text:
\section{Introduction}
This paper is an answer to the question asked by several of my colleagues:
``Can we deduce the existence of a formality quasi-isomorphism for Hochschild
cochains of ${\mathbb Q}[x^1, x^2, \dots, x^d]$ starting from Kontsevich's quasi-isomorphism
\cite{K}?'' The usual folklore argument of rational homotopy theory cannot be
directly applied here because the graded Lie algebra of polyvector fields
is infinite dimensional. Besides, it would be interesting to get a construction
whose output is a formality quasi-isomorphism for Hochschild
cochains of ${\mathbb Q}[x^1, x^2, \dots, x^d]$ which admits a ``graphical
expansion'' (as original Kontsevich's quasi-isomorphism \cite{K} does)
with all weights being rational numbers.
The goal of this paper is to produce such a construction.

Based on calculations performed in \cite{FW-irrat} by G. Felder and T. Willwacher,
it is unlikely that coefficients entering  Kontsevich's formality quasi-isomorphism \cite{K}
are all rational. In addition, the combination of T. Willwacher's theorem \cite[Theorem 1.1]{Thomas} and
the result \cite[Theorem 6.2]{stable1} indicate that the algebra of hypothetical ``motivic avatars''
of coefficients\footnote{Two comments should be made about  hypothetical ``motivic avatars''
of coefficients entering  Kontsevich's quasi-isomorphism \cite{K}. First, in order to get an algebra one
should consider integrals over strata of all dimensions. Second, the author
does not see an obvious way to define ``motivic avatars'' for these coefficients/weights.}
entering  Kontsevich's formality quasi-isomorphism \cite{K}  should be isomorphic
to the algebra of motivic avatars of multiple zeta values \cite{Goncharov}, \cite{Gon-arxiv}.

It is not surprising that the proof of the existence of the desired stable formality quasi-isomorphism
is very similar to that of the existence
of Drinfeld's associator over rational numbers \cite{Drinfeld}, \cite[Section 4.1]{Bar-Natan}.
However, there is an important difference between Drinfeld associators and stable
formality quasi-isomorphisms: unlike Drinfeld associators, stable formality quasi-isomorphisms
have a nontrivial homotopy equivalence relation. This difference adds an additional ``layer'' to
the proof. It should also be remarked that the author does not see
what would be an analog of technical Lemma \ref{lem:al-K-2-OK} for Drinfeld
associators.

The paper is organized as follows. We conclude our introduction with
notation and conventions used in this paper. In Section \ref{sec:OC-KGra-SFQ},
we recall the Kajiura-Stasheff operad ${\mathsf{OC}}$ and the operad ${\mathsf {KGra} }$ which is
assembled from graphs used in Kontsevich's paper \cite{K}. In this section,
we also recall the definition of a stable formality quasi-isomorphism (SFQ).
In Section \ref{sec:main}, we formulate the
main result (see Theorem \ref{thm:main}) which implies the existence of an SFQ
over rational numbers.  Section \ref{sec:tech} is devoted to various technical statements
which will be used later in the proof of  Theorem \ref{thm:main}.
Most of these statements are essentially borrowed from \cite{stable1}.

The proof of Theorem \ref{thm:main} goes by induction on the arities with respect
to both colors involved in the game.  In Section \ref{sec:step-by-step}, we present
all the necessary ``building blocks'' of this induction and finally, in Section \ref{sec:proof}, we assemble
these  ``building blocks'' into a proof of Theorem \ref{thm:main}.

\subsection{Notation and conventions}
We mostly follow the notation and conventions of \cite{stable1}.
${\mathbb K}$ is a field of characteristic zero and, in this note, ${\mathbb K}={\mathbb R}$ or
${\mathbb K}= {\mathbb Q}$\,.
The underlying symmetric monoidal category ${\mathfrak{C}}$ for our algebraic
structures is either the category ${\mathsf{grVect}}_{{\mathbb K}}$ of ${\mathbb Z}$-graded
${\mathbb K}$-vector spaces or the category
${\mathsf{Ch}}_{{\mathbb K}}$ of unbounded cochain complexes of ${\mathbb K}$-vector spaces.

For a cochain complex ${\cal V}$ we denote
by ${\mathbf{s}} {\cal V}$ (resp. by ${\mathbf{s}}^{-1} {\cal V}$) the suspension (resp. the
desuspension) of ${\cal V}$\,. In other words,
$$
\big({\mathbf{s}} {\cal V}\big)^{{\bullet}} = {\cal V}^{{\bullet}-1}\,,  \qquad
\big({\mathbf{s}}^{-1} {\cal V}\big)^{{\bullet}} = {\cal V}^{{\bullet}+1}\,.
$$
For a homogeneous vector $v$ in a graded vector space (or a cochain complex)
${\cal V}$, the notation $|v|$ is reserved for the degree $v$\,.

$C^{\bullet}(A)$  denotes the Hochschild cochain
complex of an associative algebra  (or more generally an $A_{\infty}$-algebra) $A$
with coefficients in $A$\,. For a commutative ring $R$ and an $R$-module $V$ we denote
by $S_R(V)$ the symmetric algebra of $V$ over $R$\,. For a vector $\xi$ in a (dg) Lie algebra
${\mathcal{L}}$, the notation ${\mathrm{a d}}_{\xi}$ is reserved for the adjoint action of $\xi$, i.e.
${\mathrm{a d}}_{\xi} (\eta) : =  [\xi, \eta]$\,.

Given an operad ${\cal O}$, we denote by $\circ_i$ the elementary
operadic insertions:
$$
\circ_i : {\cal O}(n) \otimes {\cal O}(k) \to {\cal O}(n+k-1)\,, \qquad 1 \le i \le n\,.
$$

The symmetric group on $n$ letters in denoted by $S_n$ and
the notation ${\rm S h }_{p,q}$ is reserved for the set of $(p,q)$-shuffles
in $S_{p+q}$\,. A graph is {\it directed} if each edge
carries a chosen direction. A graph ${\Gamma}$ with
$n$ vertices is called {\it labeled} if ${\Gamma}$ is equipped with a bijection between
the set of its vertices and the set $\{1,2, \dots, n\}$\,. In this paper we
consider exclusively graphs without loops (i.e. cycles of length one).

We will freely use the conventions for colored
(co)operads and colored pseudo- (co)operads from \cite[Section 2]{stable1}.
For example, for a (colored) pseudo-cooperad ${\cal C}$ and a (colored) pseudo-operad ${\cal O}$, the
notation
$$
{\mathrm{Conv}}({\cal C}, {\cal O})
$$
is reserved for the convolution dg Lie algebra \cite[Section 2.3]{stable1},
\cite[Section 4]{notes}, \cite{MVnado}.

For most of colored (co)operads and pseudo- (co)operads used in this paper
the ordinal of colors will have only two element ${\mathfrak{c}}$ and ${\mathfrak{o}}$ for
which we set ${\mathfrak{c}} < {\mathfrak{o}}$. Just as in \cite{stable1}, solid edges of colored planar trees
are the edges which carry the color ${\mathfrak{c}}$ and dashed edges are the edges which
carry the color ${\mathfrak{o}}$.

For a two colored operad ${\cal O}$, the notation ${\cal O}(n,k)^{{\mathfrak{c}}}$ (resp. ${\cal O}(n,k)^{{\mathfrak{o}}}$) is
reserved for the space of ``operations'' with $n$ inputs of the color ${\mathfrak{c}}$, $k$ inputs of
the color ${\mathfrak{o}}$, and with the output carrying the color ${\mathfrak{c}}$ (resp. ${\mathfrak{o}}$).

Using ``arity'' we can equip the convolution Lie algebra ${\mathrm{Conv}}({\cal C}, {\cal O})$
with the natural descending filtration
$$
{\mathrm{Conv}}({\cal C}, {\cal O}) =  {\cal F}_{-1}\, {\mathrm{Conv}}({\cal C}, {\cal O}) \supset {\cal F}_0\, {\mathrm{Conv}}({\cal C}, {\cal O}) \supset {\cal F}_1\, {\mathrm{Conv}}({\cal C}, {\cal O})
\supset  \dots\,,
$$
where
\begin{equation}
\label{Conv-filtr}
{\cal F}_m\, {\mathrm{Conv}}({\cal C}, {\cal O})=
\end{equation}
$$
 \big\{ f \in {\mathrm{Conv}}({\cal C}, {\cal O}) \quad \big| \quad
f \Big|_{{\cal C}({\mathbf{q}})} \,=\, 0 \quad \forall~~
\textrm{corollas} ~~ {\mathbf{q}} \quad \textrm{satisfying} ~ |{\mathbf{q}}| \le m \big\}\,,
$$
where $|{\mathbf{q}}|$ is the total number of incoming edges of the corolla ${\mathbf{q}}$\,.

This filtration is compatible with the Lie bracket and  ${\mathrm{Conv}}({\cal C}, {\cal O})$ is complete
with respect to this filtration.  Namely,
\begin{equation}
\label{complete}
{\mathrm{Conv}}({\cal C}, {\cal O}) = \lim_{m} {\mathrm{Conv}}({\cal C}, {\cal O})
~\Big/~   {\cal F}_m\, {\mathrm{Conv}}({\cal C}, {\cal O})\,.
\end{equation}

We also introduce an additional descending
filtration  ${\cal F}^{\chi}_{{\bullet}}$ on the convolution Lie algebra ${\mathrm{Conv}}({\cal C}, {\cal O})$
for each color $\chi$:
$$
{\mathrm{Conv}}({\cal C}, {\cal O}) =  {\cal F}^{\chi}_{-1}\, {\mathrm{Conv}}({\cal C}, {\cal O}) \supset
 {\cal F}^{\chi}_0\, {\mathrm{Conv}}({\cal C}, {\cal O}) \supset {\cal F}^{\chi}_1\, {\mathrm{Conv}}({\cal C}, {\cal O})
 \supset \dots\,,
$$
where
\begin{equation}
\label{Conv-filtr-chi}
{\cal F}^{\chi}_m\, {\mathrm{Conv}}({\cal C}, {\cal O})=
\end{equation}
$$
 \big\{ f \in {\mathrm{Conv}}({\cal C}, {\cal O}) \quad \big| \quad
f \Big|_{{\cal C}({\mathbf{q}})} \,=\, 0 \quad \forall~~
\textrm{corollas} ~~ {\mathbf{q}} \quad \textrm{satisfying} ~  \sharp_{\chi}({\mathbf{q}}) \le m \big\}\,,
$$
where $ \sharp_{\chi}({\mathbf{q}}) $ is the number of incoming edges of the corolla ${\mathbf{q}}$
which carry the color $\chi$.

The filtration \eqref{Conv-filtr-chi} is compatible with the Lie bracket on ${\mathrm{Conv}}({\cal C}, {\cal O})$
and  ${\mathrm{Conv}}({\cal C}, {\cal O})$ is complete with respect to
this filtration.

We denote by ${\Lambda}$ the endomorphism operad of the $1$-dimensional
vector space ${\mathbf{s}}^{-1} {\mathbb K}$ placed in degree $-1$
\begin{equation}
\label{La}
{\Lambda} = {\mathsf{E n d}}_{{\mathbf{s}}^{-1} {\mathbb K}}\,.
\end{equation}
In other words,
$$
{\Lambda}(n) = {\mathbf{s}}^{1-n} {\rm s g n}_n\,,
$$
where ${\rm s g n}_n$ is the sign representation for the symmetric group $S_n$
We observe that the collection ${\Lambda}$ is also naturally a cooperad.

