ=i,\[FW​\(u\),FW​\(v\)\],ℓ=\[u,v\],F\_\{W\}\(\\ell\)=\\begin\{cases\}W^\{\(i\)\},&\\ell=i,\\\\ \[F\_\{W\}\(u\),F\_\{W\}\(v\)\],&\\ell=\[u,v\],\\end\{cases\}and the recursion terminates when the word is a single letter\.

Evaluation: tree primitives \(multi\-ary node recursion\)\.Fix a coloured rooted treep=\(τ,c\)p=\(\\tau,c\)\. For each nodevvofτ\\tau, letC​\(v\)=\(u1,…,uk\)C\(v\)=\(u\_\{1\},\\dots,u\_\{k\}\)denote its ordered children \(intrinsic forℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}; any fixed order forℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}\)\. We compute subtree fieldsVvV\_\{v\}in postorder\. Leaves satisfyVv=W\(c​\(v\)\)V\_\{v\}=W^\{\(c\(v\)\)\}\. Ifvvhas ordered childrenC​\(v\)=\(u1,…,uk\)C\(v\)=\(u\_\{1\},\\dots,u\_\{k\}\), we set

Vv=∇Vu1,…,VukkWc​\(v\)\.V\_\{v\}=\\nabla^\{k\}\_\{V\_\{u\_\{1\}\},\\dots,V\_\{u\_\{k\}\}\}W^\{c\(v\)\}\.Here∇k\\nabla^\{k\}is the iterated covariant derivative, evaluated recursively for arbitrary vector fieldsU1,…,Uk,W∈Γ​\(T​ℳ\)U\_\{1\},\\dots,U\_\{k\},W\\in\\Gamma\(T\\mathcal\{M\}\):

∇U1,…,UkkW=∇U1\\displaystyle\\nabla^\{k\}\_\{U\_\{1\},\\dots,U\_\{k\}\}W=\\nabla\_\{U\_\{1\}\}\(∇U2,…,Ukk−1W\)\\displaystyle\\left\(\\nabla^\{k\-1\}\_\{U\_\{2\},\\dots,U\_\{k\}\}W\\right\)−∑j=2k∇U2,…,∇U1Uj,…,Ukk−1W\.\\displaystyle\-\\sum\_\{j=2\}^\{k\}\\nabla^\{k\-1\}\_\{U\_\{2\},\\dots,\\nabla\_\{U\_\{1\}\}U\_\{j\},\\dots,U\_\{k\}\}W\.Finally,FW​\(p\)=Vr,F\_\{W\}\(p\)=V\_\{r\},for the root noderr\. This is the computational form of \([5](https://arxiv.org/html/2606.05272#S2.E5)\): each internal node is evaluated by JVP\-based corrected covariant derivatives, and planarity inℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}enters through the child orderC​\(v\)C\(v\)\.

### 3\.4The Manifold Log\-ODE Method

Letℋ\\mathcal\{H\}be a connected, cocommutative graded Hopf algebra, as detailed in[SectionA\.1](https://arxiv.org/html/2606.05272#A1.SS1)\. By[TheoremA\.1](https://arxiv.org/html/2606.05272#A1.Thmtheorem1),ℋ≅𝒰​\(Prim​\(ℋ\)\)\\mathcal\{H\}\\cong\\mathcal\{U\}\(\\mathrm\{Prim\}\(\\mathcal\{H\}\)\), so logarithmic signature increments can be expressed in primitive coordinates\. For a path segmentXkX\_\{k\}, write itslogℋ\\log\_\{\\mathcal\{H\}\}\-signature asλk=∑p∈ℬℋprimλkp​p∈Prim​\(ℋ\)\\lambda\_\{k\}=\\sum\_\{p\\in\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}\}\\lambda\_\{k\}^\{p\}p\\in\\mathrm\{Prim\}\(\\mathcal\{H\}\)\. Given the vector\-field liftFWF\_\{W\}, the corresponding log\-ODE field is

Lk​\(Y\)=∑p∈ℬℋprimλkp​FW​\(p\)​\(Y\)\.L\_\{k\}\(Y\)=\\sum\_\{p\\in\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}\}\\lambda\_\{k\}^\{p\}F\_\{W\}\(p\)\(Y\)\.\(8\)The lifted fieldFW​\(p\)​\(Y\)F\_\{W\}\(p\)\(Y\)is evaluated by the recursions in[Section3\.3](https://arxiv.org/html/2606.05272#S3.SS3): shuffle primitives use the Lie bracket recursion, while tree primitives use the multi\-ary covariant derivative recursion\. The vector\-field lift is evaluated at the solver stages wheneverLkL\_\{k\}is queried\.

In Euclidean space, the segment update is obtained by solving the normalised ODEZ˙τ=Lk​\(Zτ\)\\dot\{Z\}\_\{\\tau\}=L\_\{k\}\(Z\_\{\\tau\}\),Z0=YkZ\_\{0\}=Y\_\{k\},τ∈\[0,1\]\\tau\\in\[0,1\], where the segment length is already encoded inλk\\lambda\_\{k\}\. We solve this using Heun’s second\-order method and recover log\-NCDE when, additionally,ℋ=ℋ\\mathcal\{H\}=\\operatorname\{\\mathcal\{H\}\}\.

While our method works for any connection\-equipped manifold, we make the implementation choice of using homogeneous spaces\. As such, we represent tangent vectors in frame or Lie\-algebra coordinates\. LetGGbe the Lie group acting onℳ\\mathcal\{M\}, and writeg⋅Yg\\cdot Yfor the action\. Forξ∈𝔤\\xi\\in\\mathfrak\{g\}, denote the induced fundamental vector field by

ξ\#​\(Y\)=dd​ϵ\|ϵ=0​exp⁡\(ϵ​ξ\)⋅Y\.\\xi^\{\\\#\}\(Y\)=\\left\.\\frac\{\\mathrm\{d\}\}\{\\mathrm\{d\}\\epsilon\}\\right\|\_\{\\epsilon=0\}\\exp\(\\epsilon\\xi\)\\cdot Y\.The lifted primitive evaluator returns frame coordinatesF^W​\(p\)​\(Y\)∈𝔤\\widehat\{F\}\_\{W\}\(p\)\(Y\)\\in\\mathfrak\{g\}satisfying

FW​\(p\)​\(Y\)=F^W​\(p\)​\(Y\)\#​\(Y\)\.F\_\{W\}\(p\)\(Y\)=\\widehat\{F\}\_\{W\}\(p\)\(Y\)^\{\\\#\}\(Y\)\.Consequently, the window field is represented by

L^k​\(Y\)=∑p∈ℬℋprimλkp​F^W​\(p\)​\(Y\),Lk​\(Y\)=L^k​\(Y\)\#​\(Y\)\.\\widehat\{L\}\_\{k\}\(Y\)=\\sum\_\{p\\in\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}\}\\lambda\_\{k\}^\{p\}\\widehat\{F\}\_\{W\}\(p\)\(Y\),\\quad L\_\{k\}\(Y\)=\\widehat\{L\}\_\{k\}\(Y\)^\{\\\#\}\(Y\)\.The log\-ODE on the time intervalτ∈\[0,1\]\\tau\\in\[0,1\]is then

Z˙τ=L^k​\(Zτ\)\#​\(Zτ\),Z0=Yk\.\\dot\{Z\}\_\{\\tau\}=\\widehat\{L\}\_\{k\}\(Z\_\{\\tau\}\)^\{\\\#\}\(Z\_\{\\tau\}\),\\qquad Z\_\{0\}=Y\_\{k\}\.This is a geometric numerical integration problem that may be solved usingRunge–Kutta–Munthe–KaasorCommutator free \(CF\)methods\. We selectCF​\-​EES​\(2,5\)\\mathrm\{CF\\text\{\-\}EES\}\(2,5\)for its minimal exponential design which minimises per\-step memory and compute costs\(Shmelevet al\.,[2026](https://arxiv.org/html/2606.05272#bib.bib92)\)\. We refer the reader to their text and appendices for an overview of commutator\-free methods in neural differential equations\.

In the flat case, the group action reduces to addition, and the exponential update reduces to the corresponding explicit ODE method applied to \([8](https://arxiv.org/html/2606.05272#S3.E8)\)\. Hence, the same implementation covers the EuclideanℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}case as well as the homogeneous\-spaceℋ\\operatorname\{\\mathcal\{H\}\}andℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}cases\.

### 3\.5Complexity Analysis

At truncation depthNN, the algebraic cost is governed by the primitive basis size of the chosen Hopf algebra\. Letbℋ​\(n\)b\_\{\\mathcal\{H\}\}\(n\)denote the number of degree\-nnprimitive basis elements and letddbe the path dimension\. Forℋ\\operatorname\{\\mathcal\{H\}\},b​\(n\)b\(n\)is given by Witt’s formula\(Witt,[1937](https://arxiv.org/html/2606.05272#bib.bib83)\); forℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}, primitives aredd\-decorated non\-planar rooted trees\(Göbel,[1980](https://arxiv.org/html/2606.05272#bib.bib82)\); and forℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}, primitives aredd\-decorated planar rooted trees\(Walkup,[1972](https://arxiv.org/html/2606.05272#bib.bib81)\)\. Hence

b​\(n\)\\displaystyle b\(n\)=1n​∑k∣nμ​\(k\)​dn/k,\\displaystyle=\\frac\{1\}\{n\}\\sum\_\{k\\mid n\}\\mu\(k\)\\,d^\{n/k\},bGL​\(n\)\\displaystyle b\_\{\\mathrm\{GL\}\}\(n\)=tn​dn,\\displaystyle=t\_\{n\}d^\{n\},bMKW​\(n\)\\displaystyle b\_\{\\mathrm\{MKW\}\}\(n\)=Cn−1​dn=1n​\(2​n−2n−1\)​dn,\\displaystyle=C\_\{n\-1\}d^\{n\}=\\frac\{1\}\{n\}\\binom\{2n\-2\}\{n\-1\}d^\{n\},whereμ\\muis the Möbius function andtnt\_\{n\}is the number of non\-planar rooted\-tree shapes withnnvertices\. Write

\|ℬℋprim​\(N\)\|≔∑n=1Nbℋ​\(n\)\.\|\\mathcal\{B\}\_\{\\mathcal\{H\}\}^\{\\mathrm\{prim\}\}\(N\)\|\\coloneqq\\sum\_\{n=1\}^\{N\}b\_\{\\mathcal\{H\}\}\(n\)\.Assume all models use the same atomic vector fieldfθf\_\{\\theta\}, anmm\-layermultilayer perceptron \(MLP\)with hidden widthnhn\_\{h\}and hidden\-state dimensionuu\. Forfθ:ℝu→ℝu×df\_\{\\theta\}\\colon\\mathbb\{R\}^\{u\}\\to\\mathbb\{R\}^\{u\\times d\}, one primal evaluation costs

Cθ=2​u​nh\+2​\(m−1\)​nh2\+2​u​d​nh\.C\_\{\\theta\}=2un\_\{h\}\+2\(m\-1\)n\_\{h\}^\{2\}\+2udn\_\{h\}\.Using the standard estimate that one JVP costs three primal evaluations, and reusing derivative evaluations across primitive fields, the Euclidean per\-field costs are

𝒞\(N\)\\displaystyle\\mathcal\{C\}^\{\(N\)\}=3​\|ℬprim​\(N−1\)\|​Cθ,\\displaystyle=3\\,\|\\mathcal\{B\}^\{\\mathrm\{prim\}\}\(N\-1\)\|\\,C\_\{\\theta\},𝒞GL\(N\)\\displaystyle\\mathcal\{C\}\_\{\\mathrm\{GL\}\}^\{\(N\)\}=3​\|ℬGLprim​\(N−1\)\|​Cθ\.\\displaystyle=3\\,\|\\mathcal\{B\}\_\{\\mathrm\{GL\}\}^\{\\mathrm\{prim\}\}\(N\-1\)\|\\,C\_\{\\theta\}\.
For manifold\-valued models, ordinary JVPs are replaced by covariant or coordinate directional derivatives\. Theℋ\\operatorname\{\\mathcal\{H\}\}lift evaluates each non\-letter Lyndon primitive by a Lie bracket, which requires two directional derivative evaluations\. Following the same JVP cost model, this gives

𝒞,∇\(N\)≈6​\|ℬprim​\(N−1\)\|​Cθ\.\\mathcal\{C\}\_\{\\shuffle,\\nabla\}^\{\(N\)\}\\approx 6\\,\|\\mathcal\{B\}^\{\\mathrm\{prim\}\}\(N\-1\)\|\\,C\_\{\\theta\}\.ForℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}, a degree\-nnplanar tree hasn−1n\-1edges, and each edge corresponds to one covariant JVP in the ordered tree lift\. HenceDMKW​\(N\)=∑n=1N\(n−1\)​bMKW​\(n\),D\_\{\\mathrm\{MKW\}\}\(N\)=\\sum\_\{n=1\}^\{N\}\(n\-1\)b\_\{\\mathrm\{MKW\}\}\(n\),and

𝒞MKW,∇\(N\)≈3​DMKW​\(N\)​Cθ\.\\mathcal\{C\}\_\{\\mathrm\{MKW\},\\nabla\}^\{\(N\)\}\\approx 3D\_\{\\mathrm\{MKW\}\}\(N\)C\_\{\\theta\}\.These estimates isolate the neural evaluation cost; manifold implementations also incur lower\-order geometry costs from coordinate or frame conversion, connection operations, Lie\-bracket structure terms, and group actions\.

It remains to account for solver stages\. WithJJinternal steps andRRfield evaluations per step,

𝒞ℋ,window\(N\)=J​R​𝒞ℋ\(N\),ℋ∈\{ℋ,ℋGL\},\\mathcal\{C\}\_\{\\mathcal\{H\},\\mathrm\{window\}\}^\{\(N\)\}=JR\\,\\mathcal\{C\}\_\{\\mathcal\{H\}\}^\{\(N\)\},\\quad\\mathcal\{H\}\\in\\\{\\operatorname\{\\mathcal\{H\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}\\\},so Euclidean Heun gives2​J​𝒞ℋ\(N\)2J\\,\\mathcal\{C\}\_\{\\mathcal\{H\}\}^\{\(N\)\}\. For homogeneous\-space models integrated by a commutator\-free scheme withRRfield\-evaluation stages andQQexponentials per step,

𝒞ℋ,window\(N\)≈J​R​𝒞ℋ,∇\(N\)\+J​Q​Cℳ,ℋ∈\{ℋ,ℋMKW\}\.\\mathcal\{C\}\_\{\\mathcal\{H\},\\mathrm\{window\}\}^\{\(N\)\}\\approx JR\\,\\mathcal\{C\}\_\{\\mathcal\{H\},\\nabla\}^\{\(N\)\}\+JQ\\,C\_\{\\mathcal\{M\}\},\\quad\\mathcal\{H\}\\in\\\{\\operatorname\{\\mathcal\{H\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\\\}\.HereCℳC\_\{\\mathcal\{M\}\}denotes the cost of the group exponential and action\. ForCF​\-​EES​\(2,5\)\\mathrm\{CF\\text\{\-\}EES\}\(2,5\),Q=R=3Q=R=3; in our implementation, each exponential uses a fourth\-order approximation requiring two matrix multiplications\(Sastreet al\.,[2019](https://arxiv.org/html/2606.05272#bib.bib94)\)\.

Table 2:Per\-window costs forN=2N=2\. HereP2P\_\{2\}is the primitive dimension,A2A\_\{2\}is the neural derivative multiplier so that one field evaluation costs3​A2​Cθ3A\_\{2\}C\_\{\\theta\}, and\(R,Q\)\(R,Q\)denotes field\-evaluation stages and exponentials per solver step\.ℝ,ℋ\\mathbb\{R\},\\operatorname\{\\mathcal\{H\}\}is log\-NCDE\.
### 3\.6The Branched Signature Kernel

The geometric signature kernel between two pathsx,yx,yis

kgeo​\(x,y\)=⟨SigN​\(x\),SigN​\(y\)⟩,k\_\{\\mathrm\{geo\}\}\(x,y\)=\\langle\\mathrm\{Sig\}^\{N\}\(x\),\\mathrm\{Sig\}^\{N\}\(y\)\\rangle,where the inner product is taken on the depth\-NNtruncated tensor signature space\. Signature kernels provide a principled similarity measure for stochastic processes\(Chevyrev and Oberhauser,[2022](https://arxiv.org/html/2606.05272#bib.bib65)\)and have been used to train neural SDEs through signature\-kernel scoring objectives\(Issaet al\.,[2023](https://arxiv.org/html/2606.05272#bib.bib46)\)\. In practice, training minimises

ℒgeo​\(θ\)=𝔼\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{geo\}\}\(\\theta\)=\\mathbb\{E\}\[kgeo\(X,X′\)\]X,X′∼Pθ\\displaystyle\{\}\_\{X,X^\{\\prime\}\\sim P\_\{\\theta\}\}\\big\[k\_\{\\mathrm\{geo\}\}\(X,X^\{\\prime\}\)\\big\]\(9\)−2​𝔼X∼Pθ,Y∼Pdata​\[kgeo​\(X,Y\)\],\\displaystyle\-2\\mathbb\{E\}\_\{X\\sim P\_\{\\theta\},Y\\sim P\_\{\\mathrm\{data\}\}\}\\big\[k\_\{\\mathrm\{geo\}\}\(X,Y\)\\big\],wherePdataP\_\{\\mathrm\{data\}\}denotes the data law andPθP\_\{\\theta\}the model law\. The term𝔼Y,Y′∼Pdata​\[kgeo​\(Y,Y′\)\]\\mathbb\{E\}\_\{Y,Y^\{\\prime\}\\sim P\_\{\\mathrm\{data\}\}\}\\big\[k\_\{\\mathrm\{geo\}\}\(Y,Y^\{\\prime\}\)\\big\]is constant inθ\\theta\.