For a dg operad (resp. a dg cooperad) $P$ in we denote by ${\Lambda} P$ the
dg operad (resp. the dg cooperad) which is obtained from $P$ via tensoring
with ${\Lambda}$, i.e.
\begin{equation}
\label{La-P}
{\Lambda} P (n) = {\mathbf{s}}^{1-n} P(n) \otimes {\rm s g n}_n\,.
\end{equation}

~\\

\noindent
\textbf{Anniversary note:} If you have never met Volodya Rubtsov then I
strongly suggest you visit him in the City of Angers in France.
You will meet a person who radiates an amazing amount of
generosity and charm! I would like to greet Volodya with his 60-th anniversary,
and wish him health, new brilliant mathematical ideas,
as well as many days filled with a positive spirit.

~\\[-0.3cm]

\noindent
\textbf{Acknowledgements:} I acknowledge NSF grants
DMS-0856196 and DMS-1161867 for the partial support.
A part of this text was typed during my participation in the program
``Grothendieck-Teichm\"uller Groups, Deformation and Operads''
of the Isaac Newton Institute in Cambridge, UK.
I would like to heartily thank the organizers of this wonderful and
very inspiring program.

\section{Reminder of ${\mathsf{OC}}$, ${\mathsf {KGra} }$ and stable formality quasi-isomorphisms}
\label{sec:OC-KGra-SFQ}

The definition of a stable formality quasi-isomorphism (SFQ) \cite[Section 5]{stable1}  is
based on the $2$-colored operads ${\mathsf{OC}}$ and ${\mathsf {KGra} }$\,. The 2-colored dg operad ${\mathsf{OC}}$
governs open-closed homotopy algebras introduced in \cite{OCHA} by  H. Kajiura and J. Stasheff
and the 2-colored operad ${\mathsf {KGra} }$ is ``assembled'' from graphs used in Kontsevich's paper \cite{K}.

\subsection{The Kajiura-Stasheff operad ${\mathsf{OC}}$}
As an operad in the category ${\mathsf{grVect}}$ of graded vector spaces, ${\mathsf{OC}}$
is freely generated by the 2-colored collection ${\mathfrak{o} \mathfrak{c}}$ with the following
spaces:
\begin{equation}
\label{oc-mc}
{\mathfrak{o} \mathfrak{c}}(n,0)^{{\mathfrak{c}}} = {\mathbf{s}}^{3-2n} {\mathbb K}\,, \qquad  n \ge 2\,,
\end{equation}
\begin{equation}
\label{oc-mo}
{\mathfrak{o} \mathfrak{c}}(0,k)^{{\mathfrak{o}}} = {\mathbf{s}}^{2-k}\, {\rm s g n}_k \otimes {\mathbb K}[S_k]\,, \qquad  k \ge 2\,,
\end{equation}
\begin{equation}
\label{oc-mo1}
{\mathfrak{o} \mathfrak{c}}(n,k)^{{\mathfrak{o}}} = {\mathbf{s}}^{2-2n-k}\,  {\rm s g n}_k \otimes {\mathbb K}[S_k]\,, \qquad n \ge 1\,,
\qquad k \ge 0\,,
\end{equation}
where ${\rm s g n}_k$ is the sign representation of $S_k$\,.
The remaining spaces of the collection ${\mathfrak{o} \mathfrak{c}}$ are zero.

Following \cite[Section 4]{stable1},
we represent generators of ${\mathsf{OC}}$ in ${\mathfrak{o} \mathfrak{c}}(n,0)^{{\mathfrak{c}}}$
by non-planar labeled corollas with $n$ solid incoming edges
(see figure \ref{fig:bt-n-oc-mc}). We represent  generators of ${\mathsf{OC}}$ in
${\mathfrak{o} \mathfrak{c}}(0,k)^{{\mathfrak{o}}}$ by planar labeled corollas with  $k$ dashed incoming edges
(see figure \ref{fig:bt-k-oc-mo}).
Finally, we use labeled 2-colored corollas with a planar structure given only
on the dashed edges to represent generators of ${\mathsf{OC}}$ in ${\mathfrak{o} \mathfrak{c}}(n,k)^{{\mathfrak{o}}}$
(see figure \ref{fig:bt-nk-oc-mo}).
\begin{figure}[htp]
\begin{minipage}[t]{0.3\linewidth}
\centering
\begin{tikzpicture}[scale=0.5]
\tikzstyle{w}=[circle, draw, minimum size=3, inner sep=1]
\tikzstyle{vt}=[circle, draw, fill, minimum size=0, inner sep=1]
\node[vt] (l1) at (0, 2) {};
\draw (0,2.5) node[anchor=center] {{\small $1$}};
\node[vt] (l2) at (1, 2) {};
\draw (1,2.5) node[anchor=center] {{\small $2$}};
\draw (2,1.8) node[anchor=center] {{\small $\dots$}};
\node[vt] (ln) at (3, 2) {};
\draw (3,2.5) node[anchor=center] {{\small $n$}};
\node[w] (v) at (1.5, 1) {};
\node[vt] (r) at (1.5, 0) {};
\draw (r) edge (v);
\draw (v) edge (l1);
\draw (v) edge (l2);
\draw (v) edge (ln);
\end{tikzpicture}
\caption{The non-planar corolla ${\mathsf{t}}^{{\mathfrak{c}}}_n$
representing a generator of ${\mathfrak{o} \mathfrak{c}}(n,0)^{{\mathfrak{c}}}$ } \label{fig:bt-n-oc-mc}
\end{minipage}
\hspace{0.2cm}
\begin{minipage}[t]{0.3\linewidth}
\centering
\begin{tikzpicture}[scale=0.5]
\tikzstyle{vt}=[circle, draw, fill, minimum size=0, inner sep=1]
\tikzstyle{w}=[circle, draw, minimum size=3, inner sep=1]
\node[vt] (l1) at (0, 2) {};
\draw (0,2.5) node[anchor=center] {{\small $1$}};
\node[vt] (l2) at (1, 2) {};
\draw (1,2.5) node[anchor=center] {{\small $2$}};
\draw (2,1.8) node[anchor=center] {{\small $\dots$}};
\node[vt] (lk) at (3, 2) {};
\draw (3,2.5) node[anchor=center] {{\small $k$}};
\node[w] (v) at (1.5, 1) {};
\node[vt] (r) at (1.5, 0) {};
\draw [dashed] (r) edge (v);
\draw [dashed] (v) edge (l1);
\draw [dashed] (v) edge (l2);
\draw [dashed] (v) edge (lk);
\end{tikzpicture}
\caption{The 2-colored planar corola ${\mathsf{t}}^{{\mathfrak{o}}}_k$ representing
a generator of ${\mathfrak{o} \mathfrak{c}}(0,k)^{{\mathfrak{o}}}$ } \label{fig:bt-k-oc-mo}
\end{minipage}
\hspace{0.2cm}
\begin{minipage}[t]{0.35\linewidth}
\centering
\begin{tikzpicture}[scale=0.5]
\tikzstyle{vt}=[circle, draw, fill, minimum size=0, inner sep=1]
\tikzstyle{w}=[circle, draw, minimum size=3, inner sep=1]
\node[vt] (l1) at (0, 2) {};
\draw (0,2.5) node[anchor=center] {{\small $1$}};
\draw (1.5,1.8) node[anchor=center] {{\small $\dots$}};
\node[vt] (ln) at (2, 2) {};
\draw (2,2.5) node[anchor=center] {{\small $n$}};
\node[vt] (l1mo) at (3, 2) {};
\draw (3,2.5) node[anchor=center] {{\small $1$}};
\draw (3.5,1.8) node[anchor=center] {{\small $\dots$}};
\node[vt] (lkmo) at (5, 2) {};
\draw (5,2.5) node[anchor=center] {{\small $k$}};
\node[w] (v) at (2.5, 1) {};
\node[vt] (r) at (2.5, 0) {};
\draw  [dashed] (r) edge (v);
\draw (v) edge (l1);
\draw (v) edge (ln);
\draw  [dashed] (v) edge (l1mo);
\draw  [dashed] (v) edge (lkmo);
\end{tikzpicture}
\caption{The 2-colored partially planar corola ${\mathsf{t}}^{{\mathfrak{o}}}_{n,k}$ representing
a generator of ${\mathfrak{o} \mathfrak{c}}(n,k)^{{\mathfrak{o}}}$
} \label{fig:bt-nk-oc-mo}
\end{minipage}
\end{figure}
Applying element ${\sigma} \in S_k$ to the labeled corolla ${\mathsf{t}}^{{\mathfrak{o}}}_k$
depicted on figure \ref{fig:bt-k-oc-mo} we get a basis for the vector
space ${\mathfrak{o} \mathfrak{c}}(0,k)^{{\mathfrak{o}}}$\,. Similarly, applying elements of
$({\rm  i d } , {\sigma}) \in S_n \times S_k$ to the labeled
corolla depicted on figure \ref{fig:bt-nk-oc-mo} we get a basis
for the vector space ${\mathfrak{o} \mathfrak{c}}(n,k)^{{\mathfrak{o}}}$\,.

The corollas  ${\mathsf{t}}^{{\mathfrak{c}}}_n$, ${\mathsf{t}}^{{\mathfrak{o}}}_k$ and  ${\mathsf{t}}^{{\mathfrak{o}}}_{n,k}$
carry the following degrees:
\begin{eqnarray}
\label{deg-mc-n}
| {\mathsf{t}}^{{\mathfrak{c}}}_n |  = 3 - 2n &  n \ge 2\,, \\[0.3cm]
\label{deg-mo-k}
| {\mathsf{t}}^{{\mathfrak{o}}}_k |  = 2 - k  &  k \ge 2\,, \\[0.3cm]
\label{deg-mo-nk}
| {\mathsf{t}}^{{\mathfrak{o}}}_{n,k} |  = 2 -2n - k &  ~~~~n \ge 1,~ k \ge 0\,.
\end{eqnarray}