Existing signature\-kernel implementations\(Salviet al\.,[2021](https://arxiv.org/html/2606.05272#bib.bib67); Casset al\.,[2025](https://arxiv.org/html/2606.05272#bib.bib68); Tóth,[2025](https://arxiv.org/html/2606.05272#bib.bib69); Tóthet al\.,[2025](https://arxiv.org/html/2606.05272#bib.bib70)\)compute geometric signatures, whose semimartingale limits correspond to iterated Stratonovich integrals\. For semimartingale data observed on a grid, piecewise\-linear interpolation has finite variation, so its canonical lift has zero bracket and does not expose quadratic covariation as a driver coordinate\. Consequently, an Itô or branched\-signature model using this representation must supply bracket information separately\.

###### Definition 3\.1\(Branched signature kernel objective\)\.

Fix a Hopf algebraℋ\\mathcal\{H\}, truncation depthNN, and the coordinate basis used bySigℋN\\operatorname\{Sig\}\_\{\\mathcal\{H\}\}^\{N\}; throughout,⟨⋅,⋅⟩ℋ≤N\\langle\\cdot,\\cdot\\rangle\_\{\\mathcal\{H\}\_\{\\leq N\}\}denotes the Euclidean inner product on these coordinates\. For drivers𝐗,𝐘\\mathbf\{X\},\\mathbf\{Y\}enhanced with their quadratic variation, define

kbrN​\(𝐗,𝐘\)=⟨SigℋN⁡\(𝐗\),SigℋN⁡\(𝐘\)⟩ℋ≤N\.k\_\{\\mathrm\{br\}\}^\{N\}\(\\mathbf\{X\},\\mathbf\{Y\}\)=\\left\\langle\\operatorname\{Sig\}\_\{\\mathcal\{H\}\}^\{N\}\(\\mathbf\{X\}\),\\operatorname\{Sig\}\_\{\\mathcal\{H\}\}^\{N\}\(\\mathbf\{Y\}\)\\right\\rangle\_\{\\mathcal\{H\}\_\{\\leq N\}\}\.The branched signature\-kernel objective is

ℒbr​\(θ\)=𝔼\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{br\}\}\(\\theta\)=\\mathbb\{E\}\[kbrN\(𝐗,𝐗′\)\]𝐗,𝐗′∼Pθ\\displaystyle\{\}\_\{\\mathbf\{X\},\\mathbf\{X\}^\{\\prime\}\\sim P\_\{\\theta\}\}\\big\[k\_\{\\mathrm\{br\}\}^\{N\}\(\\mathbf\{X\},\\mathbf\{X\}^\{\\prime\}\)\\big\]\(10\)−2​𝔼𝐗∼Pθ,𝐘∼Pdata​\[kbrN​\(𝐗,𝐘\)\]\.\\displaystyle\-2\\mathbb\{E\}\_\{\\mathbf\{X\}\\sim P\_\{\\theta\},\\mathbf\{Y\}\\sim P\_\{\\mathrm\{data\}\}\}\\big\[k\_\{\\mathrm\{br\}\}^\{N\}\(\\mathbf\{X\},\\mathbf\{Y\}\)\\big\]\.As in \([9](https://arxiv.org/html/2606.05272#S3.E9)\), the omitted data–data term is constant inθ\\theta\.

When bracket increments are available from the simulator, the objective uses them directly rather than reconstructing them from realised quadratic variation\. Appendix[B\.2](https://arxiv.org/html/2606.05272#A2.SS2)formalises this finite\-grid distinction\. A geometric kernel computed from path values is a kernel on the projected path law, whereas the branched kernel is computed on the enhanced law containing bracket coordinates\. A hypothetical bracket\-aware baseline built only from grid samples must first reconstruct those coordinates, and is therefore subject to realised\-covariance error\. For standard continuous semimartingales, fixed\-time realised\-covariance fluctuations are typically of orderΔn\\sqrt\{\\Delta\_\{n\}\}, so supplying analytic or simulator\-ground\-truth brackets can remove this finite\-grid error source without changing the continuous\-limit law\-matching target\.

## 4Experiments

We design our experimental evaluation to focus on regimes where standard EuclideanNRDEsstruggle\. We validateB\-NRDEacross three distinct domains, mapping each to the Hopf algebra that captures its geometry and causality\.

We first consider Euclidean rough volatility\. UtilisingℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}, we taskB\-NRDEwith learning an unconditional generative model of the rough Bergomi stochastic volatility model, evaluating performance via the fidelity of the marginal laws\. Next, we examine Stratonovich manifold dynamics underℋ\\operatorname\{\\mathcal\{H\}\}and evaluate forecasting performance\. Finally, we address manifold\-valued Itô rough paths, employingℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}to learn a generative model of covariance dynamics on thesymmetric positive definite \(SPD\)matrix manifold\. Dataset code is available in[RoughBench](https://github.com/luke-a-thompson/RoughBench), and further experimental details may be found in[AppendixD](https://arxiv.org/html/2606.05272#A4)\.

### 4\.1Baseline Models

As baselines, we employmanifold neural ODE \(M\-NODE\)\(Louet al\.,[2020](https://arxiv.org/html/2606.05272#bib.bib85)\),NCDEwith linear\(Kidgeret al\.,[2020](https://arxiv.org/html/2606.05272#bib.bib8)\), Hermite\(Morrillet al\.,[2022](https://arxiv.org/html/2606.05272#bib.bib63)\), andSavitzky–Golay \(SG\)\(Bastianet al\.,[2025](https://arxiv.org/html/2606.05272#bib.bib12)\)interpolation, dubbedNCDE,NCDE\+\+ andSG\-NCDE, respectively\. We also consider signature methodsNRDE\(Morrillet al\.,[2021b](https://arxiv.org/html/2606.05272#bib.bib3)\)and log\-NCDE\(Walkeret al\.,[2024](https://arxiv.org/html/2606.05272#bib.bib1)\), and discrete\-time GRU\(Chunget al\.,[2014](https://arxiv.org/html/2606.05272#bib.bib95)\), xLSTM\(Becket al\.,[2024](https://arxiv.org/html/2606.05272#bib.bib86)\), and stacked xLSTM, the backbone ofBecket al\.\([2024](https://arxiv.org/html/2606.05272#bib.bib86)\)\.

### 4\.2Generative Modeling of Rough Volatility Under Itô Integration

Rough volatility models posit that log\-volatility evolves with fractional regularity \(H<12H<\\tfrac\{1\}\{2\}\)\(Gatheralet al\.,[2014](https://arxiv.org/html/2606.05272#bib.bib17)\)\. A prominent example is therough Bergomi \(rBergomi\)model\(Bayeret al\.,[2015](https://arxiv.org/html/2606.05272#bib.bib19)\), where volatility is driven by a Volterra processWtH=∫0tKH​\(t−s\)​dWsW\_\{t\}^\{H\}=\\int\_\{0\}^\{t\}K\_\{H\}\(t\-s\)\\mathrm\{d\}W\_\{s\}\. Simulating these paths is a known bottleneck, as exact methods scale quadratically with the number of time steps\. Furthermore, standard signature\-based approaches are constrained to the geometric \(Stratonovich\) setting; capturing the financially\-relevant Itô integral requires augmenting the path with a lead\-lag \(Hoff\) lift\(Flintet al\.,[2016](https://arxiv.org/html/2606.05272#bib.bib18)\), doubling channel dimension fromd→2​dd\\to 2d, massively growing signature size\.

We userBergomito stress\-test the Itô capability of theB\-NRDEframework in flat geometry \(ℳ=ℝd\\mathcal\{M\}=\\mathbb\{R\}^\{d\}\)\. UnlikeNRDE, ourℋGL\\mathcal\{H\}\_\{\\mathrm\{GL\}\}formulation accommodates branched rough paths, allowingB\-NRDEto process the driving signal without lead–lag augmentation\. Equation and simulation details can be found in[SectionD\.2](https://arxiv.org/html/2606.05272#A4.SS2)\. For training, we generate a time\-augmented latent piecewise\-linear Brownian motion\(t,Zlat\)\(t,Z\_\{\\mathrm\{lat\}\}\)on\[0,1\]\[0,1\]with grid sizeΔn\\Delta\_\{n\}, and sample generated log\-price pathsX≔log⁡S^θ∼PθX\\coloneqq\\widehat\{\\log S\}\_\{\\theta\}\\sim P\_\{\\theta\}\. Ground\-truth paths areY≔log⁡S∼PdataY\\coloneqq\\log S\\sim P\_\{\\mathrm\{data\}\}\. All models minimiseℒgeo\\mathcal\{L\}\_\{\\mathrm\{geo\}\}exceptB\-NRDE, which receives three additional fine\-tuning epochs under the branched signature\-kernel score between\(t,X\)\(t,X\)and\(t,Y\)\(t,Y\), adding quadratic\-variation/covariation terms⟨Xi,Xj⟩\\langle X^\{i\},X^\{j\}\\rangleand⟨Yi,Yj⟩\\langle Y^\{i\},Y^\{j\}\\rangle; the former are the squared increments, the latter from the simulator ground truth\.

In[Table3](https://arxiv.org/html/2606.05272#S4.T3)we reportKolmogorov\-Smirnov \(KS\)distances between model\-generated and ground truth time\-marginal laws\.B\-NRDEachieves the strongest fit across three of four horizons, outperformingℋ\\operatorname\{\\mathcal\{H\}\}\-basedNRDEand log\-NCDE\.

Table 3:rBergomiKSacross time\-marginals![Refer to caption](https://arxiv.org/html/2606.05272v1/x4.png)Figure 3:rBergomisample paths generated byB\-NRDEshowing good fit to the underlying law\.
### 4\.3Sim\-to\-Real Dynamics Forecasting

Three\-dimensional motion forecasting is central to robotics and computer vision, with applications to pose prediction\(Yanget al\.,[2021](https://arxiv.org/html/2606.05272#bib.bib47); Dünkelet al\.,[2024](https://arxiv.org/html/2606.05272#bib.bib48)\)and occlusion\-robust perception\(Diet al\.,[2021](https://arxiv.org/html/2606.05272#bib.bib49)\)\. We consider sim\-to\-real forecasting of rotational dynamics, previously out of reach for signature methods due to the lack of a manifold\-compatible formulation\. As a deterministic task without an Itô dynamic, we leverage a coordinate\-freeℋ\\operatorname\{\\mathcal\{H\}\}formulation\.

![Refer to caption](https://arxiv.org/html/2606.05272v1/x5.png)Figure 4:Schematic description of the sim\-to\-real dynamics training pipeline\. Adapted fromBastianet al\.\([2025](https://arxiv.org/html/2606.05272#bib.bib12)\)\.FollowingBastianet al\.\([2025](https://arxiv.org/html/2606.05272#bib.bib12)\), we generate offline continuous\-time rotation trajectoriesR:\[0,1\]→SO​\(3\)R\\colon\[0,1\]\\to\\mathrm\{SO\}\(3\), whereR​\(t\)R\(t\)is the orientation at timett\. From each trajectory we extract sliding windows\{\(tj,R​\(tj\)\)\}j=0m\\\{\(t\_\{j\},R\(t\_\{j\}\)\)\\\}\_\{j=0\}^\{m\}and split each window into a reconstruction prefix𝒮recon\\mathcal\{S\}\_\{\\mathrm\{recon\}\}and prediction suffix𝒮pred\\mathcal\{S\}\_\{\\mathrm\{pred\}\}\. An extrapolator𝖤\\mathsf\{E\}is fit on𝒮recon\\mathcal\{S\}\_\{\\mathrm\{recon\}\}and evaluated on the full window to produce a dense input pathR~​\(t\)\\tilde\{R\}\(t\)over𝒮recon∪𝒮pred\\mathcal\{S\}\_\{\\mathrm\{recon\}\}\\cup\\mathcal\{S\}\_\{\\mathrm\{pred\}\}\. This extrapolated path is then provided to the model, which outputs predicted rotations\{R^θ​\(tj\)\}j=0m\\\{\\hat\{R\}\_\{\\theta\}\(t\_\{j\}\)\\\}\_\{j=0\}^\{m\}over the same times\. We pretrain by minimising the Frobenius discrepancy to the ground truth over the full window:

minθ​∑j∈𝒥‖R^θ​\(tj\)−R​\(tj\)‖F2,R^θ​\(tj\),R​\(tj\)∈SO​\(3\),\\min\_\{\\theta\}\\;\\sum\_\{j\\in\\mathcal\{J\}\}\\bigl\\\|\\hat\{R\}\_\{\\theta\}\(t\_\{j\}\)\-R\(t\_\{j\}\)\\bigr\\\|\_\{F\}^\{2\},\\quad\\hat\{R\}\_\{\\theta\}\(t\_\{j\}\),\\,R\(t\_\{j\}\)\\in\\mathrm\{SO\}\(3\),where𝒥=\{0,…,m\}\\mathcal\{J\}=\\\{0,\\dots,m\\\}\.

To assess sim\-to\-real generalisation, simulation\-pretrained models are evaluated on theOxford Multimotion Dataset \(OMD\)\(Judd and Gammell,[2019](https://arxiv.org/html/2606.05272#bib.bib15)\), with windows and extrapolations identical to those used for the simulation data\. The rotation geodesic error \(RGE\)\(Huynh,[2009](https://arxiv.org/html/2606.05272#bib.bib14)\), defined asRGE⁡\(R1,R2\)=2​arcsin⁡\(‖R2−R1‖F2​2\)\\operatorname\{RGE\}\(R\_\{1\},R\_\{2\}\)=2\\arcsin\\left\(\\frac\{\\\|R\_\{2\}\-R\_\{1\}\\\|\_\{F\}\}\{2\\sqrt\{2\}\}\\right\), is computed over both segments\.[Figure4](https://arxiv.org/html/2606.05272#S4.F4)illustrates the protocol\.

Table 4:Training test set Frobenius norm and RGE \(degrees\) across different motion scenarios in theOMDdataset\.MethodPretrainingFrobeniusStaticMotionTranslationMotionUnconstrainedMotionM\-NODE2\.461132\.40118\.25121\.08SO​\(3\)\\mathrm\{SO\}\(3\)\-NCDE0\.29422\.2121\.8322\.45SO​\(3\)\\mathrm\{SO\}\(3\)\-GRU0\.27120\.2520\.3921\.13xLSTM0\.23316\.6716\.7717\.36NRDE0\.0977\.457\.347\.32Stacked xLSTM0\.1107\.117\.507\.03SG\-NCDE0\.0502\.933\.312\.80B\-NRDE\(GK\)0\.0493\.233\.703\.33

Whilst SG\-NCDEattains the lowest RGE,B\-NRDEachieves comparable accuracy with only two solver steps versus 20\. The remaining small deficit is not explained by the simulation pretraining loss, suggesting modest degradation in sim\-to\-real transfer\. Crucially,B\-NRDEgeneralises the SG\-NCDEpipeline by admitting rough drivers, thereby removing theC1C^\{1\}interpolant requirement and permitting non\-smooth extrapolators such asMLPs\.

### 4\.4Itô Dynamics over the SPD Manifold

SPD\-covariance dynamics learning arises in multivariate finance\(Noureldinet al\.,[2012](https://arxiv.org/html/2606.05272#bib.bib50); Johanssonet al\.,[2023](https://arxiv.org/html/2606.05272#bib.bib55)\), medical imaging\(Pennec,[2020](https://arxiv.org/html/2606.05272#bib.bib54)\), and, more recently, Riemannian diffusion models\(Parket al\.,[2022](https://arxiv.org/html/2606.05272#bib.bib72); De Bortoliet al\.,[2022](https://arxiv.org/html/2606.05272#bib.bib73); Thorntonet al\.,[2022](https://arxiv.org/html/2606.05272#bib.bib74)\)\. Notably, finance canonically requires non\-anticipative Itô modelling to avoid lookahead bias, and diffusion models are typically formulated as Itô diffusions\.

As such, we study a mean\-reverting Itô diffusion on the SPD manifold𝕊d\+\+\\mathbb\{S\}^\{\+\+\}\_\{d\}endowed with theaffine invariant \(AI\)geometry, testingB\-NRDE’s ability to learn non\-anticipative stochastic dynamics on a curved state space while strictly preserving theSPDcone\. LetM∈𝕊d\+\+M\\in\\mathbb\{S\}^\{\+\+\}\_\{d\}and letX0∈𝕊d\+\+X\_\{0\}\\in\\mathbb\{S\}^\{\+\+\}\_\{d\}\. Consider the OU\-type Itô diffusion

d​Xt=η​logXt⁡\(M\)​d​t\+σ​Xt1/2​d​Bt​Xt1/2,\\mathrm\{d\}X\_\{t\}=\\eta\\,\\log\_\{X\_\{t\}\}\(M\)\\,\\mathrm\{d\}t\+\\sigma\\,X\_\{t\}^\{1/2\}\\,\\mathrm\{d\}B\_\{t\}\\,X\_\{t\}^\{1/2\},\(11\)whereη\>0\\eta\>0controls mean reversion,σ≥0\\sigma\\geq 0is the noise scale, andBt∈Sym​\(d\)B\_\{t\}\\in\\mathrm\{Sym\}\(d\)is a symmetric matrix Brownian motion\. In practice, the vectorised statex~t=vech​\(Xt\)∈ℝq\\tilde\{x\}\_\{t\}=\\mathrm\{vech\}\(X\_\{t\}\)\\in\\mathbb\{R\}^\{q\},d​⟨x~⟩t∈ℝq×q\\mathrm\{d\}\\langle\\tilde\{x\}\\rangle\_\{t\}\\in\\mathbb\{R\}^\{q\\times q\}withq=d​\(d\+1\)2q=\\frac\{d\(d\+1\)\}\{2\}is used throughout\.

Training minimises an expected signature\-kernel objective between the ground\-truth and generated trajectories, with log\-NCDEusing the geometric signature kernel andB\-NRDE\. Since𝕊d\+\+\\mathbb\{S\}^\{\+\+\}\_\{d\}is not totally ordered, aKSstatistic is not directly applicable\. We therefore sort the eigenvalues increasingly and report empirical time\-marginalW1W\_\{1\}distances between the eigenvalue vectorsλ​\(Xt\),λ​\(X^t\)∈ℝd\\lambda\(X\_\{t\}\),\\lambda\(\\widehat\{X\}\_\{t\}\)\\in\\mathbb\{R\}^\{d\}\.