The differential ${\cal D}$ on ${\mathsf{OC}}$ is defined by the following
equations\footnote{For a nice pictorial definition of
this differential on ${\mathsf{OC}}$ we refer the reader to \cite[Section 4.1]{stable1}.}
\begin{equation}
\label{cD-st-mc}
{\cal D} ( {\mathsf{t}}^{{\mathfrak{c}}}_n) : = - \sum_{p=2}^{n-1} \sum_{\tau \in {\rm S h }_{p, n-p}}
(\tau, {\rm  i d }) \big( {\mathsf{t}}^{{\mathfrak{c}}}_{n-p+1} \circ_{1, {\mathfrak{c}}} {\mathsf{t}}^{{\mathfrak{c}}}_{p}  \big)\,.
\end{equation}

\begin{equation}
\label{cD-st-mo-k}
{\cal D} ( {\mathsf{t}}^{{\mathfrak{o}}}_k) : = -
\sum_{p=0}^{k-2} \sum_{q= p+2}^{k}
(-1)^{p + (k-q) (q-p)} \,
{\mathsf{t}}^{{\mathfrak{o}}}_{k-q+p+1}  \circ_{p+1, {\mathfrak{o}}} {\mathsf{t}}^{{\mathfrak{o}}}_{q-p}\,.
\end{equation}

\begin{equation}
\label{cD-st-mo-nk}
{\cal D} ( {\mathsf{t}}^{{\mathfrak{o}}}_{n,k}) : =  (-1)^k
\sum_{p=2}^{n} \sum_{\tau \in {\rm S h }_{p, n-p}}
(\tau, {\rm  i d }) \big( {\mathsf{t}}^{{\mathfrak{o}}}_{n-p+1, k} \circ_{1, {\mathfrak{c}}}
{\mathsf{t}}^{{\mathfrak{c}}}_{p} \big)
\end{equation}
$$
 - \sum_{r=0}^{n}
 \sum_{\substack{ {\sigma} \in {\rm S h }_{r, n-r} \\[0.1cm] 0 \le p \le q \le k }}
(-1)^{p + (k-q)(q-p)}\, ({\sigma}, {\rm  i d }) \big( {\mathsf{t}}^{{\mathfrak{o}}}_{r, k-q+p+1} \circ_{p+1, {\mathfrak{o}}} {\mathsf{t}}^{{\mathfrak{o}}}_{n-r, q-p} \big)\,.
$$

Since the right hand sides of equations \eqref{cD-st-mc}, \eqref{cD-st-mo-k},
and \eqref{cD-st-mo-nk} are quadratic in generators, we conclude that the collection
${\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}}$ is a pseudo-cooperad and
$$
{\mathsf{OC}} = {\mathrm{Cobar} }({\mathfrak{o} \mathfrak{c}}^{\vee})\,,
$$
where ${\mathfrak{o} \mathfrak{c}}^{\vee}$ is the  2-colored coaugmented cooperad obtained from ${\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}}$
via ``adjoining the counit''.

Finally, we recall that algebras over ${\mathsf{OC}}$ (a.k.a. open-closed homotopy algebras)
are pairs of cochain complexes $({\cal V}, {\cal A})$ with
the following data\footnote{Recall that introducing a ${\Lambda}{\mathsf{Lie}}_{\infty}$-structure on ${\cal V}$ is
equivalent to introducing an $L_{\infty}$-structure on ${\mathbf{s}^{-1}\,} {\cal V}$\,.}:

\begin{itemize}

\item A ${\Lambda}{\mathsf{Lie}}_{\infty}$-structure on ${\cal V}$,

\item an $A_{\infty}$-structure on ${\cal A}$, and

\item a ${\Lambda}{\mathsf{Lie}}_{\infty}$-morphism from ${\cal V}$ to the
Hochschild cochain complex $C^{\bullet}({\cal A})$ of ${\cal A}$\,.

\end{itemize}

\subsection{The operad ${\mathsf{dGra} }$ and its $2$-colored version ${\mathsf {KGra} }$ }

To define the operad ${\mathsf{dGra} }$, we introduce a collection of
auxiliary sets $\{{\rm d g r a }(n) \}_{n \ge 1}$\,.

An element of ${\rm d g r a }_{n}$ is a directed labelled graph ${\Gamma}$
with $n$ vertices and with the additional piece
of data: the set of edges of ${\Gamma}$ is equipped with a
total order. An example of an element in ${\rm d g r a }_5$ is
shown on figure \ref{fig:exam}.
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=0.5, >=stealth']
\tikzstyle{w}=[circle, draw, minimum size=3, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=3, inner sep=1]
\node [b] (b1) at (0,0) {};
\draw (-0.4,0) node[anchor=center] {{\small $1$}};
\node [b] (b3) at (2,0) {};
\draw (2, 0.5) node[anchor=center] {{\small $3$}};
\node [b] (b2) at (6,0) {};
\draw (5.9, 0.6) node[anchor=center] {{\small $2$}};
\node [b] (b4) at (2,2) {};
\draw (2, 2.6) node[anchor=center] {{\small $4$}};
\node [b] (b5) at (5,2) {};
\draw (5, 2.6) node[anchor=center] {{\small $5$}};
\draw [->] (b3) edge (b1);
\draw (1,0.4) node[anchor=center] {{\small $i$}};
\draw [->] (b3) ..controls (3,0.5) and (5,0.5) .. (b2);
\draw (4,0.8) node[anchor=center] {{\small $ii$}};
\draw [<-] (b3) ..controls (3,-0.5) and (5,-0.5) .. (b2);
\draw (4,-0.8) node[anchor=center] {{\small $iii$}};
\draw (6.5,0) circle (0.5);
\draw (7.4,0) node[anchor=center] {{\small $iv$}};
\end{tikzpicture}
\caption{Roman numerals indicate that
$(3,1) < (3,2) < (2,3) < (2,2)$ } \label{fig:exam}
\end{figure}
Here, we use roman numerals to specify a total order on a set of edges.
For example, the roman numerals on figure \ref{fig:exam} indicate that
$(3,1) < (3,2) < (2,3) < (2,2)$\,.

Next, we introduce a collection of graded vector spaces
$\{{\mathsf{dGra} }(n) \}_{n \ge 1}$\,.
The space  ${\mathsf{dGra} }(n)$ is spanned by elements of
${\rm d g r a }_n$, modulo the relation ${\Gamma}^{{\sigma}} = (-1)^{|{\sigma}|} {\Gamma}$\,,
where the graphs ${\Gamma}^{{\sigma}}$ and ${\Gamma}$ correspond to the same
directed labelled graph but differ only by permutation ${\sigma}$
of edges. We also declare that
the degree of a graph ${\Gamma}$ in ${\mathsf{dGra} }(n)$ equals
$-e({\Gamma})$, where $e({\Gamma})$ is the number of edges in ${\Gamma}$\,.
For example, the graph ${\Gamma}$ on figure \ref{fig:exam} has
$4$ edges. Thus its degree is $-4$\,.

Following \cite{Thomas}, the collection $\{{\mathsf{dGra} }(n) \}_{n \ge 1}$
forms an operad in the category of graded vector spaces.
The symmetric group $S_n$ acts on ${\mathsf{dGra} }(n)$ in the
obvious way by rearranging labels and the operadic
multiplications are defined in terms of natural operations
of erasing vertices and attaching edges to vertices.

The operad ${\mathsf{dGra} }$ upgrades naturally to a 2-colored operad
${\mathsf {KGra} }$ whose spaces are finite linear combinations
of graphs used by M. Kontsevich in \cite{K}.

For ${\mathsf {KGra} }$, we declare that ${\mathsf {KGra} }(n,k)^{{\mathfrak{c}}} = {\mathbf{0} }  $ whenever $k \ge 1$\,.

For the space ${\mathsf {KGra} }(n,0)^{{\mathfrak{c}}}$ ($n \ge 0$) we have
\begin{equation}
\label{KGra-mc}
{\mathsf {KGra} }(n,0)^{{\mathfrak{c}}} = {\mathsf{dGra} }(n)\,.
\end{equation}

Finally, to define the space ${\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ we introduce
the auxiliary set ${\rm d g r a }_{n,k}$\,.
An element of the set ${\rm d g r a }_{n,k}$ is a directed labelled graph ${\Gamma}$
with $n$ vertices of color ${\mathfrak{c}}$, $k$ vertices of color ${\mathfrak{o}}$, and
with the following data: the set of edges of ${\Gamma}$ is equipped with a
total order.
In addition, we require that
each graph ${\Gamma} \in {\rm d g r a }_{n,k}$ has no edges originating from any vertex with
color ${\mathfrak{o}}$\,.

\begin{example}
\label{ex:KGra-mc-mo}
Figure \ref{fig:exam-mc-mo} shows an
example of a graph in ${\rm d g r a }_{2,3}$. Black (resp. white) vertices carry
the color ${\mathfrak{c}}$ (resp. ${\mathfrak{o}}$). We use separate labels for vertices of
color ${\mathfrak{c}}$ and vertices of color ${\mathfrak{o}}$\,. For example $2_{{\mathfrak{c}}}$
denotes the vertex of color ${\mathfrak{c}}$ with label $2$ and
$3_{{\mathfrak{o}}}$ denotes the vertex of color ${\mathfrak{o}}$ with label $3$\,.
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=0.5, >=stealth']
\tikzstyle{w}=[circle, draw, minimum size=4, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=4, inner sep=1]
\node [w] (w2) at (0,0) {};
\draw (0,-0.6) node[anchor=center] {{\small $2$}};
\node [w] (w1) at (1,0) {};
\draw (1,-0.6) node[anchor=center] {{\small $1$}};
\node [w] (w3) at (2,0) {};
\draw (2,-0.6) node[anchor=center] {{\small $3$}};
\node [b] (b1) at (0,2) {};
\draw (0,2.6) node[anchor=center] {{\small $1$}};
\node [b] (b2) at (2,2) {};
\draw (2,2.6) node[anchor=center] {{\small $2$}};
\draw [->] (b1) edge (w1);
\draw [->] (b1) edge (b2);
\draw [->] (b2) edge (w1);
\draw [->] (b2) edge (w3);
\end{tikzpicture}
\caption{We equip the edges with the order
$(1_{{\mathfrak{c}}}, 2_{{\mathfrak{c}}}) < (1_{{\mathfrak{c}}}, 1_{{\mathfrak{o}}}) <  (2_{{\mathfrak{c}}}, 1_{{\mathfrak{o}}})
<  (2_{{\mathfrak{c}}}, 3_{{\mathfrak{o}}}) $ } \label{fig:exam-mc-mo}
\end{figure}
\end{example}

The space ${\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ is spanned by elements of
${\rm d g r a }_{n,k}$, modulo the relation ${\Gamma}^{{\sigma}} = (-1)^{|{\sigma}|} {\Gamma}$\,,
where the graphs ${\Gamma}^{{\sigma}}$ and ${\Gamma}$ correspond to the same
directed labelled graph
 but differ only by permutation ${\sigma}$
of edges.
As above, we declare that
the degree of a graph ${\Gamma}$ in ${\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ equals $-e({\Gamma})$\,,
where $e({\Gamma})$ is the total number of edge of ${\Gamma}$\,.

The operadic structure on the resulting $2$-colored collection ${\mathsf {KGra} }$
is defined in the similar way to that on ${\mathsf{dGra} }$.
For more details, we refer the reader to \cite[Section 3]{stable1}.