In[Table5](https://arxiv.org/html/2606.05272#S4.T5)we find that the manifold\-preserving capabilities ofB\-NRDEimprove on the results of Euclidean log\-NCDEin terms of law\-matching, showing an average improvement of 56\.2%\. However, visual inspection of[Figure5](https://arxiv.org/html/2606.05272#S4.F5)shows somewhat disappointing fidelity near the initial condition, a common pitfall of signature\-based methods that are invariant to the starting value without augmentation\(Morrillet al\.,[2021a](https://arxiv.org/html/2606.05272#bib.bib80)\)\. We also find no meaningful performance difference between the branched and geometric kernels in this setting\.

Table 5:1\-Wasserstein distance between predicted and ground\-truth eigenvalue marginals on𝕊d\+\+\\mathbb\{S\}^\{\+\+\}\_\{d\}\.![Refer to caption](https://arxiv.org/html/2606.05272v1/x6.png)\(a\)Ground truth eigenvalue trajectories of[Equation11](https://arxiv.org/html/2606.05272#S4.E11)\.
![Refer to caption](https://arxiv.org/html/2606.05272v1/x7.png)\(b\)Predicted eigenvalue trajectories output byB\-NRDE\.

Figure 5:Ground truth vs\.B\-NRDEeigenvalue trajectories\.

## 5Discussion

In this work, we introducedB\-NRDE, a generalisation ofNRDEto branched rough paths, and a branched\-signature\-kernel training objective for neuralstochastic differential equations \(SDEs\)\. Our method extends signature methods beyond the geometric/Stratonovich setting to Itô, manifold, and Itô\-on\-manifold dynamics, while remaining empirically competitive\. We summarise key limitations of the current method and outline the most promising extensions\.

### 5\.1Limitations

Unlike the untruncated PDE\-based solvers available for geometric kernels\(Salviet al\.,[2021](https://arxiv.org/html/2606.05272#bib.bib67)\), branched signature kernels require explicit truncation, discarding higher\-order iterated\-integral information at greater memory and compute cost\. We expect this to worsen as the state dimensionddgrows, since the quadratic\-variation features scale asd2d^\{2\}\.

### 5\.2Future Directions

##### Numerical

Such PDE\-based solvers, or other approximation schemes, offer the most promising route to avoiding explicit truncation in our kernel\. Adaptive log\-ODE schemes could also permit larger signature windows and faster integration\(Bayeret al\.,[2023](https://arxiv.org/html/2606.05272#bib.bib88)\)\.

##### Signatures

The log\-signature quotients out the shuffle identities, projecting onto the strictly smaller Lyndon\-word basis\. No analogous projection is known explicitly for the branched Hopf algebras used here, though recent work\(Bellingeriet al\.,[2024](https://arxiv.org/html/2606.05272#bib.bib33)\)suggests one exists; this would yield smaller branched log\-signatures and improve the memory efficiency of B\-NRDE\. For scalarRDEs, such as the rough Bergomi model of[Section4\.2](https://arxiv.org/html/2606.05272#S4.SS2), the theory of multi\-index rough paths, elements of the Linares–Otto\-Tempelmayr Hopf algebra\(Linareset al\.,[2023](https://arxiv.org/html/2606.05272#bib.bib97)\), offers an even more compact representation by exploiting symmetries in the vector fields of scalar equations\(Bellingeriet al\.,[2026](https://arxiv.org/html/2606.05272#bib.bib96)\)\.

##### SPDEs

The rough\-path philosophy extends from finite\-dimensional differential equations tostochastic partial differential equations \(SPDEs\)via the theory of regularity structures\(Hairer,[2014](https://arxiv.org/html/2606.05272#bib.bib42)\)\. Much like how planarly branched rough paths enable solvingRDEsover manifolds, planar regularity structures\(Rahm,[2022](https://arxiv.org/html/2606.05272#bib.bib41)\)provide the corresponding generalisation for regularity structures andSPDEs\.[Figure6](https://arxiv.org/html/2606.05272#S5.F6)recapitulatesRahm \([2022](https://arxiv.org/html/2606.05272#bib.bib41), Fig\. 1\)and points to a natural future direction: a neural planar\-regularity\-structure model in aℝd→ℳ\\mathbb\{R\}^\{d\}\\to\\mathcal\{M\}setting, beyond theℝ→ℳ\\mathbb\{R\}\\to\\mathcal\{M\}framework ofB\-NRDEand theℝd→ℝd\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}framework of Neural Operator with Regularity Structure\(Huet al\.,[2022](https://arxiv.org/html/2606.05272#bib.bib43)\)and Deep Latent Regularity Net\(Gonget al\.,[2023](https://arxiv.org/html/2606.05272#bib.bib44)\)\.

NRDELog\-NCDEDLR\-NetNORSB\-NRDEPlanar RS?multi\-dim driversplanar,branched liftplanarlift?planar RS?Figure 6:Positioning of our method and a hypothetical planar\-regularity\-structure extension\.

## Acknowledgements

We thank Slade Matthews for his supervision, guidance, and support\. We also thank Daniil Shmelev for providing his software library, pySigLib, which we used extensively in our work, and for rapidly implementing the requested features and bug fixes needed for the experiments\.

## Impact Statement

This paper presents fundamental work in time\-series forecasting aimed at advancing the field of Machine Learning\. There are many potential societal consequences of our work, but none of which we feel must be specifically highlighted here\.

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## Appendix AThe Hopf Algebras of Rough Paths

This appendix records the algebraic structures used throughout the paper\. We first detail the formal requirements imposed on our Hopf algebras, then explain why we use\(i\)the*graded dual*of the shuffle Hopf algebra for geometric, Stratonovich\-type drivers,\(ii\)Grossman–Larson \(GL\)rather thanButcher–Connes–Kreimer \(BCK\)for branched, Itô\-type drivers, and\(iii\)the*graded dual*of theMunthe–Kaas–Wright \(MKW\)Hopf algebra for the manifold log\-ODE scheme\.We keep the presentation concise and refer toKern and Lyons \([2023](https://arxiv.org/html/2606.05272#bib.bib6)\)andManchon \([2025](https://arxiv.org/html/2606.05272#bib.bib5)\)for a more comprehensive account\.

### A\.1Hopf Algebras

##### Gradedness\.

A Hopf algebraℋ\\mathcal\{H\}is graded if it admits a decomposition

ℋ=⨁n≥0ℋn\\mathcal\{H\}=\\bigoplus\_\{n\\geq 0\}\\mathcal\{H\}\_\{n\}such that its product⋆\\starand coproductΔ\\Deltarespect degree:

ℋm⋆ℋn⊆ℋm\+n,Δ​\(ℋn\)⊆⨁m=0nℋm⊗ℋn−m\.\\mathcal\{H\}\_\{m\}\\star\\mathcal\{H\}\_\{n\}\\subseteq\\mathcal\{H\}\_\{m\+n\},\\qquad\\Delta\(\\mathcal\{H\}\_\{n\}\)\\subseteq\\bigoplus\_\{m=0\}^\{n\}\\mathcal\{H\}\_\{m\}\\otimes\\mathcal\{H\}\_\{n\-m\}\.\(12\)The componentℋn\\mathcal\{H\}\_\{n\}corresponds to levelnnsignature coordinates\. For word signatures,nnis word length; for branched signatures,nnis the number of vertices\.

##### Connectedness\.

A graded Hopf algebra is connected if the degree\-zero component is spanned by the unit:

ℋ0=span⁡\{1\}\.\\mathcal\{H\}\_\{0\}=\\operatorname\{span\}\\\{1\\\}\.For instance, in theGLHopf algebra on rooted forests, the empty forest is the unit11, and every non\-empty forest has positive degree\.

##### Truncation\.

In applications, we truncate at depthNN:

ℋ\(≤N\):=⨁n=0Nℋn\.\\mathcal\{H\}^\{\(\\leq N\)\}:=\\bigoplus\_\{n=0\}^\{N\}\\mathcal\{H\}\_\{n\}\.This truncation matches the depth of the signature or log\-signature retained in the numerical scheme\.

##### Primitives and grouplike elements\.

Primitive elements have no nontrivial coproduct:

Prim⁡\(ℋ\):=\{p∈ℋ:Δ​p=p⊗1\+1⊗p\}\.\\operatorname\{Prim\}\(\\mathcal\{H\}\):=\\\{p\\in\\mathcal\{H\}:\\Delta p=p\\otimes 1\+1\\otimes p\\\}\.For rough paths, increments are characters of the coordinate Hopf algebra; equivalently, they are grouplike elements of the graded dual\. The Hopf logarithmlog⋆\\log\_\{\\star\}, as in[Equation6](https://arxiv.org/html/2606.05272#S3.E6), maps grouplike elements to primitive elements\.

### A\.2Why cocommutativity matters: Milnor–Moore

The log\-ODE construction is expressed at the level of primitives: one takes the Hopf logarithm of a grouplike increment, obtains a Lie element, and then exponentiates the resulting truncated Lie polynomial in differential operators\. This gives a compression from the full Hopf algebra to its primitive part\. To recover the Hopf algebra from its primitives, we use the Milnor–Moore theorem\.

###### Theorem A\.1\(Milnor–Moore\(Milnor and Moore,[1965](https://arxiv.org/html/2606.05272#bib.bib90)\)\)\.

Letℋ\\mathcal\{H\}be a connected, graded, cocommutative Hopf algebra over a field of characteristic0\. Then there is a canonical Hopf algebra isomorphism

ℋ≅U​\(Prim⁡\(ℋ\)\)\.\\mathcal\{H\}\\cong U\\bigl\(\\operatorname\{Prim\}\(\\mathcal\{H\}\)\\bigr\)\.\(13\)

Consequently, we do not use the noncocommutative coordinate Hopf algebrasBCKandMKWdirectly for the primitive log\-ODE construction\. Instead, we use their cocommutative graded duals, namelyGLand\(MKW\)∘\(\\lx@glossaries@gls@link\{acronym\}\{mkw\}\{\{\{\}\}MKW\}\)^\{\\circ\}\.

### A\.3Words and Stratonovich calculus

#### A\.3\.1Words and Lyndon words

Throughout this subsection, letAAbe a finite alphabet, typicallyA=\{1,…,d\}A=\\\{1,\\ldots,d\\\}, with a fixed total order\. A*word*is a finite sequencew=i1​⋯​inw=i\_\{1\}\\cdots i\_\{n\}withik∈Ai\_\{k\}\\in A, including the empty word11whenn=0n=0\. Its length is\|w\|=n\|w\|=n, concatenation is denoted by juxtaposition, and the order onAAinduces the lexicographic order on words\.

###### Definition A\.2\(Lyndon word\)\.

A nonempty wordℓ\\ellis a*Lyndon word*if it is strictly smaller, in lexicographic order, than each of its nontrivial proper suffixes\.

###### Example A\.3\(Lyndon bracketing\)\.

LetA=\{a,b,c,d\}A=\\\{a,b,c,d\\\}witha<b<c<da<b<c<d\. The wordℓ=abac\\ell=\\texttt\{abac\}is Lyndon, since

abac<bac,abac<ac,abac<c\.\\texttt\{abac\}<\\texttt\{bac\},\\qquad\\texttt\{abac\}<\\texttt\{ac\},\\qquad\\texttt\{abac\}<\\texttt\{c\}\.Its standard factorisation isℓ=u​v\\ell=uv, withu=abu=\\texttt\{ab\}andv=acv=\\texttt\{ac\}\. Hence

\[ℓ\]=\[\[u\],\[v\]\]=\[\[ea,eb\],\[ea,ec\]\]\.\[\\ell\]=\[\[u\],\[v\]\]=\[\[e\_\{a\},e\_\{b\}\],\[e\_\{a\},e\_\{c\}\]\]\.Asℓ\\ellranges over Lyndon words, these bracketings form the Lyndon basis of the free Lie algebraPrim⁡\(ℋ\)\\operatorname\{Prim\}\(\\operatorname\{\\mathcal\{H\}\}\)\.

#### A\.3\.2Stratonovich rough paths onℝn\\mathbb\{R\}^\{n\}: the tensor Hopf algebra

We view the geometric signature as a grouplike element of the cocommutative graded dual of the shuffle Hopf algebra,

ℋ∘≔\(T​\(A\),⋆,Δunsh\),\\mathcal\{H\}^\{\\circ\}\\coloneqq\\bigl\(T\(A\),\\,\\star,\\,\\Delta\_\{\\mathrm\{unsh\}\}\\bigr\),where⋆\\staris concatenation andΔunsh\\Delta\_\{\\mathrm\{unsh\}\}is the unshuffle coproduct\. This is the graded dual of the usual shuffle Hopf algebra\(T\(A\),,Δdec\)\(T\(A\),\\shuffle,\\Delta\_\{\\mathrm\{dec\}\}\)\. For wordsu=i1​⋯​imu=i\_\{1\}\\cdots i\_\{m\}andv=j1​⋯​jnv=j\_\{1\}\\cdots j\_\{n\},

u⋆v≔i1​⋯​im​j1​⋯​jn\.u\\star v\\coloneqq i\_\{1\}\\cdots i\_\{m\}j\_\{1\}\\cdots j\_\{n\}\.The unshuffle coproduct is

Δunsh​\(i1​⋯​in\)≔∑I⊆\[n\]iI⊗iIc,iI:=ik1​⋯​ik\|I\|for​I=\{k1<⋯<k\|I\|\}\.\\Delta\_\{\\mathrm\{unsh\}\}\(i\_\{1\}\\cdots i\_\{n\}\)\\coloneqq\\sum\_\{I\\subseteq\[n\]\}i\_\{I\}\\otimes i\_\{I^\{c\}\},\\qquad i\_\{I\}:=i\_\{k\_\{1\}\}\\cdots i\_\{k\_\{\|I\|\}\}\\quad\\text\{for \}I=\\\{k\_\{1\}<\\cdots<k\_\{\|I\|\}\\\}\.It is cocommutative, since swapping tensor factors corresponds toI↔IcI\\leftrightarrow I^\{c\}\. The primitivesPrim⁡\(ℋ∘\)\\operatorname\{Prim\}\(\\mathcal\{H\}^\{\\circ\}\)form the free Lie algebra onAA, with bracket induced by the commutator of⋆\\star\. Therefore the Hopf logarithm of any geometric signature increment is a Lie element, and may be expanded in the Lyndon basis\.

###### Example A\.4\(Smooth tensor signature\)\.

LetXta=tX\_\{t\}^\{a\}=tandXtb=t2X\_\{t\}^\{b\}=t^\{2\}on\[0,1\]\[0,1\]\. Then

⟨S​\(X\),a⟩=∫01𝑑Xta=1,⟨S​\(X\),b⟩=∫01𝑑Xtb=1,\\langle S\(X\),a\\rangle=\\int\_\{0\}^\{1\}dX\_\{t\}^\{a\}=1,\\qquad\\langle S\(X\),b\\rangle=\\int\_\{0\}^\{1\}dX\_\{t\}^\{b\}=1,and

⟨S​\(X\),a​b⟩=∫0<u<v<1𝑑Xua​𝑑Xvb=∫01v​2​v​𝑑v=23,\\langle S\(X\),ab\\rangle=\\int\_\{0<u<v<1\}dX\_\{u\}^\{a\}\\,dX\_\{v\}^\{b\}=\\int\_\{0\}^\{1\}v\\,2v\\,dv=\\frac\{2\}\{3\},while

⟨S​\(X\),b​a⟩=∫0<u<v<1𝑑Xub​𝑑Xva=∫01v2​𝑑v=13\.\\langle S\(X\),ba\\rangle=\\int\_\{0<u<v<1\}dX\_\{u\}^\{b\}\\,dX\_\{v\}^\{a\}=\\int\_\{0\}^\{1\}v^\{2\}\\,dv=\\frac\{1\}\{3\}\.Thus

⟨S​\(X\),a​b\+b​a⟩=⟨S​\(X\),a⟩​⟨S​\(X\),b⟩,\\langle S\(X\),ab\+ba\\rangle=\\langle S\(X\),a\\rangle\\langle S\(X\),b\\rangle,as required by the shuffle character relation\. WithVa=∂yV\_\{a\}=\\partial\_\{y\}andVb=y​∂yV\_\{b\}=y\\partial\_\{y\}, and with the convention thata​babacts byVa​VbV\_\{a\}V\_\{b\},

Va​Vb​φ=φ′\+y​φ′′,Vb​Va​φ=y​φ′′,V\_\{a\}V\_\{b\}\\varphi=\\varphi^\{\\prime\}\+y\\varphi^\{\\prime\\prime\},\\qquad V\_\{b\}V\_\{a\}\\varphi=y\\varphi^\{\\prime\\prime\},so the two tensor coordinates act on distinct differential operators even for this one\-dimensional example\.