Several vectors of the operad ${\mathsf {KGra} }$ will play a special role in our consideration.
For this reason, we will reserve separate symbols for these vectors:
\begin{equation}
\label{G-edge}
 \begin{tikzpicture}[scale=0.5, >=stealth']
\tikzstyle{w}=[circle, draw, minimum size=4, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=4, inner sep=1]
\draw (-2,0) node[anchor=center] {{${\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} = $}};
\node [b] (b1) at (0,0) {};
\draw (0,0.6) node[anchor=center] {{\small $1$}};
\node [b] (b2) at (1.5,0) {};
\draw (1.5,0.6) node[anchor=center] {{\small $2$}};
\draw [->] (b1) edge (b2);
\draw (2.5,0) node[anchor=center] {{$+$}};
\node [b] (bb1) at (3.5,0) {};
\draw (3.5,0.6) node[anchor=center] {{\small $1$}};
\node [b] (bb2) at (5,0) {};
\draw (5,0.6) node[anchor=center] {{\small $2$}};
\draw [<-] (bb1) edge (bb2);
\end{tikzpicture}
\end{equation}
\begin{equation}
\label{binary}
{\Gamma}_{{\bullet\, \bullet}} =   \begin{tikzpicture}[scale=0.5, >=stealth']
\tikzstyle{w}=[circle, draw, minimum size=4, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=4, inner sep=1]
\node [b] (b1) at (0,0) {};
\draw (0,0.6) node[anchor=center] {{\small $1$}};
\node [b] (b2) at (1.5,0) {};
\draw (1.5,0.6) node[anchor=center] {{\small $2$}};
\end{tikzpicture}
\qquad \quad
{\Gamma}_{{\circ\, \circ}} =   \begin{tikzpicture}[scale=0.5, >=stealth']
\tikzstyle{w}=[circle, draw, minimum size=4, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=4, inner sep=1]
\node [w] (w1) at (0,0) {};
\draw (0,0.6) node[anchor=center] {{\small $1$}};
\node [w] (w2) at (1.5,0) {};
\draw (1.5,0.6) node[anchor=center] {{\small $2$}};
\draw (w1) (w2);
\end{tikzpicture}
\end{equation}
and the series of ``brooms'' for $k \ge 0$ depicted on
figure \ref{fig:brooms}.
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=0.5, >=stealth']
\tikzstyle{w}=[circle, draw, minimum size=4, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=4, inner sep=1]
\draw (-3,1.5) node[anchor=center] {{${\Gamma}^{{\mathrm{br}}}_{k}  \quad = \quad $}};
\node [b] (v) at (2.5,3) {};
\node [w] (v1) at (0,0) {};
\node [w] (v2) at (1,0) {};
\node [w] (vk) at (4,0) {};
\draw (2.5,0.7) node[anchor=center] {{\small $\dots$}};
\draw [->] (v) edge (1,1.2);
\draw  (1,1.2) edge (v1);
\draw [->] (v) edge (1.5,1);
\draw  (1.5,1) edge (v2);
\draw [->] (v) edge (3.5,1);
\draw  (3.5,1) edge (vk);
\draw (2.5,3.6) node[anchor=center] {{\small $1$}};
\draw (0,-0.6) node[anchor=center] {{\small $1$}};
\draw (1,-0.6) node[anchor=center] {{\small $2$}};
\draw (4,-0.6) node[anchor=center] {{\small $k$}};
\end{tikzpicture}
~\\[0.3cm]
\caption{Edges are ordered in this way
$(1_{{\mathfrak{c}}}, 1_{{\mathfrak{o}}}) < (1_{{\mathfrak{c}}}, 2_{{\mathfrak{o}}}) < \dots < (1_{{\mathfrak{c}}}, k_{{\mathfrak{o}}})$
} \label{fig:brooms}
\end{figure}

Note that the graph ${\Gamma}^{{\mathrm{br}}}_{0} \in {\mathsf {KGra} }(1,0)^{{\mathfrak{o}}}$
consists of a single black vertex labeled by $1$ and it has no edges.

\subsection{Stable formality quasi-isomorphisms}
\label{sec:SFQ}

We recall from \cite{stable1} that
\begin{defi}
\label{dfn:stable}
A \emph{stable formality quasi-isomorphism} (SFQ) is a morphism of
2-colored operads in the category of cochain complexes
\begin{equation}
\label{eq:stable}
F :  {\mathsf{OC}} \to {\mathsf {KGra} }
\end{equation}
satisfying the following ``boundary conditions'':
\begin{equation}
\label{Schouten}
F({\mathsf{t}}^{{\mathfrak{c}}}_n) = \begin{cases}
   {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} \qquad {\rm if} ~~ n = 2\,,  \\
   0 \qquad {\rm if} ~~  n \ge 3\,,
\end{cases}
\end{equation}
\begin{equation}
\label{mult}
F({\mathsf{t}}^{{\mathfrak{o}}}_2) = {\Gamma}_{{\circ\, \circ}}\,,
\end{equation}
and
\begin{equation}
\label{HKR}
F({\mathsf{t}}^{{\mathfrak{o}}}_{1,k}) = \frac{1}{k!} {\Gamma}^{{\mathrm{br}}}_{k}\,,
\end{equation}
where ${\mathsf{t}}^{{\mathfrak{c}}}_n$, ${\mathsf{t}}^{{\mathfrak{o}}}_k$, and ${\mathsf{t}}^{{\mathfrak{o}}}_{n,k}$ are corollas
depicted
on figures \ref{fig:bt-n-oc-mc}, \ref{fig:bt-k-oc-mo}, \ref{fig:bt-nk-oc-mo},
respectively.
\end{defi}

Following \cite[Section 5.1]{stable1}, we identify SFQs with MC elements ${\alpha}$
of the graded Lie algebra
\begin{equation}
\label{Conv-oc-o-KGra}
{\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}}, {\mathsf {KGra} })
\end{equation}
subject to the three conditions:
\begin{equation}
\label{al-Schouten}
{\alpha}({\mathbf{s}}^{-1}\, {\mathsf{t}}^{{\mathfrak{c}}}_n) = \begin{cases}
   {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} \qquad {\rm if} ~~ n = 2\,,  \\
   0 \qquad {\rm if} ~~  n \ge 3\,,
\end{cases}
\end{equation}
\begin{equation}
\label{al-mult}
{\alpha}({\mathbf{s}}^{-1}\, {\mathsf{t}}^{{\mathfrak{o}}}_2) = {\Gamma}_{{\circ\, \circ}}\,,
\end{equation}
and
\begin{equation}
\label{al-HKR}
{\alpha}({\mathbf{s}}^{-1} \, {\mathsf{t}}^{{\mathfrak{o}}}_{1,k}) = \frac{1}{k!} {\Gamma}^{{\mathrm{br}}}_{k}\,,
\end{equation}

We should remark that, since all vectors in ${\mathsf {KGra} }(0,k)^{{\mathfrak{o}}}$
have degree zero, we have
\begin{equation}
\label{al-mult-higher}
{\alpha}({\mathbf{s}}^{-1}\, {\mathsf{t}}^{{\mathfrak{o}}}_k) = 0\,,
\end{equation}
for all $k\ge 3$ and for all degree $1$ elements
${\alpha}$ in \eqref{Conv-oc-o-KGra}.

\subsection{Kontsevich's SFQ}
\label{sec:Konts}
The first example of a stable formality quasi-isomorphism
(over ${\mathbb R}$) was constructed in \cite{K} by M. Kontsevich.

In paper \cite{K}, M. Kontsevich did not use the language of operads.
However this language is very convenient for defining the action of the graph complex \cite[Section 5]{K-conj}
on formality quasi-isomorphisms and for describing the set of homotopy classes of formality
quasi-isomorphisms in the ``stable setting'' \cite{stable1}.

Let us briefly recall here Kontsevich's construction of a particular example of an SFQ.

For a pair of integers $(n,k)$, $n \ge 0$, $k \ge 0$ satisfying the inequality
$2n + k  \ge 2 $, we denote by ${\mathsf{Conf}}_{n,k}$ the configuration space of
$n$ labeled points in the upper half plane and $k$ labeled points on the
real line:
\begin{multline}
\label{Conf-n-k}
{\mathsf{Conf}}_{n,k} : =
\big\{  ( p_1, p_2, \dots, p_n; q_1, q_2, \dots, q_k) ~\big|~ p_i \in {\mathbb C}, ~~ \Im(p_i)  > 0, ~~
q_j \in {\mathbb R}, \\
p_{i_1} \neq p_{i_2}  ~ \textrm{ for } ~ i_1 \neq i_2\,,
 ~~ q_{j_1} \neq q_{j_2}  ~ \textrm{ for } ~ j_1 \neq j_2\,, \big\}\,.
\end{multline}

Let us denote by $G^{(1)}$ the 2-dimensional connected Lie group of
the following transformations of the complex plane:
\begin{equation}
\label{G-1}
G^{(1)} : = \{z \mapsto a z + b  ~|~ a, b \in {\mathbb R}, ~~ a > 0 \}\,.
\end{equation}

The condition $2n+k \ge 2$ guarantees that the diagonal action of
$G^{(1)}$ on ${\mathsf{Conf}}_{n,k}$ is free and hence the quotient
\begin{equation}
\label{C-n-k}
C_{n,k} : =  {\mathsf{Conf}}_{n,k} ~\big/~ G^{(1)}
\end{equation}
is smooth real manifold of dimension $2n+k - 2$\,.

We denote by $\overline{C}_{n,k}$ the compactification of $C_{n,k}$ constructed
by M. Kontsevich in \cite[Section 5]{K}. $\overline{C}_{n,k}$ comes with an involved
stratification which is described in great detail in {\it loc. cit.}

Let ${\Gamma}$ be a graph in ${\rm d g r a }_{n,k}$ and $e$ be an edge of ${\Gamma}$ which
originates at the black\footnote{Let us recall that, by definition of ${\rm d g r a }_{n,k}$
every edge $e$ of ${\Gamma} \in {\rm d g r a }_{n,k}$ should
originate at a vertex with color ${\mathfrak{c}}$ (i.e. black vertex).} vertex with label $i$.
To such an edge $e$, we assign a 1-form $d {\varphi}_{e}$
on ${\mathsf{Conf}}_{n,k}$ by the following rule:
\begin{itemize}

\item if the edge $e$ terminates at the black vertex with label $j$ then
$$
d {\varphi}_e : =  d {\mathrm{Arg}} (p_j - p_i) - d {\mathrm{Arg}}  (p_j - \bar{p}_i),
$$

\item  if the edge $e$ terminates at the white vertex with label $j$ then
$$
d {\varphi}_e : =  d {\mathrm{Arg}} (q_j - p_i) - d {\mathrm{Arg}}  (q_j - \bar{p}_i)\,.
$$

\end{itemize}

It is easy to see that $d {\varphi}_e$ descends to a 1-form on $C_{n,k}$.
Furthermore, $d {\varphi}_e$ extends to a smooth $1$-form on Kontsevich's
compactification $\overline{C}_{n,k}$ of $C_{n,k}$.

Using the embedding
$$
{\mathsf{Conf}}_{n,k} \subset {\mathbb C}^n \times {\mathbb R}^k
$$
we equip the manifold ${\mathsf{Conf}}_{n,k}$ with the natural orientation
which descends to $C_{n,k}$ and extends to $\overline{C}_{n,k}$.

To every element ${\Gamma} \in {\rm d g r a }_{n,k}$, we assign the following  weight:
\begin{equation}
\label{W-G}
W_{{\Gamma}} : = \frac{1}{(2\pi)^{2n + k -2}} ~\int_{\overline{C}^+_{n,k}}~ \bigwedge_{e \in E({\Gamma})}  d {\varphi}_{e}\,,
\end{equation}
where $\overline{C}^+_{n,k}$ is a connected component of $\overline{C}_{n,k}$ which is the closure of
the subspace of configurations singled out by the condition
$$
q_1 < q_2 < \dots < q_k\,,
$$
and the order of $1$-forms in $\displaystyle \bigwedge_{e \in E({\Gamma})}  d {\varphi}_{e}$
agrees with the total order on the set of edges of ${\Gamma}$\,. It is clear that the weight
$W_{{\Gamma}}$ for  ${\Gamma} \in {\rm d g r a }_{n,k}$ is non-zero only if the total number of
edges of ${\Gamma}$ equals $2n+ k - 2$\,.