### A\.4Rooted trees and Itô calculus

#### A\.4\.1Non\-planar and planar rooted trees

###### Definition A\.5\(Rooted tree\)\.

A rooted tree is a finite directed acyclic graph with a distinguished root and optional decoration by elements of an alphabetAA\. The tree with root decorated byiiand childrenτ1,…,τn\\tau\_\{1\},\\ldots,\\tau\_\{n\}is denoted

\[τ1,…,τn\]i=iτ1…τn\.\[\\tau\_\{1\},\\dots,\\tau\_\{n\}\]\_\{i\}=\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to33\.23pt\{\\vbox to18\.58pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\} \{\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@lineto\{\-11\.62495pt\}\{10\.76385pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\{\}\{\}\{\}\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@lineto\{3\.87498pt\}\{6\.45831pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \{\{\}\{\}\}\{\{\}\}\{\}\{\{\}\}\{\}\\pgfsys@moveto\{4\.26773pt\}\{10\.76385pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.51773pt\}\{8\.26385pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{$\\dots$\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{\}\{\{\}\}\{\} \{\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@lineto\{11\.62495pt\}\{10\.76385pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{13\.12495pt\}\{10\.76385pt\}\\pgfsys@curveto\{13\.12495pt\}\{11\.59229pt\}\{12\.45338pt\}\{12\.26385pt\}\{11\.62495pt\}\{12\.26385pt\}\\pgfsys@curveto\{10\.79652pt\}\{12\.26385pt\}\{10\.12495pt\}\{11\.59229pt\}\{10\.12495pt\}\{10\.76385pt\}\\pgfsys@curveto\{10\.12495pt\}\{9\.93542pt\}\{10\.79652pt\}\{9\.26385pt\}\{11\.62495pt\}\{9\.26385pt\}\\pgfsys@curveto\{12\.45338pt\}\{9\.26385pt\}\{13\.12495pt\}\{9\.93542pt\}\{13\.12495pt\}\{10\.76385pt\}\\pgfsys@closepath\\pgfsys@moveto\{11\.62495pt\}\{10\.76385pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{11\.62495pt\}\{10\.76385pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{13\.50871pt\}\{10\.18747pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny$\\tau\_\{n\}$\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\.The order\|τ\|\|\\tau\|is the number of vertices, and a forest is a finite product of rooted trees\.

We distinguish*non\-planar*rooted trees, whose children are unordered, from*planar*rooted trees, whose children are ordered from left to right\. Thus, in the planar setting,

ijk≠ikj,\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.28pt\{\\vbox to13\.07pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.46155pt\}\{\-1\.6994pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.9462pt\}\{6\.87498pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny k\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\neq\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.28pt\{\\vbox to13\.07pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ 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\{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ 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\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.29642pt\}\{7\.39778pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\},whereas these two trees agree in the non\-planar setting\.

#### A\.4\.2Pre\-Lie grafting and Euclidean elementary differentials

The Euclidean branched theory is governed by a pre\-Lie product\. A pre\-Lie algebra is a vector space with a bilinear product⊳\\trianglerightsatisfying

x⊳\(y⊳z\)−\(x⊳y\)⊳z=y⊳\(x⊳z\)−\(y⊳x\)⊳z\.x\\triangleright\(y\\triangleright z\)\-\(x\\triangleright y\)\\triangleright z=y\\triangleright\(x\\triangleright z\)\-\(y\\triangleright x\)\\triangleright z\.For vector fields onℝn\\mathbb\{R\}^\{n\}, this product is

U⊳V≔D​V​\[U\]\.U\\triangleright V\\coloneqq DV\[U\]\.The free pre\-Lie algebra is represented by non\-planar rooted trees, with product given by grafting\. Forτ,σ∈𝒰​𝒯\\tau,\\sigma\\in\\mathcal\{UT\},

τ↷σ≔∑v∈V​\(σ\)τ↷vσ,\\tau\\curvearrowright\\sigma\\coloneqq\\sum\_\{v\\in V\(\\sigma\)\}\\tau\\curvearrowright\_\{v\}\\sigma,whereτ↷vσ\\tau\\curvearrowright\_\{v\}\\sigmaattaches the root ofτ\\tauto the vertexvvofσ\\sigma\. The Euclidean elementary differential map satisfies

F​\(i\)=Vi,F​\(\[τ1,…,τk\]i\)=Dk​Vi​\(F​\(τ1\),…,F​\(τk\)\),F\(\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to7\.14pt\{\\vbox to3\.94pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 0\.87694pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.46155pt\}\{1\.14604pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{i\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\)=V\_\{i\},\\qquad F\(\[\\tau\_\{1\},\\ldots,\\tau\_\{k\}\]\_\{i\}\)=D^\{k\}V\_\{i\}\\bigl\(F\(\\tau\_\{1\}\),\\ldots,F\(\\tau\_\{k\}\)\\bigr\),and the grafting identity

F​\(τ↷σ\)=F​\(τ\)⊳F​\(σ\)\.F\(\\tau\\curvearrowright\\sigma\)=F\(\\tau\)\\triangleright F\(\\sigma\)\.

#### A\.4\.3Branched rough paths inℝn\\mathbb\{R\}^\{n\}:GLas the cocommutative dual ofBCK

Itô integration introduces quadratic variation corrections, and word\-indexed shuffle coordinates do not close under the Itô chain rule\. Branched rough paths restore closure by indexing the expansion by rooted trees\(Hairer and Kelly,[2014](https://arxiv.org/html/2606.05272#bib.bib16)\)\. The coordinate Hopf algebra is traditionally the noncocommutativeBCKHopf algebra, while the primitive log\-ODE construction uses its cocommutative graded dual, theGLHopf algebra\(Grossman and Larson,[1989](https://arxiv.org/html/2606.05272#bib.bib24)\)\.

TheGLproduct is the Guin–Oudom product extending pre\-Lie grafting\. IfΔ⊔⊔​α=∑α\(1\)⊗α\(2\)\\Delta\_\{\\sqcup\\sqcup\}\\alpha=\\sum\\alpha\_\{\(1\)\}\\otimes\\alpha\_\{\(2\)\}denotes the cocommutative unshuffle coproduct on forests, then

α⋆GLβ=∑α\(1\)​\(α\(2\)↷β\),\\alpha\\star\_\{\\mathrm\{GL\}\}\\beta=\\sum\\alpha\_\{\(1\)\}\\bigl\(\\alpha\_\{\(2\)\}\\curvearrowright\\beta\\bigr\),with the standard extension of↷\\curvearrowrightfrom trees to forests\(Oudom and Guin,[2004](https://arxiv.org/html/2606.05272#bib.bib64)\)\. In particular, for treesτ,σ∈𝒰​𝒯\\tau,\\sigma\\in\\mathcal\{UT\},

τ⋆GLσ=τ​σ\+τ↷σ\.\\tau\\star\_\{\\mathrm\{GL\}\}\\sigma=\\tau\\sigma\+\\tau\\curvearrowright\\sigma\.For example,

m↷ij=ijm\+ijm\.\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to9\.08pt\{\\vbox to5\.88pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower\-0\.09254pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{1\.69489pt\}\{1\.76906pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{m\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\curvearrowright\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to17\.9pt\{\\vbox to13\.02pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-21\.52771pt\}\{17\.22217pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-19\.83282pt\}\{16\.14578pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-11\.93515pt\}\{9\.54813pt\}\\pgfsys@lineto\{\-20\.35641pt\}\{16\.28513pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\+\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.28pt\{\\vbox to13\.52pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.46155pt\}\{\-1\.6994pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.29642pt\}\{7\.39778pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.4509pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\.The Hopf algebra

ℋGL=\(S​\(𝒰​𝒯\),⋆GL,Δ⊔⊔\)\\mathcal\{H\}\_\{\\mathrm\{GL\}\}=\\bigl\(S\(\\mathcal\{UT\}\),\\star\_\{\\mathrm\{GL\}\},\\Delta\_\{\\sqcup\\sqcup\}\\bigr\)is connected, graded, and cocommutative\. Hence[TheoremA\.1](https://arxiv.org/html/2606.05272#A1.Thmtheorem1)identifies it with the enveloping Hopf algebra of its primitive Lie algebra\.

###### Example A\.6\(Smooth branched signature\)\.

LetXta=tX\_\{t\}^\{a\}=tandXtb=t2X\_\{t\}^\{b\}=t^\{2\}on\[0,1\]\[0,1\]\. The canonical smooth branched lift is defined recursively by

⟨𝐗0,t,i⟩=Xti−X0i,⟨𝐗0,t,\[τ1,…,τk\]i⟩=∫0t∏r=1k⟨𝐗0,s,τr⟩​d​Xsi\.\\langle\\mathbf\{X\}\_\{0,t\},\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to6\.82pt\{\\vbox to3\.62pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 1\.03758pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{1\.20705pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{i\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\rangle=X\_\{t\}^\{i\}\-X\_\{0\}^\{i\},\\qquad\\langle\\mathbf\{X\}\_\{0,t\},\[\\tau\_\{1\},\\ldots,\\tau\_\{k\}\]\_\{i\}\\rangle=\\int\_\{0\}^\{t\}\\prod\_\{r=1\}^\{k\}\\langle\\mathbf\{X\}\_\{0,s\},\\tau\_\{r\}\\rangle\\,dX\_\{s\}^\{i\}\.Thus

⟨𝐗,ba⟩=∫01s​d​\(s2\)=23,⟨𝐗,ab⟩=∫01s2​𝑑s=13,\\langle\\mathbf\{X\},\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to18\.12pt\{\\vbox to12\.39pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-2\.07666pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.42668pt\}\{\-1\.7361pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny b\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-9\.26385pt\}\{9\.43951pt\}\{\-9\.93542pt\}\{10\.11108pt\}\{\-10\.76385pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-11\.59229pt\}\{10\.11108pt\}\{\-12\.26385pt\}\{9\.43951pt\}\{\-12\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-12\.26385pt\}\{7\.78265pt\}\{\-11\.59229pt\}\{7\.11108pt\}\{\-10\.76385pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-9\.93542pt\}\{7\.11108pt\}\{\-9\.26385pt\}\{7\.78265pt\}\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-8\.87292pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny a\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\rangle=\\int\_\{0\}^\{1\}s\\,d\(s^\{2\}\)=\\frac\{2\}\{3\},\\qquad\\langle\\mathbf\{X\},\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to17\.3pt\{\\vbox to12\.39pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.7pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{1\.89093pt\}\{\-1\.07639pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny a\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-9\.26385pt\}\{9\.43951pt\}\{\-9\.93542pt\}\{10\.11108pt\}\{\-10\.76385pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-11\.59229pt\}\{10\.11108pt\}\{\-12\.26385pt\}\{9\.43951pt\}\{\-12\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-12\.26385pt\}\{7\.78265pt\}\{\-11\.59229pt\}\{7\.11108pt\}\{\-10\.76385pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-9\.93542pt\}\{7\.11108pt\}\{\-9\.26385pt\}\{7\.78265pt\}\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-8\.33717pt\}\{6\.87498pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny b\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\rangle=\\int\_\{0\}^\{1\}s^\{2\}\\,ds=\\frac\{1\}\{3\},and, since the non\-planar forest product is commutative,

⟨𝐗,bab⟩=∫01s​s2​d​\(s2\)=25\.\\langle\\mathbf\{X\},\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.5pt\{\\vbox to12\.76pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-2\.07666pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.42668pt\}\{\-1\.7361pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny b\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.25485pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny a\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.7191pt\}\{6\.87498pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny b\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\rangle=\\int\_\{0\}^\{1\}s\\,s^\{2\}\\,d\(s^\{2\}\)=\\frac\{2\}\{5\}\.WithVa=∂yV\_\{a\}=\\partial\_\{y\}andVb=y2​∂yV\_\{b\}=y^\{2\}\\partial\_\{y\}onℝ\\mathbb\{R\},

F​\(ba\)=D​Vb​\[Va\]=2​y​∂y,F​\(bab\)=D2​Vb​\[Va,Vb\]=2​y2​∂y\.F\\left\(\\,\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to18\.12pt\{\\vbox to12\.39pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-2\.07666pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.42668pt\}\{\-1\.7361pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny b\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-9\.26385pt\}\{9\.43951pt\}\{\-9\.93542pt\}\{10\.11108pt\}\{\-10\.76385pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-11\.59229pt\}\{10\.11108pt\}\{\-12\.26385pt\}\{9\.43951pt\}\{\-12\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-12\.26385pt\}\{7\.78265pt\}\{\-11\.59229pt\}\{7\.11108pt\}\{\-10\.76385pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-9\.93542pt\}\{7\.11108pt\}\{\-9\.26385pt\}\{7\.78265pt\}\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-8\.87292pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny a\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\right\)=DV\_\{b\}\[V\_\{a\}\]=2y\\partial\_\{y\},\\qquad F\\left\(\\,\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.5pt\{\\vbox to12\.76pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-2\.07666pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.42668pt\}\{\-1\.7361pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny b\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.25485pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny a\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.7191pt\}\{6\.87498pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny b\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\right\)=D^\{2\}V\_\{b\}\[V\_\{a\},V\_\{b\}\]=2y^\{2\}\\partial\_\{y\}\.The same rooted\-tree coordinates therefore encode both the Itô\-type branched signature and the Euclidean elementary differentials appearing in the expansion\.

#### A\.4\.4Post\-Lie grafting and manifold elementary differentials

On manifolds, the composition of differential operators is not represented by the Euclidean pre\-Lie product alone\. For a smooth connection∇\\nabla, the MKW elementary differentials used below are defined through total covariant derivatives\. When∇\\nablahas zero curvature and parallel torsionTT, this construction admits the post\-Lie interpretation

U⊳V≔∇UV,\[U,V\]∇≔−T​\(U,V\)\.U\\triangleright V\\coloneqq\\nabla\_\{U\}V,\\qquad\[U,V\]\_\{\\nabla\}\\coloneqq\-T\(U,V\)\.The induced Lie bracket is

⟦U,V⟧=U⊳V−V⊳U\+\[U,V\]∇,\\llbracket U,V\\rrbracket=U\\triangleright V\-V\\triangleright U\+\[U,V\]\_\{\\nabla\},which agrees with the Jacobi bracket of vector fields\. Under the same flat/parallel\-torsion assumptions, the post\-Lie identities are

x⊳\[y,z\]∇=\[x⊳y,z\]∇\+\[y,x⊳z\]∇,x\\triangleright\[y,z\]\_\{\\nabla\}=\[x\\triangleright y,z\]\_\{\\nabla\}\+\[y,x\\triangleright z\]\_\{\\nabla\},and

\[x,y\]∇⊳z=x⊳\(y⊳z\)−\(x⊳y\)⊳z−y⊳\(x⊳z\)\+\(y⊳x\)⊳z\.\[x,y\]\_\{\\nabla\}\\triangleright z=x\\triangleright\(y\\triangleright z\)\-\(x\\triangleright y\)\\triangleright z\-y\\triangleright\(x\\triangleright z\)\+\(y\\triangleright x\)\\triangleright z\.Planar rooted trees give the free combinatorial model of the resulting post\-Lie calculus\. The relevant grafting operation is left grafting:

τ↷ℓσ≔∑v∈V​\(σ\)τ↷ℓ,vσ,\\tau\\curvearrowright\_\{\\ell\}\\sigma\\coloneqq\\sum\_\{v\\in V\(\\sigma\)\}\\tau\\curvearrowright\_\{\\ell,v\}\\sigma,whereτ↷ℓ,vσ\\tau\\curvearrowright\_\{\\ell,v\}\\sigmaattaches the root ofτ\\tautovvas the left\-most child\. This convention records the order of covariant differentiations\. At first order,

F​\(ij\)=Vj⊳Vi=∇VjVi\.F\\left\(\\,\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to17\.9pt\{\\vbox to13\.02pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.46155pt\}\{\-1\.6994pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-9\.26385pt\}\{9\.43951pt\}\{\-9\.93542pt\}\{10\.11108pt\}\{\-10\.76385pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-11\.59229pt\}\{10\.11108pt\}\{\-12\.26385pt\}\{9\.43951pt\}\{\-12\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-12\.26385pt\}\{7\.78265pt\}\{\-11\.59229pt\}\{7\.11108pt\}\{\-10\.76385pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-9\.93542pt\}\{7\.11108pt\}\{\-9\.26385pt\}\{7\.78265pt\}\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-7\.91449pt\}\{7\.39778pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\right\)=V\_\{j\}\\triangleright V\_\{i\}=\\nabla\_\{V\_\{j\}\}V\_\{i\}\.For the manifold log\-ODE construction, the essential algebraic property is the pseudo\-bialgebra identity

\#​\(α⋆MKWβ\)=\#​\(α\)∘\#​\(β\),\\\#\(\\alpha\\star\_\{\\mathrm\{MKW\}\}\\beta\)=\\\#\(\\alpha\)\\circ\\\#\(\\beta\),where\#\\\#is defined on ordered forests by total covariant derivatives\. This identity holds for the MKW Hopf algebra with any smooth connection; the flat/parallel\-torsion assumption is only needed when one wants to interpret the primitive structure as a post\-Lie morphism\.

###### Example A\.7\(Second\-order chain rule\)\.

For one\-node trees,

j⋆MKWi=j​i\+ij\.\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to7\.71pt\{\\vbox to4\.51pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 0\.58885pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.98993pt\}\{1\.69315pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{j\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\star\_\{\\mathrm\{MKW\}\}\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to6\.82pt\{\\vbox to3\.62pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 1\.03758pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{1\.20705pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{i\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}=\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to7\.71pt\{\\vbox to4\.51pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 0\.58885pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.98993pt\}\{1\.69315pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{j\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to6\.82pt\{\\vbox to3\.62pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 1\.03758pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{1\.20705pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{i\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\+\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to17\.58pt\{\\vbox to12\.68pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{\-1\.6384pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-7\.77393pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\.Applying\#\\\#to a test functionφ∈C∞​\(ℳ\)\\varphi\\in C^\{\\infty\}\(\\mathcal\{M\}\)gives

\#​\(j⋆MKWi\)​φ=∇2φ​\(Vj,Vi\)\+\(∇VjVi\)​φ\.\\\#\(\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to7\.71pt\{\\vbox to4\.51pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 0\.58885pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.98993pt\}\{1\.69315pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{j\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\star\_\{\\mathrm\{MKW\}\}\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to6\.82pt\{\\vbox to3\.62pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 1\.03758pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{1\.20705pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{i\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\)\\varphi=\\nabla^\{2\}\\varphi\(V\_\{j\},V\_\{i\}\)\+\(\\nabla\_\{V\_\{j\}\}V\_\{i\}\)\\varphi\.By the covariant chain rule,

∇2φ​\(Vj,Vi\)\+\(∇VjVi\)​φ=Vj​\(Vi​φ\)\.\\nabla^\{2\}\\varphi\(V\_\{j\},V\_\{i\}\)\+\(\\nabla\_\{V\_\{j\}\}V\_\{i\}\)\\varphi=V\_\{j\}\(V\_\{i\}\\varphi\)\.Thus the grafting termij\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to17\.58pt\{\\vbox to12\.68pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{\-1\.6384pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-9\.26385pt\}\{9\.43951pt\}\{\-9\.93542pt\}\{10\.11108pt\}\{\-10\.76385pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-11\.59229pt\}\{10\.11108pt\}\{\-12\.26385pt\}\{9\.43951pt\}\{\-12\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-12\.26385pt\}\{7\.78265pt\}\{\-11\.59229pt\}\{7\.11108pt\}\{\-10\.76385pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-9\.93542pt\}\{7\.11108pt\}\{\-9\.26385pt\}\{7\.78265pt\}\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-7\.77393pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}is exactly the correction needed for⋆MKW\\star\_\{\\mathrm\{MKW\}\}to represent composition of differential operators\.

#### A\.4\.5Branched rough paths onℳ\\mathcal\{M\}: the Munthe–Kaas–Wright Hopf algebra

LetAAdenote the alphabet of driver labels, and letOF⁡\(A\)\\operatorname\{OF\}\(A\)be the vector space spanned byAA\-decorated ordered forests\. We writeτ​σ\\tau\\sigmafor ordered concatenation of forests and

\[τ1,…,τk\]i\[\\tau\_\{1\},\\ldots,\\tau\_\{k\}\]\_\{i\}for the planar tree with root labeliiand ordered childrenτ1,…,τk\\tau\_\{1\},\\ldots,\\tau\_\{k\}\. Thusijk≠ikj\.\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.28pt\{\\vbox to13\.07pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.46155pt\}\{\-1\.6994pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.29642pt\}\{7\.39778pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ 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\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\neq\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.28pt\{\\vbox to13\.07pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} 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\}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\.

LetℋMKW\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}denote the MKW coordinate Hopf algebra of ordered forests\. Its product is the shuffle product of ordered forests, and its coproduct is the full left\-admissible\-cut coproduct\. Branched signatures on manifolds are characters ofℋMKW\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\. For the log\-ODE construction, we use the graded dual

ℋMKW≔ℋMKW∘=⨁n≥0ℋMKW,n∗\.\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\\coloneqq\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}^\{\\circ\}=\\bigoplus\_\{n\\geq 0\}\\mathcal\{H\}\_\{\\mathrm\{MKW\},n\}^\{\*\}\.In this dual Hopf algebra, the coproduct is the cocommutative deshuffle coproduct on words of ordered forests, i\.e\. the sum over all order\-preserving splittings into two subwords\. SinceℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}is connected, graded, and cocommutative over a field of characteristic zero,[TheoremA\.1](https://arxiv.org/html/2606.05272#A1.Thmtheorem1)gives

ℋMKW≅U​\(Prim⁡\(ℋMKW\)\)\.\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\\cong U\\bigl\(\\operatorname\{Prim\}\(\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\)\\bigr\)\.The primitives form a Lie algebra under the commutator of⋆MKW\\star\_\{\\mathrm\{MKW\}\}\. Equipped with the left\-grafting operation, this primitive structure is the free post\-Lie algebra onAA, which is the primitive algebra underlying the manifold log\-ODE construction ofKern and Lyons \([2023](https://arxiv.org/html/2606.05272#bib.bib6)\)\.

The product onℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}is the Guin–Oudom product associated with left grafting\. If

Δunsh​α=∑α\(1\)⊗α\(2\)\\Delta\_\{\\mathrm\{unsh\}\}\\alpha=\\sum\\alpha\_\{\(1\)\}\\otimes\\alpha\_\{\(2\)\}denotes the unshuffle coproduct, then

α⋆MKWβ=∑α\(1\)​\(α\(2\)↷ℓβ\),\\alpha\\star\_\{\\mathrm\{MKW\}\}\\beta=\\sum\\alpha\_\{\(1\)\}\\bigl\(\\alpha\_\{\(2\)\}\\curvearrowright\_\{\\ell\}\\beta\\bigr\),with the standard post\-Lie extension of↷ℓ\\curvearrowright\_\{\\ell\}from trees to ordered forests\. In particular, for single treesτ,σ\\tau,\\sigma,

τ⋆MKWσ=τ​σ\+τ↷ℓσ\.\\tau\\star\_\{\\mathrm\{MKW\}\}\\sigma=\\tau\\sigma\+\\tau\\curvearrowright\_\{\\ell\}\\sigma\.For example,

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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.46155pt\}\{\-1\.6994pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-9\.26385pt\}\{9\.43951pt\}\{\-9\.93542pt\}\{10\.11108pt\}\{\-10\.76385pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-11\.59229pt\}\{10\.11108pt\}\{\-12\.26385pt\}\{9\.43951pt\}\{\-12\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-12\.26385pt\}\{7\.78265pt\}\{\-11\.59229pt\}\{7\.11108pt\}\{\-10\.76385pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-9\.93542pt\}\{7\.11108pt\}\{\-9\.26385pt\}\{7\.78265pt\}\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-7\.91449pt\}\{7\.39778pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}, where the new child is inserted as the left\-most child\. Hence the planar orderimj\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.28pt\{\\vbox to13\.52pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.46155pt\}\{\-1\.6994pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.4509pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.29642pt\}\{7\.39778pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}is distinguished fromijm\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.28pt\{\\vbox to13\.52pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.9685pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ 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\}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.46155pt\}\{\-1\.6994pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ 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\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.4509pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\.

###### Example A\.8\(Smooth planarly branched signature\)\.

LetXta=tX\_\{t\}^\{a\}=tandXtb=t2X\_\{t\}^\{b\}=t^\{2\}on\[0,1\]\[0,1\]\. For the smooth, ordered lift,

⟨𝐗0,t,a​b⟩=∫0<u<v<t𝑑Xua​𝑑Xvb=23​t3,\\langle\\mathbf\{X\}\_\{0,t\},\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to6\.54pt\{\\vbox to3\.4pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 1\.14545pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{1\.89093pt\}\{1\.76906pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{a\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to7\.35pt\{\\vbox to4\.15pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 0\.76878pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.42668pt\}\{1\.10934pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{b\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\rangle=\\int\_\{0<u<v<t\}dX\_\{u\}^\{a\}\\,dX\_\{v\}^\{b\}=\\frac\{2\}\{3\}t^\{3\},whereas

⟨𝐗0,t,b​a⟩=∫0<u<v<t𝑑Xub​𝑑Xva=13​t3\.\\langle\\mathbf\{X\}\_\{0,t\},\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to7\.35pt\{\\vbox to4\.15pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 0\.76878pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.42668pt\}\{1\.10934pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{b\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to6\.54pt\{\\vbox to3\.4pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 1\.14545pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{1\.89093pt\}\{1\.76906pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{a\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\rangle=\\int\_\{0<u<v<t\}dX\_\{u\}^\{b\}\\,dX\_\{v\}^\{a\}=\\frac\{1\}\{3\}t^\{3\}\.Consequently,

⟨𝐗,bab⟩=∫01⟨𝐗0,t,a​b⟩​𝑑Xtb=415,\\langle\\mathbf\{X\},\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to23\.5pt\{\\vbox to12\.76pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-2\.07666pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ 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\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\rangle=\\int\_\{0\}^\{1\}\\langle\\mathbf\{X\}\_\{0,t\},\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to7\.35pt\{\\vbox to4\.15pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 0\.76878pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.42668pt\}\{1\.10934pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{b\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to6\.54pt\{\\vbox to3\.4pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 1\.14545pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{1\.5pt\}\{3\.67387pt\}\{0\.82843pt\}\{4\.34544pt\}\{0\.0pt\}\{4\.34544pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{4\.34544pt\}\{\-1\.5pt\}\{3\.67387pt\}\{\-1\.5pt\}\{2\.84544pt\}\\pgfsys@curveto\{\-1\.5pt\}\{2\.01701pt\}\{\-0\.82843pt\}\{1\.34544pt\}\{0\.0pt\}\{1\.34544pt\}\\pgfsys@curveto\{0\.82843pt\}\{1\.34544pt\}\{1\.5pt\}\{2\.01701pt\}\{1\.5pt\}\{2\.84544pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{2\.84544pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{1\.89093pt\}\{1\.76906pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{a\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\rangle\\,dX\_\{t\}^\{b\}=\\frac\{2\}\{15\}\.Thus the ordered trees are distinguished by the signature\. Their sum recovers the corresponding unordered coefficient2/52/5\.

###### Example A\.9\(Failure of ordinary planar cuts\)\.

A naive extension of BCK to planar trees, using ordinary admissible cuts, does not represent covariant elementary differentiation\. It is enough to see this at a pointy∈ℳy\\in\\mathcal\{M\}\. Choose local vector fieldsVi,Vj,VmV\_\{i\},V\_\{j\},V\_\{m\}, withVj​\(y\)≠0V\_\{j\}\(y\)\\neq 0, whose first covariant derivatives atyysatisfy

∇VjVi=Vj,∇VmVj=0,∇VmVi=0,∇VjVm=Vj,\\nabla\_\{V\_\{j\}\}V\_\{i\}=V\_\{j\},\\qquad\\nabla\_\{V\_\{m\}\}V\_\{j\}=0,\\qquad\\nabla\_\{V\_\{m\}\}V\_\{i\}=0,\\qquad\\nabla\_\{V\_\{j\}\}V\_\{m\}=V\_\{j\},atyy\. All identities below are evaluated at this point\. For MKW left grafting,

m↷ℓij=imj\+ijm\.\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to7\.81pt\{\\vbox to4\.61pt\{\\pgfpicture\\makeatletter\\hbox\{\\thinspace\\lower 0\.54002pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\}\\pgfsys@moveto\{0\.0pt\}\{2\.84544pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{1\.76099pt\}\{1\.76906pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny\{m\}\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\curvearrowright\_\{\\ell\}\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to17\.58pt\{\\vbox to12\.68pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{\-1\.6384pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.3848pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.15585pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\+\\,\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to28\.34pt\{\\vbox to21\.34pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 23\.2277pt\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{\-1\.6384pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-9\.26385pt\}\{9\.43951pt\}\{\-9\.93542pt\}\{10\.11108pt\}\{\-10\.76385pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-11\.59229pt\}\{10\.11108pt\}\{\-12\.26385pt\}\{9\.43951pt\}\{\-12\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-12\.26385pt\}\{7\.78265pt\}\{\-11\.59229pt\}\{7\.11108pt\}\{\-10\.76385pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-9\.93542pt\}\{7\.11108pt\}\{\-9\.26385pt\}\{7\.78265pt\}\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-7\.77393pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\{\{\}\}\}\}\{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-20\.02771pt\}\{17\.22217pt\}\\pgfsys@curveto\{\-20\.02771pt\}\{18\.0506pt\}\{\-20\.69928pt\}\{18\.72217pt\}\{\-21\.52771pt\}\{18\.72217pt\}\\pgfsys@curveto\{\-22\.35614pt\}\{18\.72217pt\}\{\-23\.02771pt\}\{18\.0506pt\}\{\-23\.02771pt\}\{17\.22217pt\}\\pgfsys@curveto\{\-23\.02771pt\}\{16\.39374pt\}\{\-22\.35614pt\}\{15\.72217pt\}\{\-21\.52771pt\}\{15\.72217pt\}\\pgfsys@curveto\{\-20\.69928pt\}\{15\.72217pt\}\{\-20\.02771pt\}\{16\.39374pt\}\{\-20\.02771pt\}\{17\.22217pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-21\.52771pt\}\{17\.22217pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-21\.52771pt\}\{17\.22217pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-19\.76672pt\}\{16\.14578pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-11\.93515pt\}\{9\.54813pt\}\\pgfsys@lineto\{\-20\.35641pt\}\{16\.28513pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\.Under the elementary differential map this gives

F​\(imj\)\+F​\(ijm\)=∇2Vi​\(Vm,Vj\)\+∇∇VmVjVi\.F\\left\(\\,\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to22\.96pt\{\\vbox to12\.72pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{\-1\.6384pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.3848pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.15585pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\right\)\+F\\left\(\\,\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to28\.34pt\{\\vbox to21\.34pt\{\\pgfpicture\\makeatletter\\hbox\{\\hskip 23\.2277pt\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-9\.26385pt\}\{9\.43951pt\}\{\-9\.93542pt\}\{10\.11108pt\}\{\-10\.76385pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-11\.59229pt\}\{10\.11108pt\}\{\-12\.26385pt\}\{9\.43951pt\}\{\-12\.26385pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-12\.26385pt\}\{7\.78265pt\}\{\-11\.59229pt\}\{7\.11108pt\}\{\-10\.76385pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-9\.93542pt\}\{7\.11108pt\}\{\-9\.26385pt\}\{7\.78265pt\}\{\-9\.26385pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-10\.76385pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-7\.77393pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\{\{\}\}\}\}\{\}\{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.1713pt\}\{0\.93704pt\}\\pgfsys@lineto\{\-9\.59256pt\}\{7\.67404pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \}\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-19\.76672pt\}\{16\.14578pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-11\.93515pt\}\{9\.54813pt\}\\pgfsys@lineto\{\-20\.35641pt\}\{16\.28513pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\right\)=\\nabla^\{2\}V\_\{i\}\(V\_\{m\},V\_\{j\}\)\+\\nabla\_\{\\nabla\_\{V\_\{m\}\}V\_\{j\}\}V\_\{i\}\.Using

∇2Vi​\(U,W\)=∇U\(∇WVi\)−∇∇UWVi,\\nabla^\{2\}V\_\{i\}\(U,W\)=\\nabla\_\{U\}\(\\nabla\_\{W\}V\_\{i\}\)\-\\nabla\_\{\\nabla\_\{U\}W\}V\_\{i\},we obtain

∇2Vi​\(Vm,Vj\)\+∇∇VmVjVi=∇Vm\(∇VjVi\)=∇VmVj=0\.\\nabla^\{2\}V\_\{i\}\(V\_\{m\},V\_\{j\}\)\+\\nabla\_\{\\nabla\_\{V\_\{m\}\}V\_\{j\}\}V\_\{i\}=\\nabla\_\{V\_\{m\}\}\(\\nabla\_\{V\_\{j\}\}V\_\{i\}\)=\\nabla\_\{V\_\{m\}\}V\_\{j\}=0\.Thus MKW left grafting gives exactly the covariant derivative of∇VjVi\\nabla\_\{V\_\{j\}\}V\_\{i\}in theVmV\_\{m\}\-direction\.

By contrast, ordinary planar Connes–Kreimer cuts allow the right child ofijm\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to22\.96pt\{\\vbox to12\.72pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{\-1\.6384pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ 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\}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.15585pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.3848pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}to be cut independently, so the graded dual also produces the right\-sibling insertionijm\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to22\.96pt\{\\vbox to12\.72pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{\-1\.6384pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.15585pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.3848pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\. Its elementary differential is

F​\(ijm\)=∇2Vi​\(Vj,Vm\)\.F\\left\(\\,\\mathord\{\{\\vbox\{\\hbox\{ \\hbox to22\.96pt\{\\vbox to12\.72pt\{\\pgfpicture\\makeatletter\\hbox\{\\qquad\\lower\-1\.80786pt\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@setlinewidth\{\\the\\pgflinewidth\}\\pgfsys@invoke\{ \}\\nullfont\\hbox to0\.0pt\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@roundcap\\pgfsys@invoke\{ \} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\{\{\{\{\}\}\}\}\{\}\{\}\\hbox\{\\hbox\{\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{1\.5pt\}\{0\.82843pt\}\{0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\{1\.5pt\}\\pgfsys@curveto\{\-0\.82843pt\}\{1\.5pt\}\{\-1\.5pt\}\{0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\\pgfsys@curveto\{\-1\.5pt\}\{\-0\.82843pt\}\{\-0\.82843pt\}\{\-1\.5pt\}\{0\.0pt\}\{\-1\.5pt\}\\pgfsys@curveto\{0\.82843pt\}\{\-1\.5pt\}\{1\.5pt\}\{\-0\.82843pt\}\{1\.5pt\}\{0\.0pt\}\\pgfsys@closepath\\pgfsys@moveto\{0\.0pt\}\{0\.0pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{0\.0pt\}\{0\.0pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{2\.5412pt\}\{\-1\.6384pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny i\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-13\.15585pt\}\{7\.45879pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny j\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\}\{\{\}\}\{\{\}\} \{\{\}\} \{\{\{\}\{\}\{\{\}\}\{\}\}\}\{\{\{\}\}\}\{\{\{\{\}\}\{\{\}\}\}\}\{\{\}\}\{\{\{ \}\}\}\\hbox\{\\hbox\{\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\{\}\\pgfsys@moveto\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-14\.64578pt\}\{9\.43951pt\}\{\-15\.31735pt\}\{10\.11108pt\}\{\-16\.14578pt\}\{10\.11108pt\}\\pgfsys@curveto\{\-16\.97421pt\}\{10\.11108pt\}\{\-17\.64578pt\}\{9\.43951pt\}\{\-17\.64578pt\}\{8\.61108pt\}\\pgfsys@curveto\{\-17\.64578pt\}\{7\.78265pt\}\{\-16\.97421pt\}\{7\.11108pt\}\{\-16\.14578pt\}\{7\.11108pt\}\\pgfsys@curveto\{\-15\.31735pt\}\{7\.11108pt\}\{\-14\.64578pt\}\{7\.78265pt\}\{\-14\.64578pt\}\{8\.61108pt\}\\pgfsys@closepath\\pgfsys@moveto\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@fillstroke\\pgfsys@invoke\{ \} \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-16\.14578pt\}\{8\.61108pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\}\\hbox\{\{\\pgfsys@beginscope\\pgfsys@invoke\{ \}\{\{\}\{\{\{\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{\}\{ \}\{\{\{\{\}\}\\pgfsys@beginscope\\pgfsys@invoke\{ \}\\pgfsys@transformcm\{1\.0\}\{0\.0\}\{0\.0\}\{1\.0\}\{\-14\.3848pt\}\{7\.5347pt\}\\pgfsys@invoke\{ \}\\hbox\{\{\\definecolor\{pgfstrokecolor\}\{rgb\}\{0,0,0\}\\pgfsys@color@rgb@stroke\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\pgfsys@color@rgb@fill\{0\}\{0\}\{0\}\\pgfsys@invoke\{ \}\\hbox\{\{\\tiny m\}\} \}\}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \{ \{\{\}\}\{\}\{\{\}\} \{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\}\{\{\{\{\{\}\}\{\}\{\}\{\}\{\}\{\{\}\}\}\}\}\{\{\}\}\{\}\{\}\{\}\{\}\\pgfsys@moveto\{\-1\.32353pt\}\{0\.70589pt\}\\pgfsys@lineto\{\-14\.82225pt\}\{7\.9052pt\}\\pgfsys@stroke\\pgfsys@invoke\{ \} \}\\pgfsys@invoke\{ \}\\pgfsys@endscope\}\}\} \\pgfsys@invoke\{ \}\\pgfsys@endscope\{\{ \{\}\{\}\{\}\{\}\{\}\}\}\{\}\{\}\\hss\}\\pgfsys@discardpath\\pgfsys@invoke\{ \}\\pgfsys@endscope\\hss\}\}\\endpgfpicture\}\}\}\}\}\}\\right\)=\\nabla^\{2\}V\_\{i\}\(V\_\{j\},V\_\{m\}\)\.But

∇2Vi​\(Vj,Vm\)=∇Vj\(∇VmVi\)−∇∇VjVmVi=0−∇VjVi=−Vj\.\\nabla^\{2\}V\_\{i\}\(V\_\{j\},V\_\{m\}\)=\\nabla\_\{V\_\{j\}\}\(\\nabla\_\{V\_\{m\}\}V\_\{i\}\)\-\\nabla\_\{\\nabla\_\{V\_\{j\}\}V\_\{m\}\}V\_\{i\}=0\-\\nabla\_\{V\_\{j\}\}V\_\{i\}=\-V\_\{j\}\.Hence the ordinary planar extension contributes a nonzero elementary differential to a composition for which the covariant chain rule gives zero\.