Let us observe that the set ${\rm d g r a }_{n,k}$ carries an obvious equivalence
relation: two elements ${\Gamma}$ and ${\Gamma}'$ in ${\rm d g r a }_{n,k}$ are equivalent if and
only if they have the same underlying labeled (colored) graph. We denote
by
$$
[{\rm d g r a }_{n,k}]
$$
the set of corresponding equivalence classes.

Finally we define a degree $1$ element ${\alpha}^K \in {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}}, {\mathsf {KGra} })$ by
setting
\begin{equation}
\label{al-K-Schouten}
{\alpha}^K({\mathbf{s}}^{-1}\, {\mathsf{t}}^{{\mathfrak{c}}}_n) : = \begin{cases}
   {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} \qquad {\rm if} ~~ n = 2\,,  \\
   0 \qquad {\rm if} ~~  n \ge 3\,,
\end{cases}
\end{equation}
\begin{equation}
\label{al-K-mult}
{\alpha}^K({\mathbf{s}}^{-1}\, {\mathsf{t}}^{{\mathfrak{o}}}_2) : = {\Gamma}_{{\circ\, \circ}}\,,
\end{equation}
and
\begin{equation}
\label{al-K-main}
{\alpha}^K ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k}) : = \sum_{z \in [{\rm d g r a }_{n,k} ] } W_{{\Gamma}_z} \, {\Gamma}_z\,,
\end{equation}
where ${\Gamma}_{z}$ is any representative of the equivalence class $z \in [{\rm d g r a }_{n,k}]$\,.

We observe that, since the vector
$$
W_{{\Gamma}} \, {\Gamma} \in {\mathbb K} \L {\rm d g r a }_{n,k} \rangle
$$
depends only on the equivalence class of ${\Gamma}$\,, the right hand side
of \eqref{al-K-main} does not depend on the choice of representatives
${\Gamma}_z$\,.

A direct computation shows that the weights of the ``brooms''
depicted on figure \ref{fig:brooms} are given by the formula
$$
W_{{\Gamma}^{{\mathrm{br}}}_k}  = \frac{1}{k!}\,.
$$
Hence ${\alpha}^K$ satisfies all the required ``boundary conditions''
\eqref{al-Schouten}, \eqref{al-mult}, \eqref{al-HKR}\,.

Following the line of arguments of \cite[Section 6.4]{K} one can
show that ${\alpha}^K$ satisfies the MC equation
$$
[{\alpha}^K, {\alpha}^K] = 0
$$
in  ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}}, {\mathsf {KGra} })$\,.
Thus ${\alpha}^K$ gives us an SFQ.
\begin{remark}
\label{rem:Weight}
The weight of a graph ${\Gamma} \in {\rm d g r a }_{n,k}$ defined in \cite[Section 6.2]{K}
comes with additional factors. These factors are absent in \eqref{W-G}
because our identification between polyvector fields and ``functions'' on the odd
cotangent bundle is different from the one used by M. Kontsevich
in \cite[Section 6.3]{K}.
\end{remark}

\subsubsection{On connectedness condition for SFQs}
\label{sec:connected}

Let ${\alpha}^K$ be the MC element of \eqref{Conv-oc-o-KGra} constructed
above, $n \ge 1$ and ${\Gamma}$ be a disconnected graph in ${\rm d g r a }_{n,k}$
with $2n+k-2$ edges.

It is not hard to see that, for every point
$P \in C_{n,k}$, the restrictions of $1$-forms
$$
\{d {\varphi}_{e}\}_{e \in E({\Gamma})}
$$
do not span the cotangent space $T^*_P C_{n,k}$\,.

Therefore, for every $n \ge 1$, the linear combination
\begin{equation}
\label{al-K-n-k}
{\alpha}^K ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k})
\end{equation}
involves only connected graphs.

In this paper we assume that degree zero vectors
$$
\xi \in {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}}, {\mathsf {KGra} })
$$
entering the definition of homotopy equivalence \cite[Definition 5.2]{stable1}
satisfy the additional condition: {\it all graphs in the linear combinations}
$$
\xi({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k})
$$
are connected.

Similarly, instead of using the full directed graph complex ${\mathsf{dfGC}}$ (see
\cite[Appendix K]{Thomas} or \cite[Section 6.1]{stable1})
we will only use its ``connected part'' ${\mathsf{dfGC}}_{{\mathrm {conn} }}$\,.

Thus we will deal exclusively with SFQs whose MC elements ${\alpha}$ satisfy the
additional condition:
\begin{cond}
\label{cond:connected}
For every $n \ge 1$ and $k \ge 0$ the linear combination
$$
{\alpha} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k})
$$
involves only connected graphs.
\end{cond}
By the argument given above, Kontsevich's SFQ satisfies this condition.

\section{The main theorem}
\label{sec:main}

Thanks to boundary conditions \eqref{al-Schouten}, \eqref{al-mult},
\eqref{al-HKR} and identity \eqref{al-mult-higher}, we know that
${\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{c}}}_n)$ and ${\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{k'})$ and
${\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{1,k})$ have rational coefficients for all
$n \ge 2$, $k' \ge 2$ and $k \ge 0$\,.

The main result of this note is the following theorem:
\begin{thm}
\label{thm:main}
There exists a sequence of MC elements
\begin{equation}
\label{al-sequence}
\{ {\alpha}_m  \}_{m \ge 1}
\end{equation}
in the dg Lie algebra \eqref{Conv-oc-o-KGra} (with real coefficients) which satisfy the
following properties:
\begin{itemize}

\item Each ${\alpha}_m$ satisfies boundary
conditions\footnote{I.e. each MC element ${\alpha}_m$ corresponds to an SFQ.}
 \eqref{al-Schouten}, \eqref{al-mult}, and \eqref{al-HKR},

\item  ${\alpha}_1 : = {\alpha}^K$,

\item ${\alpha}_m - {\alpha}_{m-1} \in {\cal F}^{{\mathfrak{c}}}_{m-1} {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$, and

\item ${\alpha}_m({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k})$ has rational coefficients for all $n \le m$\,.

\end{itemize}
\end{thm}

Since the filtration  ${\cal F}^{{\mathfrak{c}}}_{{\bullet}} {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ on
${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ is complete, Theorem \ref{thm:main} implies that
\begin{cor}
\label{cor:overQ}
There exists a stable formality quasi-isomorphism defined over rational
numbers. $\qed$
\end{cor}

A proof of Theorem \ref{thm:main} is given in Section \ref{sec:proof}
at the end of the paper. Before proceeding to a construction of the desired
sequence of MC elements, we present some required technical statements.
Most of these statements are essentially borrowed from \cite{stable1}.

\section{Technical statements}
\label{sec:tech}

\subsection{The operator ${\partial}^{{\mathrm {Hoch} }}$ and its properties}
\label{sec:diff-Hoch}

Just as in \cite{stable1}, vectors $c \in {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ singled out
by the following properties will play a special role:
\begin{pty}
\label{P:one}
All white vertices in each graph of the
linear combination $c$ have valency one.
\end{pty}
\begin{pty}
\label{P:anti-symm}
For every ${\sigma} \in S_k$ we have
\begin{equation}
\label{app-anti-symm}
({\rm  i d } , {\sigma})\, (c) = (-1)^{|{\sigma}|} c\,.
\end{equation}
\end{pty}
For example, the ``brooms'' ${\Gamma}^{{\mathrm{br}}}_k$ depicted
on figure \ref{fig:brooms} obviously satisfy these properties.

We denote by $\Pi{\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ the subspace of all
vectors in ${\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ satisfying Properties \ref{P:one} and
\ref{P:anti-symm}
\begin{equation}
\label{Pi-KGra}
\Pi{\mathsf {KGra} }(n,k)^{{\mathfrak{o}}} : = \big\{ c \in {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}} ~|~ c  \textrm{ satisfies Properties \ref{P:one} and
\ref{P:anti-symm} }   \big\}\,.
\end{equation}

For every vector $c \in  {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$, we denote by $\Pi_1(c)$
the linear combination of graphs in ${\rm d g r a }_{n,k}$ which is obtained from
$c$ by retaining only graphs whose all white vertices are univalent.
The assignment
\begin{equation}
\label{Pi-1}
c \mapsto \Pi_1(c)
\end{equation}
is obviously a linear projection from ${\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ onto the
the subspace of vectors in ${\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ which satisfy Property \ref{P:one}.

Composing $\Pi_1$ with the alternation operator
\begin{equation}
\label{Alt-mo}
{\mathrm {Alt}}^{{\mathfrak{o}}} =  \frac{1}{k!} \sum_{{\sigma} \in S_k}
(-1)^{|{\sigma}|} ({\rm  i d }, {\sigma})  ~ : ~  {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}} \to {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}
\end{equation}
we get a canonical projection
\begin{equation}
\label{Pi}
\Pi := {\mathrm {Alt}}^{{\mathfrak{o}}} \circ \Pi_1 ~:~ {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}} \to  \Pi{\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}\,.
\end{equation}

Let us recall from \cite{stable1} that, for every given $n$, the direct sum
\begin{equation}
\label{KGra-s}
\bigoplus_{k \ge 0} {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}
\end{equation}
carries the degree $0$ endomorphism ${\partial}^{{\mathrm {Hoch} }}$ defined by
the formula
\begin{equation}
\label{pa-Hoch}
{\partial}^{{\mathrm {Hoch} }} ({\gamma}) =
{\Gamma}_{{\circ\, \circ}} \,\c_{2, {\mathfrak{o}}}\, {\gamma} +
\sum_{i=1}^k (-1)^i {\gamma} \,\c_{i, {\mathfrak{o}}}\, {\Gamma}_{{\circ\, \circ}} +
(-1)^{k+1}{\Gamma}_{{\circ\, \circ}} \,\c_{1, {\mathfrak{o}}}\, {\gamma}\,,
\end{equation}
$$
{\gamma} \in {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}\,.
$$

A direct computation shows that
$$
{\partial}^{{\mathrm {Hoch} }} \circ {\partial}^{{\mathrm {Hoch} }} = 0
$$
and hence the direct sum \eqref{KGra-s} can be turned
into a cochain complex by redefining the degrees of terms
${\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ in the obvious way.