## Appendix BFurther Mathematical Details

### B\.1Vector Field Lift Pseudocode

In this section, we provide pseudocode for the vector field lifts of[Section3\.2](https://arxiv.org/html/2606.05272#S3.SS2)\. For vector fieldsG,U1,…,UmG,U\_\{1\},\\dots,U\_\{m\}, write

TCD∇​\(G;U1,…,Um\):=\(∇mG\)​\(U1,…,Um\),\\mathrm\{TCD\}\_\{\\nabla\}\(G;U\_\{1\},\\dots,U\_\{m\}\):=\(\\nabla^\{m\}G\)\(U\_\{1\},\\dots,U\_\{m\}\),where the total covariant derivative is given recursively by

\(∇0G\)​\(y\)\\displaystyle\(\\nabla^\{0\}G\)\(y\)=G​\(y\),\\displaystyle=G\(y\),\(∇mG\)​\(U1,…,Um\)​\(y\)\\displaystyle\(\\nabla^\{m\}G\)\(U\_\{1\},\\dots,U\_\{m\}\)\(y\)=∇U1\(\(∇m−1G\)​\(U2,…,Um\)\)⁡\(y\)\\displaystyle=\\nabla\_\{U\_\{1\}\}\\big\(\(\\nabla^\{m\-1\}G\)\(U\_\{2\},\\dots,U\_\{m\}\)\\big\)\(y\)−∑r=2m\(∇m−1G\)​\(U2,…,∇U1Ur,…,Um\)​\(y\)\.\\displaystyle\\quad\-\\sum\_\{r=2\}^\{m\}\(\\nabla^\{m\-1\}G\)\(U\_\{2\},\\dots,\\nabla\_\{U\_\{1\}\}U\_\{r\},\\dots,U\_\{m\}\)\(y\)\.
Algorithm 2ℋ\\operatorname\{\\mathcal\{H\}\}Vector Field LiftInput:driver vector fields

\{Wi∈Γ​\(T​ℳ\)\}i=0d−1\\\{W\_\{i\}\\in\\Gamma\(T\\mathcal\{M\}\)\\\}\_\{i=0\}^\{d\-1\}, represented in a chosen frame by coefficient functions

wi:ℳ→ℝnw\_\{i\}:\\mathcal\{M\}\\to\\mathbb\{R\}^\{n\}, Hopf algebra

ℋ=ℋ⁡\(d,N\)\\mathcal\{H\}=\\operatorname\{\\mathcal\{H\}\}\(d,N\)with cached Lyndon basis

BB, geometry

\(ℳ,∇\)\(\\mathcal\{M\},\\nabla\)
Precompute \(once\):Lyndon words up to depth

NN; for each length\-one basis index

kk, store its letter

rkr\_\{k\}; for each non\-letter basis index

kk, store child indices

Ck=\(p,q\)C\_\{k\}=\(p,q\)from the standard factorisation

Initialise empty cache

\{Fk\}k=0\|B\|−1\\\{F\_\{k\}\\\}\_\{k=0\}^\{\|B\|\-1\}
Memoised builder:

Build⁡\(k\)\\operatorname\{Build\}\(k\)returns cached

FkF\_\{k\}if present

Ck←B\.children​\[k\]C\_\{k\}\\leftarrow B\.\\mathrm\{children\}\[k\]

if

Ck=\(\)C\_\{k\}=\(\)then

i←rki\\leftarrow r\_\{k\}

Fk←WiF\_\{k\}\\leftarrow W\_\{i\}

else

\(p,q\)←Ck\(p,q\)\\leftarrow C\_\{k\}

Fp←Build⁡\(p\),Fq←Build⁡\(q\)F\_\{p\}\\leftarrow\\operatorname\{Build\}\(p\),\\;\\;F\_\{q\}\\leftarrow\\operatorname\{Build\}\(q\)

Fk←\[Fp,Fq\]JacF\_\{k\}\\leftarrow\[F\_\{p\},F\_\{q\}\]\_\{\\mathrm\{Jac\}\}

equivalently, if using torsion

TT,

Fk=∇FpFq−∇FqFp−T​\(Fp,Fq\)F\_\{k\}=\\nabla\_\{F\_\{p\}\}F\_\{q\}\-\\nabla\_\{F\_\{q\}\}F\_\{p\}\-T\(F\_\{p\},F\_\{q\}\)
endif

Cache and return

FkF\_\{k\}
for

k=0k=0to

\|B\|−1\|B\|\-1do

Fk←Build⁡\(k\)F\_\{k\}\\leftarrow\\operatorname\{Build\}\(k\)

endfor

Output:lifted fields

\{Fk\}k=0\|B\|−1\\\{F\_\{k\}\\\}\_\{k=0\}^\{\|B\|\-1\}aligned with the truncated Lyndon basis of

ℋ\\operatorname\{\\mathcal\{H\}\}

Algorithm 3ℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\},ℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}Vector Field LiftInput:driver vector fields

\{Wi∈Γ​\(T​ℳ\)\}i=0d−1\\\{W\_\{i\}\\in\\Gamma\(T\\mathcal\{M\}\)\\\}\_\{i=0\}^\{d\-1\}, represented in a chosen frame by coefficient functions

wi:ℳ→ℝnw\_\{i\}:\\mathcal\{M\}\\to\\mathbb\{R\}^\{n\}, Hopf algebra

ℋ∈\{ℋGL⁡\(d,N\),ℋMKW⁡\(d,N\)\}\\mathcal\{H\}\\in\\\{\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}\(d,N\),\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\(d,N\)\\\}with cached rooted\-tree basis

BB, geometry

\(ℳ,∇\)\(\\mathcal\{M\},\\nabla\)
Convention:for

ℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}, use the Euclidean or flat torsion\-free setting so that unordered children give symmetric elementary differentials; for

ℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}, use ordered children and total covariant derivatives for the chosen connection

∇\\nabla
Precompute \(once\):enumerate canonical non\-planar rooted trees for

ℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}or planar rooted trees for

ℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}up to depth

NN; for each basis index

kk, store root colour

rkr\_\{k\}and child indices

CkC\_\{k\}
Initialise empty cache

\{Fk\}k=0\|B\|−1\\\{F\_\{k\}\\\}\_\{k=0\}^\{\|B\|\-1\}
Memoised builder:

Build⁡\(k\)\\operatorname\{Build\}\(k\)returns cached

FkF\_\{k\}if present

Ck←B\.children​\[k\]C\_\{k\}\\leftarrow B\.\\mathrm\{children\}\[k\],

i←rki\\leftarrow r\_\{k\}
if

Ck=\(\)C\_\{k\}=\(\)then

Fk←WiF\_\{k\}\\leftarrow W\_\{i\}

else

\(p1,…,pm\)←Ck\(p\_\{1\},\\dots,p\_\{m\}\)\\leftarrow C\_\{k\}

Fpj←Build⁡\(pj\)F\_\{p\_\{j\}\}\\leftarrow\\operatorname\{Build\}\(p\_\{j\}\)for

j=1,…,mj=1,\\dots,m
if

ℋ=ℋGL\\mathcal\{H\}=\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}then

Fk​\(y\)←Dm​wi​\(y\)​\[Fp1​\(y\),…,Fpm​\(y\)\]F\_\{k\}\(y\)\\leftarrow D^\{m\}w\_\{i\}\(y\)\[F\_\{p\_\{1\}\}\(y\),\\dots,F\_\{p\_\{m\}\}\(y\)\]

else

Fk​\(y\)←TCD∇​\(Wi;Fp1,…,Fpm\)​\(y\)F\_\{k\}\(y\)\\leftarrow\\mathrm\{TCD\}\_\{\\nabla\}\(W\_\{i\};F\_\{p\_\{1\}\},\\dots,F\_\{p\_\{m\}\}\)\(y\)

endif

endif

Cache and return

FkF\_\{k\}
for

k=0k=0to

\|B\|−1\|B\|\-1do

Fk←Build⁡\(k\)F\_\{k\}\\leftarrow\\operatorname\{Build\}\(k\)

endfor

Output:lifted fields

\{Fk\}k=0\|B\|−1\\\{F\_\{k\}\\\}\_\{k=0\}^\{\|B\|\-1\}aligned with the truncated rooted\-tree basis of

ℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}or

ℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}

### B\.2Finite\-grid role of bracket channels

This appendix formalises the finite\-grid distinction between the geometric signature kernel and the proposed branched signature kernel\. The claim is not that geometric signatures have a different infinite\-resolution law\-identifiability target\. Rather, at a fixed observation grid, a geometric kernel computed from path values is a kernel on the projected path law, while our branched kernel is computed on the enhanced law containing quadratic variation and covariation coordinates\.

Letπn=\{0=t0<t1<⋯<tNn=T\}\\pi\_\{n\}=\\\{0=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{N\_\{n\}\}=T\\\}be the observation grid\. DefineΩn:=\(ℝd\)Nn\+1\\Omega\_\{n\}:=\(\\mathbb\{R\}^\{d\}\)^\{N\_\{n\}\+1\},𝒜n:=\(Sym​\(d\)\)Nn\+1\\mathcal\{A\}\_\{n\}:=\(\\mathrm\{Sym\}\(d\)\)^\{N\_\{n\}\+1\}, andΩ¯n:=Ωn×𝒜n\\bar\{\\Omega\}\_\{n\}:=\\Omega\_\{n\}\\times\\mathcal\{A\}\_\{n\}\. We writex¯=\(x,a\)∈Ω¯n\\bar\{x\}=\(x,a\)\\in\\bar\{\\Omega\}\_\{n\}, wherex=\(xt0,…,xtNn\)x=\(x\_\{t\_\{0\}\},\\ldots,x\_\{t\_\{N\_\{n\}\}\}\)are path values anda=\(at0,…,atNn\)a=\(a\_\{t\_\{0\}\},\\ldots,a\_\{t\_\{N\_\{n\}\}\}\)are bracket values\. For a semimartingaleXX, the supplied enhanced observation is\(Xπn,Aπn\)\(X\_\{\\pi\_\{n\}\},A\_\{\\pi\_\{n\}\}\), whereAt:=⟨X⟩tA\_\{t\}:=\\langle X\\rangle\_\{t\}\. LetΠ:Ω¯n→Ωn\\Pi:\\bar\{\\Omega\}\_\{n\}\\to\\Omega\_\{n\}be the projectionΠ​\(x,a\)=x\\Pi\(x,a\)=x\. Throughout,\|⋅\|\|\\cdot\|denotes the Euclidean norm onℝd\\mathbb\{R\}^\{d\}and the Frobenius norm onSym​\(d\)\\mathrm\{Sym\}\(d\)\.

For a kernelk​\(u,v\)=⟨φ​\(u\),φ​\(v\)⟩𝒦k\(u,v\)=\\langle\\varphi\(u\),\\varphi\(v\)\\rangle\_\{\\mathcal\{K\}\}, write

MMDk​\(P,Q\):=‖𝔼U∼P​φ​\(U\)−𝔼V∼Q​φ​\(V\)‖𝒦\.\\mathrm\{MMD\}\_\{k\}\(P,Q\):=\\left\\\|\\mathbb\{E\}\_\{U\\sim P\}\\varphi\(U\)\-\\mathbb\{E\}\_\{V\\sim Q\}\\varphi\(V\)\\right\\\|\_\{\\mathcal\{K\}\}\.In the training objective, we omit the data–data term in the squared MMD since it is constant inθ\\theta\.

###### Proposition B\.1\(Projected geometric kernels and bracket visibility\)\.

LetkgeoNk\_\{\\mathrm\{geo\}\}^\{N\}be a finite\-depth geometric signature kernel computed from the piecewise\-linear interpolation ofx∈Ωnx\\in\\Omega\_\{n\}, with feature mapφgeoN:Ωn→ℋbrN\\varphi\_\{\\mathrm\{geo\}\}^\{N\}:\\Omega\_\{n\}\\to\\mathcal\{H\}\_\{\\mathrm\{br\}\}^\{N\}\. Extend it to enhanced observations by ignoring brackets,k¯geoN​\(\(x,a\),\(y,b\)\):=kgeoN​\(x,y\)\\bar\{k\}\_\{\\mathrm\{geo\}\}^\{N\}\(\(x,a\),\(y,b\)\):=k\_\{\\mathrm\{geo\}\}^\{N\}\(x,y\)\. Then, for any enhanced lawsP,Q∈𝒫​\(Ω¯n\)P,Q\\in\\mathcal\{P\}\(\\bar\{\\Omega\}\_\{n\}\),

MMDk¯geoN​\(P,Q\)=MMDkgeoN​\(Π\#​P,Π\#​Q\)\.\\mathrm\{MMD\}\_\{\\bar\{k\}\_\{\\mathrm\{geo\}\}^\{N\}\}\(P,Q\)=\\mathrm\{MMD\}\_\{k\_\{\\mathrm\{geo\}\}^\{N\}\}\(\\Pi\_\{\\\#\}P,\\Pi\_\{\\\#\}Q\)\.In particular, ifΠ\#​P=Π\#​Q\\Pi\_\{\\\#\}P=\\Pi\_\{\\\#\}Q, thenMMDk¯geoN​\(P,Q\)=0\\mathrm\{MMD\}\_\{\\bar\{k\}\_\{\\mathrm\{geo\}\}^\{N\}\}\(P,Q\)=0\.

LetφbrN:Ω¯n→ℋbrN\\varphi\_\{\\mathrm\{br\}\}^\{N\}:\\bar\{\\Omega\}\_\{n\}\\to\\mathcal\{H\}\_\{\\mathrm\{br\}\}^\{N\}be the truncated branched, or bracket\-augmented, feature map, and letkbrN​\(x¯,y¯\)=⟨φbrN​\(x¯\),φbrN​\(y¯\)⟩ℋbrNk\_\{\\mathrm\{br\}\}^\{N\}\(\\bar\{x\},\\bar\{y\}\)=\\langle\\varphi\_\{\\mathrm\{br\}\}^\{N\}\(\\bar\{x\}\),\\varphi\_\{\\mathrm\{br\}\}^\{N\}\(\\bar\{y\}\)\\rangle\_\{\\mathcal\{H\}\_\{\\mathrm\{br\}\}^\{N\}\}\. Suppose that a scalar bracket statisticb:Ω¯n→ℝb:\\bar\{\\Omega\}\_\{n\}\\to\\mathbb\{R\}appears as a linear coordinate of this feature map: there existsvb∈ℋbrNv\_\{b\}\\in\\mathcal\{H\}\_\{\\mathrm\{br\}\}^\{N\}such thatb​\(x¯\)=⟨φbrN​\(x¯\),vb⟩b\(\\bar\{x\}\)=\\langle\\varphi\_\{\\mathrm\{br\}\}^\{N\}\(\\bar\{x\}\),v\_\{b\}\\ranglefor allx¯∈Ω¯n\\bar\{x\}\\in\\bar\{\\Omega\}\_\{n\}\. Then

\|𝔼P​b−𝔼Q​b\|≤‖vb‖ℋbrN​MMDkbrN​\(P,Q\)\.\\bigl\|\\mathbb\{E\}\_\{P\}b\-\\mathbb\{E\}\_\{Q\}b\\bigr\|\\leq\\\|v\_\{b\}\\\|\_\{\\mathcal\{H\}\_\{\\mathrm\{br\}\}^\{N\}\}\\mathrm\{MMD\}\_\{k\_\{\\mathrm\{br\}\}^\{N\}\}\(P,Q\)\.Consequently, ifΠ\#​P=Π\#​Q\\Pi\_\{\\\#\}P=\\Pi\_\{\\\#\}Qbut𝔼P​b≠𝔼Q​b\\mathbb\{E\}\_\{P\}b\\neq\\mathbb\{E\}\_\{Q\}b, then the path\-only geometric MMD is zero while the branched MMD is strictly positive\.