We claim that
\begin{claim}
\label{cl:Pi-Hoch}
For every vector ${\gamma} \in {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$
\begin{equation}
\label{Pi-Hoch}
{\partial}^{{\mathrm {Hoch} }} (\Pi({\gamma})) =  \Pi ( {\partial}^{{\mathrm {Hoch} }} {\gamma}) = 0\,,
\end{equation}
$\Pi({\gamma})$ belongs to the image of ${\partial}^{{\mathrm {Hoch} }}$ if and only if $\Pi({\gamma}) = 0$\,.
Furthermore, for every vector $c \in \ker({\partial}^{{\mathrm {Hoch} }})$ the difference
$$
c - \Pi(c)
$$
belongs to the image of ${\partial}^{{\mathrm {Hoch} }}$\,.
\end{claim}
\begin{proof}
The proof of equations \eqref{Pi-Hoch} is straightforward. The remaining two
statements follow directly from Proposition A.1 in \cite[Appendix A]{stable1}.
\end{proof}

\subsection{The operator ${\mathfrak{d}}$ and its properties}
Let us recall that a {\it pike} is a univalent black vertex $v$ of ${\Gamma} \in {\rm d g r a }_{n,k}$
whose adjacent edge terminates at $v$\,.
An example of a graph with a pike is shown on figure \ref{fig:pike}.
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=0.5, >=stealth']
\tikzstyle{w}=[circle, draw, minimum size=4, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=4, inner sep=1]
\node [b] (b1) at (-0.5,2) {};
\draw (-0.5,2.6) node[anchor=center] {{\small $1$}};
\node [b] (b2) at (1,2) {};
\draw (1,2.6) node[anchor=center] {{\small $2$}};
\node [b] (b3) at (2.5,2) {};
\draw (2.5,2.6) node[anchor=center] {{\small $3$}};
\node [w] (w1) at (-1,0) {};
\draw (-1,-0.6) node[anchor=center] {{\small $1$}};
\node [w] (w2) at (0,0) {};
\draw (0,-0.6) node[anchor=center] {{\small $2$}};
\node [w] (w3) at (1,0) {};
\draw (1,-0.6) node[anchor=center] {{\small $3$}};
\node [w] (w4) at (2,0) {};
\draw (2,-0.6) node[anchor=center] {{\small $4$}};
\draw [->] (b2) edge (b1);
\draw [->] (b3) edge (b2);
\draw [->] (b2) edge (w1);
\draw [->] (b2) edge (w2);
\draw [->] (b2) edge (w3);
\draw [->] (b2) edge (w4);
\end{tikzpicture}
~\\[0.3cm]
\caption{The black vertex with label $1$ is a pike but the black vertex
with label $3$ is not} \label{fig:pike}
\end{figure}

Let us consider the direct sum of spaces of invariants
\begin{equation}
\label{Hg}
\bigoplus_{n,k} \Big( \Pi {\mathsf {KGra} }(n, k)^{{\mathfrak{o}}} \Big)^{S_n}
\end{equation}
and introduce the following families of cycles
\begin{equation}
\label{tau-n-i}
\tau_{n,i} = (i, i+1, i+2, \dots, n-1, n)~  \in~ S_n\,,
\end{equation}
\begin{equation}
\label{si-k-i}
{\sigma}_{k,i} = (i, i-1,  \dots, 2, 1) ~ \in ~ S_k\,.
\end{equation}

Next, we recall the following endomorphism of \eqref{Hg} from
\cite[Appendix B]{stable1}
\begin{equation}
\label{md}
{\mathfrak{d}}({\gamma}) =  k \, \sum_{i=1}^{n+1} \, (\tau_{n+1,i}, {\rm  i d })
\big(\, {\gamma}  \,\c_{1, {\mathfrak{o}}}\, {\Gamma}^{{\mathrm{br}}}_{0}
\, \big)\,, \qquad  {\gamma} \in \big( \Pi{\mathsf {KGra} }(n,k)^{{\mathfrak{o}}} \big)^{S_n}\,.
\end{equation}
Notice that, since the graph ${\Gamma}^{{\mathrm{br}}}_0$ consists of a single
black vertex and has no edges, the insertion $\c_{1, {\mathfrak{o}}}$
of ${\Gamma}^{{\mathrm{br}}}_{0}$ replaces the white vertex with label $1$
by a black vertex with label $n+1$ and shifts
the labels on the remaining white vertices
down by $1$\,.

Since the linear combination
${\gamma} \in \Pi {\mathsf {KGra} }(n, k)^{{\mathfrak{o}}}$ is anti-symmetric with respect
to permutations of labels on white vertices, we have
\begin{equation}
\label{md-sq}
{\mathfrak{d}}^2  = 0\,.
\end{equation}
Thus, shifting the degrees of summands in \eqref{Hg} in an appropriate
way, one can turn  \eqref{Hg} into a cochain complex with ${\mathfrak{d}}$ being
the differential. This cochain complex was analyzed in \cite[Appendix B]{stable1}.

For our purposes, we need another endomorphism ${\mathfrak{d}}^*$ of
\eqref{Hg} which we will now define.
Let ${\gamma} $ be a vector in  $\big( \Pi{\mathsf {KGra} }(n,k)^{{\mathfrak{o}}} \big)^{S_n} $\,.
To compute ${\mathfrak{d}}^*({\gamma})$ we follow these steps:

\begin{itemize}

\item First, we omit in ${\gamma}$ all graphs for which the black
vertex with label $1$ is not a pike. We denote
the resulting linear combination in
$ {\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ by ${\gamma}'$\,.

\item Second, we replace the black vertex with label $1$
in each graph of ${\gamma}'$ by a white vertex and shift all labels
on black vertices down by $1$\,.
We assign label $1$
to this additional white vertex and shift the labels of the remaining
white vertices up by $1$\,. We denote the resulting
linear combination in
$ {\mathsf {KGra} }(n-1,k+1)^{{\mathfrak{o}}}$ by ${\gamma}''$\,.

\item Finally, we set
\begin{equation}
\label{md-star-ga}
{\mathfrak{d}}^*({\gamma}) = \sum_{i = 1}^{k+1} \frac{(-1)^{i-1}}{k+1}
({\rm  i d } , {\sigma}_{k+1, i}) ({\gamma}'')\,.
\end{equation}

\end{itemize}
Note that the linear combination ${\gamma}'$ is invariant with respect to the
action of the group $S_{\{2, 3, \dots, n\}} \times \{{\rm  i d }\}$\,. Hence, the linear combination
${\mathfrak{d}}^*({\gamma})$ is $S_{n-1}$-invariant. Furthermore, ${\mathfrak{d}}^*({\gamma})$ obviously
satisfies Properties \ref{P:one} and \ref{P:anti-symm}.

Lemma B.3 from \cite[Appendix B]{stable1} says that for every
$$
{\gamma} \in \big( \Pi{\mathsf {KGra} }(n,k)^{{\mathfrak{o}}} \big)^{S_n}
$$
we have
\begin{equation}
\label{md-md-star}
{\mathfrak{d}} {\mathfrak{d}}^* ({\gamma}) + {\mathfrak{d}}^* {\mathfrak{d}} ({\gamma}) = k {\gamma} + \sum_{r \ge 1} r \, {\gamma}_r\,,
\end{equation}
where ${\gamma}_r$ is obtained from the linear combination ${\gamma}$ be retaining only the graphs
with exactly $r$ pikes.

\subsection{An auxiliary lemma}
\label{sec:lemma}
To construct the MC element ${\alpha}_2$ in sequence \eqref{al-sequence},
we need the following lemma:
\begin{lem}
\label{lem:al-K-2-OK}
If ${\alpha}^{K}$ is the MC element of  ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ defined
by equations \eqref{al-K-Schouten}, \eqref{al-K-mult}, and \eqref{al-K-main}
then
\begin{equation}
\label{al-K-2-OK}
\Pi \big( {\alpha}^{K}({\mathbf{s}^{-1}\,} {\mathsf{t}}_{2,k}) \big) = 0
\end{equation}
for every $k \ge 0$\,.
\end{lem}
\begin{proof}
Since the case $k=0$ is already established in \cite[Subsection 7.3.1.1]{K},
we proceed by induction on $k$.

Let us observe that for every $k \ge 0$, the vector ${\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}_{2,k})$ in
${\mathsf {KGra} }(2, k)^{{\mathfrak{o}}}$ has degree $-2-k$\,. Therefore each graph in the
linear combination  ${\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}_{2,k})$ has $k+2$ edges and hence every
graph in  ${\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}_{2,k})$ does not have a pike.

Next we assume, by induction, that equation \eqref{al-K-2-OK} holds for every $k < q$\,.
Moreover, since every graph in $\Pi({\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}_{2,q}))$ has $q+2$ edges,  $\Pi({\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}_{2,q}))$
must be a linear combination of graphs
shown on figure \ref{fig:G-2-q}.
\begin{figure}[htp]
\centering
\begin{tikzpicture}[scale=0.5, >=stealth']
\tikzstyle{w}=[circle, draw, minimum size=4, inner sep=1]
\tikzstyle{b}=[circle, draw, fill, minimum size=4, inner sep=1]
\node [b] (b1) at (1,4) {};
\draw (1,4.6) node[anchor=center] {{\small $1$}};
\node [b] (b2) at (3,4) {};
\draw (3,4.6) node[anchor=center] {{\small $2$}};

\node [w] (vi1) at (-2,0) {};
\draw (-2,-0.6) node[anchor=center] {{\small $i_1$}};
\node [w] (vi2) at (-1,0) {};
\draw (-1,-0.6) node[anchor=center] {{\small $i_2$}};
\draw (0,0) node[anchor=center] {{$\dots$}};
\node [w] (vip) at (1,0) {};
\draw (1,-0.6) node[anchor=center] {{\small $i_p$}};

\node [w] (vj1) at (3,0) {};
\draw (3,-0.6) node[anchor=center] {{\small $j_1$}};
\node [w] (vj2) at (4,0) {};
\draw (4,-0.6) node[anchor=center] {{\small $j_2$}};
\draw (5,0) node[anchor=center] {{$\dots$}};
\node [w] (vjqp) at (6,0) {};
\draw (6.4,-0.6) node[anchor=center] {{\small $j_{q-p}$}};
\draw [->] (b1) ..controls (1.5,4.5) and (2.5,4.5) .. (b2);
\draw [<-] (b1) ..controls (1.5,3.5) and (2.5,3.5) .. (b2);
\draw [->] (b1) edge (vi1);
\draw [->] (b1) edge (vi2);
\draw [->] (b1) edge (vip);
\draw [->] (b2) edge (vj1);
\draw [->] (b2) edge (vj2);
\draw [->] (b2) edge (vjqp);
\end{tikzpicture}
~\\[0.3cm]
\caption{Here $\{i_1, i_2,  \dots, i_p\} \sqcup
\{j_1, j_2, \dots, j_{q-p}\}$ is a partition of the set
$\{1, 2, \dots, q\}$ } \label{fig:G-2-q}
\end{figure}

Using the above inductive assumption, boundary conditions \eqref{al-Schouten},
\eqref{al-mult}, \eqref{al-HKR} and identity \eqref{al-mult-higher},  we see that
$\mu \circ ( {\alpha}^{K} {\mathbf{s}^{-1}\,} \otimes  {\alpha}^{K} {\mathbf{s}^{-1}\,})\circ  {\cal D}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{3, q-1}) $
is the following sum
\begin{equation}
\label{with-pikes}
(-1)^{q-1}  {\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, q-1}) \circ_{1, {\mathfrak{c}}} {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} +
\end{equation}
\begin{equation}
\label{Hoch-line}
{\Gamma}_{{\circ\, \circ}} \circ_{2, {\mathfrak{o}}}  {\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{3, q-2})  +
\sum_{p=1}^{q-2} (-1)^{p}
{\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{3, q-2}) \circ_{p, {\mathfrak{o}}} {\Gamma}_{{\circ\, \circ}}
+ (-1)^{q-1} {\Gamma}_{{\circ\, \circ}} \circ_{1, {\mathfrak{o}}}  {\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{3, q-2})
\end{equation}
\begin{equation}
\label{pike-line}
+ \sum_{p=1}^q (-1)^p \big({\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, q}) \circ_{p, {\mathfrak{o}}} {\Gamma}^{{\mathrm{br}}}_0
+ \textrm{ cyclic. perm. in }\{1_{{\mathfrak{c}}}, 2_{{\mathfrak{c}}}, 3_{{\mathfrak{c}}}\}  \big)+
\end{equation}
\begin{equation}
\label{the-rest}
+ \sum_{k = 1}^{q-1} \sum_{p = 1}^{q-k} \frac{(-1)^{p+ k (q-p-k)}}{k!} {\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, q-k}) \circ_{p, {\mathfrak{o}}} {\Gamma}^{{\mathrm{br}}}_k
\end{equation}
\begin{equation}
\label{the-rest1}
+ \sum_{k = 1}^{q-1} \sum_{p = 1}^{q-k} \frac{(-1)^{p+ k (q-p-k)}}{(q-k)!}
{\Gamma}^{{\mathrm{br}}}_{q-k} \circ_{p, {\mathfrak{o}}}  {\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, k})\,.
\end{equation}

Since ${\alpha}^K$ is a MC element in ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$,
$$
\mu \circ ( {\alpha}^{K} {\mathbf{s}^{-1}\,} \otimes  {\alpha}^{K} {\mathbf{s}^{-1}\,})\circ  {\cal D}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{3, q-1}) = 0\,.
$$
Hence, applying the projection $\Pi$ \eqref{Pi} to the sum of expressions
\eqref{with-pikes}, \eqref{Hoch-line}, \eqref{pike-line}, \eqref{the-rest}, and
\eqref{the-rest1}, we must get zero.

Expression \eqref{Hoch-line} belongs to the image of ${\partial}^{{\mathrm {Hoch} }}$ \eqref{pa-Hoch}.
Hence, due to \eqref{Pi-Hoch}, this expression
may be discarded. Applying $\Pi$ to expression \eqref{with-pikes} we get zero
by the inductive assumption.

Retaining only graphs with pikes in the linear combination
\begin{equation}
\label{Pi-pike-line}
 \sum_{p=1}^q (-1)^p \big(\Pi({\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, q}) \circ_{p, {\mathfrak{o}}} {\Gamma}^{{\mathrm{br}}}_0)
+ \textrm{ cyclic. perm. in }\{1_{{\mathfrak{c}}}, 2_{{\mathfrak{c}}}, 3_{{\mathfrak{c}}}\}  \big)
\end{equation}
we get the sum
\begin{equation}
\label{pike-Pi-pike-line}
(- q) \big( \Pi({\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, q})) \big) \circ_{1, {\mathfrak{o}}} {\Gamma}^{{\mathrm{br}}}_0
+ \textrm{ cyclic. perm. in }\{1_{{\mathfrak{c}}}, 2_{{\mathfrak{c}}}, 3_{{\mathfrak{c}}}\}\,.
\end{equation}

Finally, it is easy to see that expressions  \eqref{the-rest} and
\eqref{the-rest1} do not involve graphs with pikes.

Thus we conclude that the sum in \eqref{pike-Pi-pike-line} must be zero.
Since the linear combination $\Pi({\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, q}))$ does not involve graphs with
pikes, the latter conclusion implies that
\begin{equation}
\label{al-K-2-upshot}
\Pi( {\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, q}) ) =0\,.
\end{equation}

Lemma \ref{lem:al-K-2-OK} is proved.
\end{proof}

\begin{remark}
\label{rem:to-Stokes}
We would like to observe that, for large values of $q$, it is hard to prove equation
\eqref{al-K-2-upshot} by direct computations of integrals entering Kontsevich's
formula. Of course, at the end of the day, equation \eqref{al-K-2-upshot}  is
a consequence of the Stokes theorem for the compactified configuration spaces
$\overline{C}_{n,k}$\,.
\end{remark}

\section{Correcting Kontsevich's MC element ${\alpha}^{K}$ step by step}
\label{sec:step-by-step}

\subsection{Construction of the MC element ${\alpha}_2$}
\label{sec:al-2}

Evaluating the left hand side of the MC equation for ${\alpha}^{K}$ on ${\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2,k}$
and using boundary conditions \eqref{al-Schouten},
\eqref{al-mult}, \eqref{al-HKR} together with identity \eqref{al-mult-higher}, we
deduce that
\begin{equation}
\label{level-2-k}
{\partial}^{{\mathrm {Hoch} }} {\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2,k-1}) = -
\frac{(-1)^k}{k!} {\Gamma}^{{\mathrm{br}}}_k \circ_{1, {\mathfrak{c}}} {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} +
\end{equation}
$$
+ \sum_{0 \le p \le q \le k} \frac{(-1)^{p + (k-q)(q-p)}}{(k-q+p+1)!  (q-p)! } \big( {\Gamma}^{{\mathrm{br}}}_{k-q+p+1} \circ_{p+1}
{\Gamma}^{{\mathrm{br}}}_{q-p} + ({\sigma}_{12}, {\rm  i d })  ({\Gamma}^{{\mathrm{br}}}_{k-q+p+1} \circ_{p+1} {\Gamma}^{{\mathrm{br}}}_{q-p} ) \big)\,,
$$
where ${\sigma}_{12}$ is the transposition $(1,2)$ and ${\partial}^{{\mathrm {Hoch} }}$ is defined in \eqref{pa-Hoch}.

We observe that for every triple $n,k,e$ the set ${\rm d g r a }_{n,k}$ contains finitely many graphs
with $e$ edges. Hence, for every fixed $k \ge 1$, equation \eqref{level-2-k} is a finite size
inhomogeneous linear system of equations on coefficients in ${\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2,k-1})$\,. Furthermore,
since all coefficients in the right hand side of  \eqref{level-2-k} are rational, and the system admits a solution
over reals, it also admits a solution over rationals.

Let us denote by $\beta_{2,k-1}$ a degree $-k-1$ element in ${\mathsf {KGra} }(2, k-1)^{{\mathfrak{o}}}$ which
solves \eqref{level-2-k}\,, i.e.
\begin{equation}
\label{level-2-k-beta}
{\partial}^{{\mathrm {Hoch} }} \beta_{2,k-1} = -
\frac{(-1)^k}{k!} {\Gamma}^{{\mathrm{br}}}_k \circ_{1, {\mathfrak{c}}} {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} +
\end{equation}
$$
+ \sum_{0 \le p \le q \le k} \frac{(-1)^{p + (k-q)(q-p)}}{(k-q+p+1)!  (q-p)!} \big( {\Gamma}^{{\mathrm{br}}}_{k-q+p+1} \circ_{p+1}
{\Gamma}^{{\mathrm{br}}}_{q-p} + ({\sigma}_{12}, {\rm  i d })  ({\Gamma}^{{\mathrm{br}}}_{k-q+p+1} \circ_{p+1} {\Gamma}^{{\mathrm{br}}}_{q-p} ) \big)
$$

For every $k \ge 1$ we choose a solution $\beta_{2,k-1}$ of \eqref{level-2-k-beta}
which satisfies the additional property
\begin{equation}
\label{Pi-beta-2}
\Pi(\beta_{2, k-1}) = 0\,.
\end{equation}
This is possible because $\Pi$ projects vectors of ${\mathsf {KGra} }(n,k)^{{\mathfrak{o}}}$ onto
cocycles of the differential ${\partial}^{{\mathrm {Hoch} }}$\,.

Due to Lemma \ref{lem:al-K-2-OK} and equations \eqref{level-2-k},
\eqref{level-2-k-beta}, \eqref{Pi-beta-2},
the difference
\begin{equation}
\label{difference}
{\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, k}) - \beta_{2, k}
\end{equation}
is ${\partial}^{{\mathrm {Hoch} }}$-closed and satisfies the additional condition
\begin{equation}
\label{Pi-beta-al-2}
\Pi ({\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, k}) - \beta_{2, k} ) = 0
\end{equation}
for every $k \ge 0$\,.

Therefore, Claim \ref{cl:Pi-Hoch} implies that for every $k \ge 1$ there exists a vector
$\xi_{2, k-1} \in {\mathsf {KGra} }(2, k-1)^{{\mathfrak{o}}}$ of degree $-2-k$ such that
\begin{equation}
\label{difference-exact}
{\alpha}^K({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2, k}) - \beta_{2, k} = {\partial}^{{\mathrm {Hoch} }} (\xi_{2, k-1})\,.
\end{equation}

So we define a degree zero vector $\xi_2$ in ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}}, {\mathsf {KGra} })$ by setting
\begin{equation}
\label{xi-2}
\begin{array}{c}
\xi_2({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2,k}) = \xi_{2, k}\,, \qquad \forall~~ k \ge 0\,, \\[0.3cm]
\xi_2({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k}) = 0\,, \qquad \forall~~ n \neq 2, ~~ k \ge 0\,, \\[0.3cm]
\xi_2({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{k'}) = 0\,, \qquad \forall ~~ k' \ge 2\,, \\[0.3cm]
\xi_2({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{c}}}_{n'}) = 0\,, \qquad \forall ~~ n' \ge 2\,.
\end{array}
\end{equation}

Finally, we set
\begin{equation}
\label{al-2-dfn}
{\alpha}_2 : = \exp([\xi_2, ~ ]) {\alpha}^K\,.
\end{equation}

Since $\xi_2$ satisfies the conditions of \cite[Proposition 5.1]{stable1}, the
new MC element ${\alpha}_2$ satisfies boundary conditions   \eqref{al-Schouten},
\eqref{al-mult}, \eqref{al-HKR}. Therefore, the difference
$$
{\alpha}_2 - {\alpha}^K
$$
belongs to ${\cal F}^{{\mathfrak{c}}}_1 {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$\,.

Furthermore, equation \eqref{difference-exact} implies that
$$
{\alpha}_2({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{2,k}) = \beta_{2,k}\,.
$$

Thus the desired MC element ${\alpha}_2$ is constructed.

\subsection{Pikes in ${\alpha}_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$ can be ``killed''}
Let $m$ be an integer $\ge 3$ and assume that the desired MC elements
${\alpha}_{m'}$ in \eqref{al-sequence} are constructed for all $m' \le m-1$\,.

We need to show that ${\alpha}_{m-1}$ is isomorphic to another MC element
${\alpha}'_{m-1}$ for which all graphs in the linear combination ${\alpha}'_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$
do not have pikes and
$$
{\alpha}'_{m-1} - {\alpha}_{m-1} \in {\cal F}^{{\mathfrak{c}}}_{m-1} {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}}, {\mathsf {KGra} })\,.
$$

Let us denote by ${\alpha}^r_{m-1}$ the linear combination in $\big( {\mathsf {KGra} }(m,0)^{{\mathfrak{o}}}\big)^{S_m}$
which is obtained from ${\alpha}_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$ by retaining only graphs with
exactly $r$ pikes. Since ${\alpha}^r_{m-1}$ does not have white vertices at all, we conclude that
${\alpha}^r_{m-1}$ belongs to the graded vector space \eqref{Hg} and
$$
{\mathfrak{d}} ({\alpha}^r_{m-1}) = 0\,,
$$
where the operator ${\mathfrak{d}}$ is defined in \eqref{md}.