###### Proof\.

The lifted geometric feature map isφ¯geoN​\(x,a\):=φgeoN​\(x\)=φgeoN​\(Π​\(x,a\)\)\\bar\{\\varphi\}\_\{\\mathrm\{geo\}\}^\{N\}\(x,a\):=\\varphi\_\{\\mathrm\{geo\}\}^\{N\}\(x\)=\\varphi\_\{\\mathrm\{geo\}\}^\{N\}\(\\Pi\(x,a\)\)\. Hence

𝔼X¯∼P​φ¯geoN​\(X¯\)=𝔼X∼Π\#​P​φgeoN​\(X\),\\mathbb\{E\}\_\{\\bar\{X\}\\sim P\}\\bar\{\\varphi\}\_\{\\mathrm\{geo\}\}^\{N\}\(\\bar\{X\}\)=\\mathbb\{E\}\_\{X\\sim\\Pi\_\{\\\#\}P\}\\varphi\_\{\\mathrm\{geo\}\}^\{N\}\(X\),and the same identity holds withQQin place ofPP\. Taking the Hilbert norm of the difference gives

MMDk¯geoN​\(P,Q\)=MMDkgeoN​\(Π\#​P,Π\#​Q\)\.\\mathrm\{MMD\}\_\{\\bar\{k\}\_\{\\mathrm\{geo\}\}^\{N\}\}\(P,Q\)=\\mathrm\{MMD\}\_\{k\_\{\\mathrm\{geo\}\}^\{N\}\}\(\\Pi\_\{\\\#\}P,\\Pi\_\{\\\#\}Q\)\.IfΠ\#​P=Π\#​Q\\Pi\_\{\\\#\}P=\\Pi\_\{\\\#\}Q, the right\-hand side is zero\.

For the branched claim, defineμP:=𝔼X¯∼P​φbrN​\(X¯\)\\mu\_\{P\}:=\\mathbb\{E\}\_\{\\bar\{X\}\\sim P\}\\varphi\_\{\\mathrm\{br\}\}^\{N\}\(\\bar\{X\}\)andμQ:=𝔼Y¯∼Q​φbrN​\(Y¯\)\\mu\_\{Q\}:=\\mathbb\{E\}\_\{\\bar\{Y\}\\sim Q\}\\varphi\_\{\\mathrm\{br\}\}^\{N\}\(\\bar\{Y\}\)\. Sinceb​\(x¯\)=⟨φbrN​\(x¯\),vb⟩b\(\\bar\{x\}\)=\\langle\\varphi\_\{\\mathrm\{br\}\}^\{N\}\(\\bar\{x\}\),v\_\{b\}\\rangle,

𝔼P​b−𝔼Q​b=⟨μP−μQ,vb⟩\.\\mathbb\{E\}\_\{P\}b\-\\mathbb\{E\}\_\{Q\}b=\\langle\\mu\_\{P\}\-\\mu\_\{Q\},v\_\{b\}\\rangle\.Cauchy’s inequality gives\|𝔼P​b−𝔼Q​b\|≤‖μP−μQ‖​‖vb‖\\bigl\|\\mathbb\{E\}\_\{P\}b\-\\mathbb\{E\}\_\{Q\}b\\bigr\|\\leq\\\|\\mu\_\{P\}\-\\mu\_\{Q\}\\\|\\\|v\_\{b\}\\\|, and‖μP−μQ‖=MMDkbrN​\(P,Q\)\\\|\\mu\_\{P\}\-\\mu\_\{Q\}\\\|=\\mathrm\{MMD\}\_\{k\_\{\\mathrm\{br\}\}^\{N\}\}\(P,Q\)\. If𝔼P​b≠𝔼Q​b\\mathbb\{E\}\_\{P\}b\\neq\\mathbb\{E\}\_\{Q\}b, this inequality forcesMMDkbrN​\(P,Q\)\>0\\mathrm\{MMD\}\_\{k\_\{\\mathrm\{br\}\}^\{N\}\}\(P,Q\)\>0\. ∎

This is a finite\-depth statement: we only claim visibility of the bracket coordinates included inφbrN\\varphi\_\{\\mathrm\{br\}\}^\{N\}, not characteristicness ofkbrNk\_\{\\mathrm\{br\}\}^\{N\}for arbitrary laws onΩ¯n\\bar\{\\Omega\}\_\{n\}\.

The previous proposition compares the actual geometric and branched kernels\. A separate question is what happens when bracket channels are not supplied\. In that case, a grid\-only method can become bracket\-aware only by reconstructing brackets from the observed path values\.

LetRn:Ωn→𝒜nR\_\{n\}:\\Omega\_\{n\}\\to\\mathcal\{A\}\_\{n\}be a bracket estimator\. The realised quadratic covariation estimator is the canonical example:

\[Rn​\(x\)\]tm:=∑k=0m−1\(xtk\+1−xtk\)​\(xtk\+1−xtk\)⊤,m=0,…,Nn\.\[R\_\{n\}\(x\)\]\_\{t\_\{m\}\}:=\\sum\_\{k=0\}^\{m\-1\}\(x\_\{t\_\{k\+1\}\}\-x\_\{t\_\{k\}\}\)\(x\_\{t\_\{k\+1\}\}\-x\_\{t\_\{k\}\}\)^\{\\top\},\\qquad m=0,\\ldots,N\_\{n\}\.Define the supplied and reconstructed enhanced laws byP¯nsup:=Law​\(Xπn,Aπn\)\\bar\{P\}\_\{n\}^\{\\mathrm\{sup\}\}:=\\mathrm\{Law\}\(X\_\{\\pi\_\{n\}\},A\_\{\\pi\_\{n\}\}\)andP¯nrec:=Law​\(Xπn,Rn​\(Xπn\)\)\\bar\{P\}\_\{n\}^\{\\mathrm\{rec\}\}:=\\mathrm\{Law\}\(X\_\{\\pi\_\{n\}\},R\_\{n\}\(X\_\{\\pi\_\{n\}\}\)\)\.

###### Proposition B\.2\(Cost of reconstructing bracket coordinates\)\.

EquipΩ¯n\\bar\{\\Omega\}\_\{n\}withdΩ¯n​\(\(x,a\),\(y,b\)\):=maxm⁡\|xtm−ytm\|\+maxm⁡\|atm−btm\|d\_\{\\bar\{\\Omega\}\_\{n\}\}\(\(x,a\),\(y,b\)\):=\\max\_\{m\}\|x\_\{t\_\{m\}\}\-y\_\{t\_\{m\}\}\|\+\\max\_\{m\}\|a\_\{t\_\{m\}\}\-b\_\{t\_\{m\}\}\|\. Let𝒟n⊆Ω¯n\\mathcal\{D\}\_\{n\}\\subseteq\\bar\{\\Omega\}\_\{n\}contain the supports ofP¯nsup\\bar\{P\}\_\{n\}^\{\\mathrm\{sup\}\}andP¯nrec\\bar\{P\}\_\{n\}^\{\\mathrm\{rec\}\}\. Iff:Ω¯n→ℝf:\\bar\{\\Omega\}\_\{n\}\\to\\mathbb\{R\}isLfL\_\{f\}\-Lipschitz on𝒟n\\mathcal\{D\}\_\{n\}, then

\|𝔼​f​\(Xπn,Rn​\(Xπn\)\)−𝔼​f​\(Xπn,Aπn\)\|≤Lf​𝔼​maxm⁡\|Rn​\(Xπn\)tm−Atm\|\.\\left\|\\mathbb\{E\}f\(X\_\{\\pi\_\{n\}\},R\_\{n\}\(X\_\{\\pi\_\{n\}\}\)\)\-\\mathbb\{E\}f\(X\_\{\\pi\_\{n\}\},A\_\{\\pi\_\{n\}\}\)\\right\|\\leq L\_\{f\}\\,\\mathbb\{E\}\\max\_\{m\}\\left\|R\_\{n\}\(X\_\{\\pi\_\{n\}\}\)\_\{t\_\{m\}\}\-A\_\{t\_\{m\}\}\\right\|\.IfφbrN\\varphi\_\{\\mathrm\{br\}\}^\{N\}isLNL\_\{N\}\-Lipschitz on𝒟n\\mathcal\{D\}\_\{n\}with respect todΩ¯nd\_\{\\bar\{\\Omega\}\_\{n\}\}, then

MMDkbrN​\(P¯nrec,P¯nsup\)≤LN​𝔼​maxm⁡\|Rn​\(Xπn\)tm−Atm\|\.\\mathrm\{MMD\}\_\{k\_\{\\mathrm\{br\}\}^\{N\}\}\(\\bar\{P\}\_\{n\}^\{\\mathrm\{rec\}\},\\bar\{P\}\_\{n\}^\{\\mathrm\{sup\}\}\)\\leq L\_\{N\}\\,\\mathbb\{E\}\\max\_\{m\}\\left\|R\_\{n\}\(X\_\{\\pi\_\{n\}\}\)\_\{t\_\{m\}\}\-A\_\{t\_\{m\}\}\\right\|\.

###### Proof\.

Couple the supplied and reconstructed enhanced observations by using the same grid pathXπnX\_\{\\pi\_\{n\}\}\. Since their path coordinates are identical,

dΩ¯n​\(\(Xπn,Rn​\(Xπn\)\),\(Xπn,Aπn\)\)=maxm⁡\|Rn​\(Xπn\)tm−Atm\|\.d\_\{\\bar\{\\Omega\}\_\{n\}\}\\bigl\(\(X\_\{\\pi\_\{n\}\},R\_\{n\}\(X\_\{\\pi\_\{n\}\}\)\),\(X\_\{\\pi\_\{n\}\},A\_\{\\pi\_\{n\}\}\)\\bigr\)=\\max\_\{m\}\\left\|R\_\{n\}\(X\_\{\\pi\_\{n\}\}\)\_\{t\_\{m\}\}\-A\_\{t\_\{m\}\}\\right\|\.The two coupled observations lie in𝒟n\\mathcal\{D\}\_\{n\}almost surely, so the Lipschitz bound forffgives the first claim after taking expectations\.

For the MMD bound, Jensen’s inequality and Lipschitzness ofφbrN\\varphi\_\{\\mathrm\{br\}\}^\{N\}on𝒟n\\mathcal\{D\}\_\{n\}give

MMDkbrN​\(P¯nrec,P¯nsup\)\\displaystyle\\mathrm\{MMD\}\_\{k\_\{\\mathrm\{br\}\}^\{N\}\}\(\\bar\{P\}\_\{n\}^\{\\mathrm\{rec\}\},\\bar\{P\}\_\{n\}^\{\\mathrm\{sup\}\}\)≤𝔼​‖φbrN​\(Xπn,Rn​\(Xπn\)\)−φbrN​\(Xπn,Aπn\)‖\\displaystyle\\leq\\mathbb\{E\}\\left\\\|\\varphi\_\{\\mathrm\{br\}\}^\{N\}\(X\_\{\\pi\_\{n\}\},R\_\{n\}\(X\_\{\\pi\_\{n\}\}\)\)\-\\varphi\_\{\\mathrm\{br\}\}^\{N\}\(X\_\{\\pi\_\{n\}\},A\_\{\\pi\_\{n\}\}\)\\right\\\|≤LN​𝔼​maxm⁡\|Rn​\(Xπn\)tm−Atm\|\.\\displaystyle\\leq L\_\{N\}\\,\\mathbb\{E\}\\max\_\{m\}\\left\|R\_\{n\}\(X\_\{\\pi\_\{n\}\}\)\_\{t\_\{m\}\}\-A\_\{t\_\{m\}\}\\right\|\.∎

###### Proposition B\.5\(Primitives lift to vector fields\)\.

Letℋ\\mathcal\{H\}be a bialgebra and letF:ℋ→DF:\\mathcal\{H\}\\to Dbe a pseudo\-bialgebra map in the sense of[Definition2\.2](https://arxiv.org/html/2606.05272#S2.Thmtheorem2)\. Ifp∈ℋp\\in\\mathcal\{H\}is primitive, i\.e\.Δ​p=p⊗1\+1⊗p\\Delta p=p\\otimes 1\+1\\otimes p, thenF​\(p\)∈Γ​\(T​M\)F\(p\)\\in\\Gamma\(TM\)\.

###### Proof\.

It suffices to show thatF​\(p\)F\(p\)is a derivation ofC∞​\(M\)C^\{\\infty\}\(M\)\. Fix arbitraryϕ,ψ∈C∞​\(M\)\\phi,\\psi\\in C^\{\\infty\}\(M\)\. By property \(b\) of[Definition2\.2](https://arxiv.org/html/2606.05272#S2.Thmtheorem2),

F​\(p\)​\(ϕ⋅ψ\)=F​\(Δ​p\)​\(ψ⊗ϕ\),F\(p\)\(\\phi\\cdot\\psi\)=F\(\\Delta p\)\(\\psi\\otimes\\phi\),where, for simple tensors,F​\(a⊗b\)​\(ψ⊗ϕ\)F\(a\\otimes b\)\(\\psi\\otimes\\phi\)denotes\(F​\(a\)​ψ\)⋅\(F​\(b\)​ϕ\)\(F\(a\)\\psi\)\\cdot\(F\(b\)\\phi\)\. Sinceppis primitive,

F​\(Δ​p\)​\(ψ⊗ϕ\)\\displaystyle F\(\\Delta p\)\(\\psi\\otimes\\phi\)=F​\(p⊗1\+1⊗p\)​\(ψ⊗ϕ\)\\displaystyle=F\(p\\otimes 1\+1\\otimes p\)\(\\psi\\otimes\\phi\)=\(F​\(p\)​ψ\)⋅F​\(1\)​ϕ\+F​\(1\)​ψ⋅\(F​\(p\)​ϕ\)\.\\displaystyle=\(F\(p\)\\psi\)\\cdot F\(1\)\\phi\+F\(1\)\\psi\\cdot\(F\(p\)\\phi\)\.Property \(a\) givesF​\(1\)=IdF\(1\)=\\mathrm\{Id\}\. Hence

F​\(p\)​\(ϕ⋅ψ\)\\displaystyle F\(p\)\(\\phi\\cdot\\psi\)=\(F​\(p\)​ψ\)⋅ϕ\+ψ⋅\(F​\(p\)​ϕ\)\\displaystyle=\(F\(p\)\\psi\)\\cdot\\phi\+\\psi\\cdot\(F\(p\)\\phi\)=\(F​\(p\)​ϕ\)⋅ψ\+ϕ⋅\(F​\(p\)​ψ\),\\displaystyle=\(F\(p\)\\phi\)\\cdot\\psi\+\\phi\\cdot\(F\(p\)\\psi\),where the last equality uses commutativity of pointwise multiplication\. ThusF​\(p\)F\(p\)satisfies the Leibniz rule, and thereforeF​\(p\)∈Γ​\(T​M\)F\(p\)\\in\\Gamma\(TM\)\. ∎

## Appendix CFurther Experiments

### C\.1Recovering Log\-NCDE

As stated in[Section3\.4](https://arxiv.org/html/2606.05272#S3.SS4), BNRDE recovers log\-NCDE whenℋ=ℋ\\mathcal\{H\}=\\operatorname\{\\mathcal\{H\}\}andℳ=ℝ\\mathcal\{M\}=\\mathbb\{R\}\. Results on PPG\-DaLiA\(Reisset al\.,[2019](https://arxiv.org/html/2606.05272#bib.bib89)\)confirm this, with both models attaining an identical 0\.134mean squared error \(MSE\)after 10 epochs\.

### C\.2Signature Computation Times

We report signature precomputation times for the relevant pySigLib log signature computations below, in[Figure7](https://arxiv.org/html/2606.05272#A3.F7)\.

![Refer to caption](https://arxiv.org/html/2606.05272v1/x8.png)\(a\)Planarly branched log\-signature,ℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\.
![Refer to caption](https://arxiv.org/html/2606.05272v1/x9.png)\(b\)Branched log\-signature,ℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}\.
![Refer to caption](https://arxiv.org/html/2606.05272v1/x10.png)\(c\)Log\-signature,ℋ\\operatorname\{\\mathcal\{H\}\}\.

Figure 7:Milliseconds to compute log\-signatures forℋ∈\{ℋ,ℋGL,ℋMKW\}\\mathcal\{H\}\\in\\\{\\operatorname\{\\mathcal\{H\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\},\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}\\\}with path dimensiond∈\[2,24\]d\\in\[2,24\]and signature truncation depthN∈\{2,3\}N\\in\\\{2,3\\\}\. The custom CUDA kernels implemented in pySigLib do not support tree counts greater than 1024, producing the blank cells in[Figure7\(a\)](https://arxiv.org/html/2606.05272#A3.F7.sf1)and[Figure7\(b\)](https://arxiv.org/html/2606.05272#A3.F7.sf2)\.

## Appendix DExperimental Details

### D\.1Software and Hardware Details

##### Software details

All experiments were conducted on Python 3\.13 using[JAX](https://github.com/jax-ml/jax)0\.8\.1\(Bradburyet al\.,[2018](https://arxiv.org/html/2606.05272#bib.bib28)\)\. We use[Diffrax](https://github.com/patrick-kidger/diffrax)0\.7\.1\(Kidgeret al\.,[2021](https://arxiv.org/html/2606.05272#bib.bib11)\)for its Euclidean differential equation framework,[Equinox](https://github.com/patrick-kidger/equinox)0\.13\.2\(Kidger and Garcia,[2021](https://arxiv.org/html/2606.05272#bib.bib29)\)as our neural network framework,[Optax](https://github.com/google-deepmind/optax)0\.2\.6\(DeepMindet al\.,[2020](https://arxiv.org/html/2606.05272#bib.bib30)\)for its implementation of the Muon optimizer\(Jordanet al\.,[2024](https://arxiv.org/html/2606.05272#bib.bib91)\), and[Cyreal](https://github.com/smorad/cyreal)0\.1\.5\(Morad,[2026](https://arxiv.org/html/2606.05272#bib.bib31)\)for dataloading\. Alllogℋ\\log\_\{\\mathcal\{H\}\}\-signatures are formed using a[custom PySigLib fork](https://github.com/luke-a-thompson/pySigLibax)\(Shmelev and Salvi,[2025](https://arxiv.org/html/2606.05272#bib.bib71)\), with the log\-ODE method provided by[Roughrax](https://github.com/luke-a-thompson/roughrax)and geometric numerical integrators provided by[Georax](https://github.com/luke-a-thompson/georax)\(Shmelevet al\.,[2026](https://arxiv.org/html/2606.05272#bib.bib92)\)\.

##### Released packages

We release two JAX packages to support the present work: one for the log\-ODE method on manifolds and one containing the benchmark datasets used\.

Table 6:New packages introduced to support the present work\.
##### Hardware details

All model training and evaluation was performed on an NVIDIA RTX 5080 \(16GB\) with CUDA 13\.1, an AMD Ryzen 9 9950X3D processor running on Ubuntu 24\.04 with 64GB of system memory\.

### D\.2Rough Volatility

Rough Bergomi \(rBergomi\)is traditionally formulated as a system of coupled Volterrastochastic differential equations \(SDEs\)with rough kernels:

St\\displaystyle S\_\{t\}=S0\+∫0tSs​vs​𝑑Bs,\\displaystyle=S\_\{0\}\+\\int\_\{0\}^\{t\}S\_\{s\}\\sqrt\{v\_\{s\}\}\\,dB\_\{s\},vt\\displaystyle v\_\{t\}=ξ0​\(t\)​exp⁡\(ηΓ​\(H\+12\)​∫0t\(t−s\)H−12​𝑑Ws−η22​Γ​\(H\+12\)2​∫0t\(t−s\)2​H−1​𝑑s\)\.\\displaystyle=\\xi\_\{0\}\(t\)\\exp\\left\(\\frac\{\\eta\}\{\\Gamma\\bigl\(H\+\\tfrac\{1\}\{2\}\\bigr\)\}\\int\_\{0\}^\{t\}\(t\-s\)^\{H\-\\tfrac\{1\}\{2\}\}\\,dW\_\{s\}\-\\frac\{\\eta^\{2\}\}\{2\\,\\Gamma\\bigl\(H\+\\tfrac\{1\}\{2\}\\bigr\)^\{2\}\}\\int\_\{0\}^\{t\}\(t\-s\)^\{2H\-1\}\\,ds\\right\)\.Here,StS\_\{t\}denotes the underlying asset price,vtv\_\{t\}denotes the instantaneous variance process,η\\etais the vol\-of\-vol, andH∈\(0,12\)H\\ \\in\(0,\\frac\{1\}\{2\}\)is the Hurst exponent controlling volatility roughness\. The forward variance curveξ0​\(t\)\\xi\_\{0\}\(t\)satisfies𝔼​\[vt\]=ξ0​\(t\)\\mathbb\{E\}\[v\_\{t\}\]=\\xi\_\{0\}\(t\)and the correlation of the two driversWt,BtW\_\{t\},B\_\{t\}is given byCorr⁡\(d​Wt,d​Bt\)=ρ∈\[−1,1\]\\operatorname\{Corr\}\(dW\_\{t\},dB\_\{t\}\)=\\rho\\in\[\-1,1\]\.

To simulate ourrBergomisample paths, we cast the model into therough differential equation \(RDE\)framework of\(Bonesiniet al\.,[2024](https://arxiv.org/html/2606.05272#bib.bib22)\),

St\\displaystyle S\_\{t\}=S0\+∫0tSu​v0​exp⁡\{ν​Vu−ν2​u2​H2​Γ​\(H\+12\)2\}​𝑑𝐖u−∫0t12​Su​v0​exp⁡\{2​ν​Vu−ν2​u2​HΓ​\(H\+12\)2\}​𝑑u\\displaystyle=S\_\{0\}\+\\int\_\{0\}^\{t\}S\_\{u\}\\sqrt\{v\_\{0\}\}\\exp\\left\\\{\\nu V\_\{u\}\-\\frac\{\\nu^\{2\}u^\{2H\}\}\{2\\Gamma\(H\+\\frac\{1\}\{2\}\)^\{2\}\}\\right\\\}d\\mathbf\{W\}\_\{u\}\-\\int\_\{0\}^\{t\}\\frac\{1\}\{2\}S\_\{u\}v\_\{0\}\\exp\\left\\\{2\\nu V\_\{u\}\-\\frac\{\\nu^\{2\}u^\{2H\}\}\{\\Gamma\(H\+\\frac\{1\}\{2\}\)^\{2\}\}\\right\\\}du\(14\)Vt\\displaystyle V\_\{t\}=1Γ​\(H\+12\)​∫0t\(t−u\)H−12​\(ρ​d​Wu\+1−ρ2​d​Bu\),\\displaystyle=\\frac\{1\}\{\\Gamma\(H\+\\frac\{1\}\{2\}\)\}\\int\_\{0\}^\{t\}\(t\-u\)^\{H\-\\frac\{1\}\{2\}\}\\left\(\\rho dW\_\{u\}\+\\sqrt\{1\-\\rho^\{2\}\}dB\_\{u\}\\right\),where𝐖t\\mathbf\{W\}\_\{t\}is the signature ofWtW\_\{t\}andVtV\_\{t\}is a Riemann\-Liouville volatility process computed following\(Bennedsenet al\.,[2017](https://arxiv.org/html/2606.05272#bib.bib21)\)\. We then solve[Equation14](https://arxiv.org/html/2606.05272#A4.E14)as a Wong\-Zakaiordinary differential equation \(ODE\)driven by a two\-dimensional lead\-lag path\(Wt,Vt\)\(W\_\{t\},V\_\{t\}\), yielding convergence to the Itô solution\. We use the calibration results ofCallum \([2023](https://arxiv.org/html/2606.05272#bib.bib20)\)and setv0=0\.04,ν=1\.991,H=0\.25,ρ=−0\.848v\_\{0\}=0\.04,\\ \\nu=1\.991,\\ H=0\.25,\\ \\rho=\-0\.848which shows aMSEof3\.73×10−53\.73\\times 10^\{\-5\}of model\-generated implied volatilities over SPX compared to market implied volatilities on 30/05/2022\. We employ the hybrid scheme ofBennedsenet al\.\([2017](https://arxiv.org/html/2606.05272#bib.bib21)\)to generate the Riemann\-Liouville fractional Brownian driver\.

### D\.3Sim\-to\-Real Dynamics Forecasting

In this section, we describe our implementation of the sim\-to\-real dynamics forecasting experiment developed byBastianet al\.\([2025](https://arxiv.org/html/2606.05272#bib.bib12)\)\.

We begin by generating theFREE\_ROTATIONsynthetic dataset available at their[official repository](https://github.com/bastianlb/forecasting-rotational-dynamics/tree/main)\. We then convert to\.npyand convert the \(quaternion\-last\) data into a timeseries ofSO​\(3\)\\mathrm\{SO\}\(3\)rotation matrices and form stride\-one length\-twenty sliding windows across the time series and flatten theℝ3×3\\mathbb\{R\}^\{3\\times 3\}matrices intoℝ9\\mathbb\{R\}^\{9\}vectors for feeding into the models\. This produces a dataset of shapeℝbatch⋅damping×windows⋅20×9\\mathbb\{R\}^\{\\text\{batch\}\\cdot\\text\{damping\}\\times\\text\{windows\}\\cdot 20\\times 9\}\.

Each window is divided into two segments,pred∈ℝ12×9\\texttt\{pred\}\\in\\mathbb\{R\}^\{12\\times 9\}, andrecon∈ℝ8×9\\texttt\{recon\}\\in\\mathbb\{R\}^\{8\\times 9\}, the ground truthreconis discarded, and an extrapolator fit topredproduces a newreconwhich is concatenated with the ground truthpred\. We list the extrapolators here:

1. 1\.SO​\(3\)\\mathrm\{SO\}\(3\)\-neural controlled differential equation \(NCDE\): Hermite polynomial
2. 2\.SO​\(3\)\\mathrm\{SO\}\(3\)\-GRU: Hermite polynomial
3. 3\.SG\-NCDE: Weighted Savitsky\-Golay polynomial
4. 4\.Branched neural rough differential equation \(B\-NRDE\):multilayer perceptron \(MLP\)

To save parameters in the readoutMLP, all models output vectors inℝ6\\mathbb\{R\}^\{6\}, which are transformed back intoℝ3×3\\mathbb\{R\}^\{3\\times 3\}rotation matrices via a Gram\-Schmidt procedure\(Zhouet al\.,[2020](https://arxiv.org/html/2606.05272#bib.bib32)\)\.

### D\.4Mean\-reverting Dynamics over the SPD Manifold under Affine\-invariant Geometry

We simulate[Equation11](https://arxiv.org/html/2606.05272#S4.E11)on𝕊d\+\+\\mathbb\{S\}^\{\+\+\}\_\{d\}withd=3d=3over horizonT=1T=1on a uniform grid ofN=1025N=1025time points, with initial conditionX0=I3X\_\{0\}=I\_\{3\}\. We set the \(matrix\-valued\) parameter

Σ=\[0\.180\.04−0\.020\.020\.140\.060\.000\.080\.16\],A=\[0\.000\.200\.00−0\.100\.000\.100\.00−0\.150\.00\],γ=0\.25,ε=0\.1,σ=0\.4\.\\Sigma=\\begin\{bmatrix\}0\.18&0\.04&\-0\.02\\\\ 0\.02&0\.14&0\.06\\\\ 0\.00&0\.08&0\.16\\end\{bmatrix\},\\qquad A=\\begin\{bmatrix\}0\.00&0\.20&0\.00\\\\ \-0\.10&0\.00&0\.10\\\\ 0\.00&\-0\.15&0\.00\\end\{bmatrix\},\\qquad\\gamma=0\.25,\\qquad\\varepsilon=0\.1,\\qquad\\sigma=0\.4\.Define

Q:=Σ⊤​Σ,M:=b:=\(d−1\)​Q\+ε​Id,H:=−γ​Id\+A\.Q:=\\Sigma^\{\\top\}\\Sigma,\\qquad M:=b:=\(d\-1\)Q\+\\varepsilon I\_\{d\},\\qquad H:=\-\\gamma I\_\{d\}\+A\.For the affine\-invariant mean reversion we use the \(PSD\) stiffness

K:=Π𝕊d\+​\(sym​\(−H\)\),η:=1d​tr​\(K\),K:=\\Pi\_\{\\mathbb\{S\}^\{\+\}\_\{d\}\}\\bigl\(\\mathrm\{sym\}\(\-H\)\\bigr\),\\qquad\\eta:=\\tfrac\{1\}\{d\}\\mathrm\{tr\}\(K\),whereΠ𝕊d\+\\Pi\_\{\\mathbb\{S\}^\{\+\}\_\{d\}\}denotes eigenvalue clipping to𝕊d\+\\mathbb\{S\}^\{\+\}\_\{d\}\. We take the symmetric Brownian driverBt∈Sym​\(d\)B\_\{t\}\\in\\mathrm\{Sym\}\(d\)with covariation

d​⟨Ba​b,Bc​d⟩t=12​\(δa​c​δb​d\+δa​d​δb​c\)​d​t,\\mathrm\{d\}\\langle B\_\{ab\},B\_\{cd\}\\rangle\_\{t\}=\\tfrac\{1\}\{2\}\\bigl\(\\delta\_\{ac\}\\delta\_\{bd\}\+\\delta\_\{ad\}\\delta\_\{bc\}\\bigr\)\\,\\mathrm\{d\}t,\(corresponding to independent diagonal entries and off\-diagonals scaled by1/21/\\sqrt\{2\}\), and use the identity correlation in the symmetric degrees of freedom\.

##### Analytic quadratic variation\.

For the Itô diffusion

d​Xt=η​logXt⁡\(M\)​d​t\+σ​Xt1/2​d​Bt​Xt1/2,\\mathrm\{d\}X\_\{t\}=\\eta\\,\\log\_\{X\_\{t\}\}\(M\)\\,\\mathrm\{d\}t\+\\sigma\\,X\_\{t\}^\{1/2\}\\,\\mathrm\{d\}B\_\{t\}\\,X\_\{t\}^\{1/2\},the entrywise quadratic covariation ofXtX\_\{t\}is

d​⟨Xi​j,Xm​n⟩t=σ22​\(Xi​m​Xj​n\+Xi​n​Xj​m\)​d​t,1≤i,j,m,n≤d\.\\mathrm\{d\}\\langle X\_\{ij\},X\_\{mn\}\\rangle\_\{t\}=\\frac\{\\sigma^\{2\}\}\{2\}\\Bigl\(X\_\{im\}X\_\{jn\}\+X\_\{in\}X\_\{jm\}\\Bigr\)\\,\\mathrm\{d\}t,\\qquad 1\\leq i,j,m,n\\leq d\.Writingxt=vech​\(Xt\)∈ℝqx\_\{t\}=\\mathrm\{vech\}\(X\_\{t\}\)\\in\\mathbb\{R\}^\{q\}withq=d​\(d\+1\)2q=\\frac\{d\(d\+1\)\}\{2\}, the corresponding quadratic variation is obtained by the linear projection

d​⟨x⟩t=E​d​⟨vec​\(X\)⟩t​E⊤,\\mathrm\{d\}\\langle x\\rangle\_\{t\}=E\\,\\mathrm\{d\}\\langle\\mathrm\{vec\}\(X\)\\rangle\_\{t\}\\,E^\{\\top\},whereE∈ℝq×d2E\\in\\mathbb\{R\}^\{q\\times d^\{2\}\}selects the lower\-triangular coordinates consistent with ourvech\\mathrm\{vech\}convention\. In the generated dataset we store per\-step incrementsΔ​⟨x⟩k≈\(tk\+1−tk\)​\(d​⟨x⟩t/d​t\)\|t=tk\\Delta\\langle x\\rangle\_\{k\}\\approx\(t\_\{k\+1\}\-t\_\{k\}\)\\,\\bigl\(\\mathrm\{d\}\\langle x\\rangle\_\{t\}/\\mathrm\{d\}t\\bigr\)\\big\|\_\{t=t\_\{k\}\}\.

### D\.5Numerical Integration

NCDE\-type baselines are solved with Tsit5\(Tsitouras,[2011](https://arxiv.org/html/2606.05272#bib.bib36)\)using automatic initial stepsizing as described inHaireret al\.\([2008](https://arxiv.org/html/2606.05272#bib.bib35), Sec\. 2\.4\)and PID step size control controller followingSöderlind \([2002](https://arxiv.org/html/2606.05272#bib.bib37)\)andHairer and Wanner \([2002](https://arxiv.org/html/2606.05272#bib.bib39), Sec\. 4\.2\)\. Euclidean signature\-based models, including log\-NCDE, NRDE, and B\-NRDE, in the rBergomi task use Heun with a fixed step size\. Manifold B\-NRDE uses theCF​\-​EES​\(2,5\)\\mathrm\{CF\\text\{\-\}EES\}\(2,5\)commutator\-free integrator\. Backpropagation through the solver proceeds by the discretise\-then\-optimise method\(Maet al\.,[2021](https://arxiv.org/html/2606.05272#bib.bib38)\)\.

### D\.6Hyperparameters

All models are trained for 100 epochs with batch size 1024, gradient clipping at 1\.0, and a 0\.8/0\.1/0\.1 train/validation/test split\. We use Muon with learning rate5×10−45\\times 10^\{\-4\},\(β1,β2\)=\(0\.9,0\.999\)\(\\beta\_\{1\},\\beta\_\{2\}\)=\(0\.9,0\.999\), weight decay1×10−61\\times 10^\{\-6\}, andϵ=1×10−8\\epsilon=1\\times 10^\{\-8\}\. PID tolerancesrtol=1×10−3\\mathrm\{rtol\}=1\\times 10^\{\-3\}andatol=1×10−3\\mathrm\{atol\}=1\\times 10^\{\-3\}, with minimum step size1×10−41\\times 10^\{\-4\}\. Unless otherwise stated, models use initial\-condition and vector\-field MLPs of width 32, depth 2, and latent state dimension 32\. Signature depth is always taken atN=2N=2\. Hyperparameters differing between models are listed in[Table7](https://arxiv.org/html/2606.05272#A4.T7)\.

Table 7:Hyperparameters that vary across experiments/models\.ExperimentHyperparameterModelGRUxLSTMStacked xLSTMNCDENCDE\+\+log\-NCDEneural rough differential equation \(NRDE\)B\-NRDERoughBergomiActivation———SoftplusSoftplusSoftplusSoftplusLipSwishFinal activation———Identitytanh\\tanhtanh\\tanhtanh\\tanhtanh\\tanhControl interpolation———LinearHermiteLinearLinearLinearSignature window—————161616Hopf algebra—————ℋ\\operatorname\{\\mathcal\{H\}\}ℋ\\operatorname\{\\mathcal\{H\}\}ℋGL\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{GL\}\}\}Signature kernel———GeometricGeometricGeometricGeometricBranchedSO​\(3\)\\mathrm\{SO\}\(3\)DynamicsActivationSoftplus——SoftplusSoftplusSoftplusSoftplusLipSwishFinal activation————————Control interpolation———HermiteHermiteHermiteHermite—Signature window—————444Hopf algebra—————ℋ\\operatorname\{\\mathcal\{H\}\}ℋ\\operatorname\{\\mathcal\{H\}\}ℋ\\operatorname\{\\mathcal\{H\}\}Signature kernel———GeometricGeometricGeometricGeometricBranchedExtrapolator———HermiteHermite,Savitzky–Golay \(SG\)HermiteHermiteMLPsymmetric positive definite \(SPD\)DynamicsActivation————————Final activation————————Control interpolation————————Signature window—————646464Hopf algebra—————ℋ\\operatorname\{\\mathcal\{H\}\}ℋ\\operatorname\{\\mathcal\{H\}\}ℋMKW\\operatorname\{\\mathcal\{H\}\_\{\\mathrm\{MKW\}\}\}Signature kernel—————GeometricGeometricBranched