Hence, using \eqref{md-md-star}, we deduce that for
\begin{equation}
\label{chi-1m}
\chi_{m-1}  = - \sum_{r \ge 1} \frac{1}{r} {\mathfrak{d}}^* ({\alpha}^r_{m-1})
\end{equation}
the linear combination
\begin{equation}
\label{al-1m-plus}
{\alpha}_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0}) + {\mathfrak{d}} (\chi_{m-1})
\end{equation}
does not involve graphs with pikes.

Next, using $\chi_{m-1}$ we define a degree zero vector
\begin{equation}
\label{xi}
\xi \in {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })
\end{equation}
by setting
\begin{equation}
\label{xi-pikes}
\xi ({\mathbf{s}}^{-1}\,{\mathsf{t}}^{{\mathfrak{o}}}_{m-1,1}) =  \chi_{m-1}\,,
\qquad
\xi ({\mathbf{s}}^{-1}\,{\mathsf{t}}^{{\mathfrak{c}}}_{n_1}) = \xi ({\mathbf{s}}^{-1}\, {\mathsf{t}}^{{\mathfrak{o}}}_{k_1}) =
\xi({\mathbf{s}}^{-1}\, {\mathsf{t}}^{{\mathfrak{o}}}_{n,k}) = 0
\end{equation}
for all $n_1$, $k_1$ and for all pairs $(n,k) \neq (m-1,1)$\,.

Then, we act by $\exp({\mathrm{a d}}_{\xi})$ on the MC element
${\alpha}_{m-1}$ and get a new MC element
\begin{equation}
\label{al-1m-pr}
{\alpha}'_{m-1} : = \exp({\mathrm{a d}}_{\xi})\, {\alpha}_{m-1}\,.
\end{equation}

It is not hard to see that
\begin{equation}
\label{new-al-1m}
\exp({\mathrm{a d}}_{\xi})\, {\alpha}_{m-1} \big( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0}\big) =
{\alpha}_{m-1} \big( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0}\big) +
[\xi, {\alpha}_{m-1}] \big( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0}\big)
\end{equation}
and
$$
[\xi, {\alpha}_{m-1}] \big( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0}\big) = {\mathfrak{d}}( \chi_{m-1})\,.
$$

Therefore, due to the above observation, the linear
combination ${\alpha}'_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0})$ does not involve pikes.

It is easy to see that, for every $1 \le n < m-1$ and for every $k \ge 0$
$$
({\alpha}'_{m-1} - {\alpha}_{m-1} )\big( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k} \big) = 0\,.
$$

It is also easy to see that for every $k \neq 2$
\begin{equation}
\label{al-pr-al-1m-k}
({\alpha}'_{m-1} - {\alpha}_{m-1} )\big( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1,k} \big) = 0\,.
\end{equation}

Finally, we observe that each graph in the linear
combination $\chi_{m-1}$ \eqref{chi-1m} has the univalent
white vertex. Hence,
$$
{\partial}^{{\mathrm {Hoch} }} \chi_{m-1} = 0
$$
and, using this identity, it is easy to deduce that
\begin{equation}
\label{al-pr-al-1m}
({\alpha}'_{m-1} - {\alpha}_{m-1} )\big( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1,2} \big) = 0\,.
\end{equation}

Thus we proved the following technical statement:
\begin{claim}
\label{cl:kill-pikes}
The SFQ corresponding to the MC element ${\alpha}_{m-1}$ is homotopy
equivalent to an SFQ which corresponds to a MC element ${\alpha}'_{m-1}$
satisfying following properties:
\begin{itemize}

\item the linear combination ${\alpha}'_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$ does
not involve graphs with pikes;

\item the difference ${\alpha}'_{m-1} - {\alpha}_{m-1}$ belongs to
${\cal F}^{{\mathfrak{c}}}_{m-1} {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$\,.

\end{itemize}
\end{claim}
\qed

\subsection{Adjusting ${\alpha}_{m-1}$ by the action of ${\mathsf{dfGC}}_{{\mathrm {conn} }}$}
\label{sec:adj-by-dfGC}

Due to Claim \ref{cl:kill-pikes} we may assume, without loss of generality,
that  ${\alpha}_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$ does not involve graphs with pikes.

Evaluating the left hand side of the MC equation for ${\alpha}_{m-1}$ on ${\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m+1, 0}$
and using the usual boundary conditions on ${\alpha}_{m-1}$
we get the following identity
\begin{equation}
\label{al-1m-m1}
\sum_{\tau \in {\rm S h }_{2, m-1}}
(\tau, {\rm  i d }) \big ( {\alpha}_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0}) \circ_{1, {\mathfrak{c}}} {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} \big)
\end{equation}
\begin{equation}
\label{arrow-in}
-  \sum_{i=1}^{m+1}
({\sigma}_{m+1, i}, {\rm  i d }) \big( {\Gamma}^{{\mathrm{br}}}_{1}  \circ_{1, {\mathfrak{o}}} {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0}) \big)
\end{equation}
\begin{equation}
\label{arrow-out}
-  \sum_{i=1}^{m+1}
(\tau_{m+1, i}, {\rm  i d }) \big( {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 1})  \circ_{1, {\mathfrak{o}}} {\Gamma}^{{\mathrm{br}}}_{0} \big)
\end{equation}
\begin{equation}
\label{rational-line}
- \sum_{r=2}^{m-1} \sum_{{\sigma} \in {\rm S h }_{r, m+1-r}}
({\sigma}, {\rm  i d }) \big( {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{r, 1})  \circ_{1, {\mathfrak{o}}} {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m+1-r, 0}) \big) = 0 \,,
\end{equation}
where ${\sigma}_{m+1, i}$ is the cycle $(i, i-1, \dots, 2, 1) \in S_{m+1}$ and
$\tau_{m+1, i}$ is the cycle $(i, i+1, \dots, m, m+1) \in S_{m+1}$\,.

Since the graph ${\Gamma}^{{\mathrm{br}}}_{0} \in {\mathsf {KGra} }(1,0)^{{\mathfrak{o}}}$
is a single black vertex labeled by $1$, the linear combination in line \eqref{arrow-out}
involves only graphs with pikes.

Graphs with pikes do not appear in line \eqref{rational-line} because all
graphs in $ {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k})$ for $n \ge 1$ are connected
and each graph in ${\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m+1-r, 0}) \big)$ has at least $2$
black vertices.

On the other hand, the linear combination ${\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0})$ does not
involve graphs with pikes. Hence the sum in line  \eqref{arrow-out} must cancel
all the contributions with pikes coming from the insertion of ${\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}}$ into
vertices of graphs in  ${\alpha}_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$ in line \eqref{al-1m-m1}.

Thus the first three lines combine into
\begin{equation}
\label{sum-3-lines}
- {\Gamma}^{{\mathrm{br}}}_0 \circ_{1, {\mathfrak{c}}} [{\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}}, {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0})]
\end{equation}
and the whole identity can be rewritten as
\begin{equation}
\label{al-1m-m1-new}
{\Gamma}^{{\mathrm{br}}}_0 \circ_{1, {\mathfrak{c}}} [{\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}}, {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0})]   =
\end{equation}
$$
- \sum_{r=2}^{m-1} \sum_{{\sigma} \in {\rm S h }_{r, m+1-r}}
({\sigma}, {\rm  i d }) \big( {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{r, 1})  \circ_{1, {\mathfrak{o}}} {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m+1-r, 0}) \big) \,,
$$
where $ {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0})$ is viewed as a degree $0$ vector in ${\mathsf{dfGC}}_{{\mathrm {conn} }}$.

By the inductive assumption, all coefficients in the right hand side of \eqref{al-1m-m1-new}
are rational. Therefore,  \eqref{al-1m-m1-new} is a finite size inhomogeneous linear system
with rational right hand side.

Since this system admits a solution over reals, it also admits a solution over rationals.
In other words, there exists a degree $2-2m$ vector
$$
\beta_{m,0} \in ({\mathsf{dGra} }_{{\mathrm {conn} }}(m))^{S_m} ({\mathbb Q})
$$
which also solves the equation
\begin{equation}
\label{beta-m-0-eq}
{\Gamma}^{{\mathrm{br}}}_0 \circ_{1, {\mathfrak{c}}} [{\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}}, \beta_{m,0}]   =
\end{equation}
$$
- \sum_{r=2}^{m-1} \sum_{{\sigma} \in {\rm S h }_{r, m+1-r}}
({\sigma}, {\rm  i d }) \big( {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{r, 1})  \circ_{1, {\mathfrak{o}}} {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m+1-r, 0}) \big) = 0 \,.
$$

Therefore, the difference $ {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0}) - \beta_{m,0}$ is a degree
zero cocycle in ${\cal F}_{m-1} {\mathsf{dfGC}}$.

Thus we proved the following statement:
\begin{claim}
\label{cl:adj-by-dfGC}
Let us suppose that an SFQ is given by
a MC element ${\alpha}_{m-1}$ which satisfies the following
properties:
\begin{itemize}

\item for every $n \le m-1$ and $k \ge 0$ the linear combination
${\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k})$ has only rational coefficients,

\item the linear combination ${\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0})$ does
not involve graphs with pikes.

\end{itemize}
Then there exists a degree $0$ cocycle ${\gamma} \in  {\mathsf{dfGC}}$ such that
the new MC element $\wt{{\alpha}}_{m-1}: =\exp({\mathrm{a d}}_{{\gamma}})({\alpha}_{m-1})$ satisfies the
properties:
\begin{itemize}

\item $\wt{{\alpha}}_{m-1} - {\alpha}_{m-1}\in {\cal F}^{{\mathfrak{c}}}_{m-1}  {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$,
in particular,  for every $n \le m-1$ and $k \ge 0$ the linear combination
$\wt{{\alpha}}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k})$ has rational coefficients;

\item all coefficients in the linear combination $\wt{{\alpha}}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0})$
are also rational.

\end{itemize}

\end{claim}
\qed

\subsection{Replacing coefficients in ${\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k})$ for $k \ge 1$ by rational numbers}
\label{sec:working-up-mo}

Let $m$ be, as above, a fixed integer $\ge 3$\,. Our previous work shows that
there exists a MC element ${\alpha} \in {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ satisfying
``boundary conditions'' \eqref{al-Schouten},
\eqref{al-mult}, \eqref{al-HKR} and such that
\begin{itemize}

\item all coefficients in ${\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k'})$ are rational for $n \le m-1$ and all $k' \ge 0$

\item all coefficients in ${\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$ are rational as well.

\end{itemize}

Let $k$ be a positive integer such that all coefficients in the linear combination
${\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k'})$ are rational for $k' < k$\,. We will show that, playing with
the homotopy equivalence, we can get  a MC element ${\alpha}'$ such that
$$
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k_1}) = {\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k_1}) \qquad \forall~ n \le m-1, ~k_1\ge 0
$$
$$
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k'}) = {\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k'}) \qquad \forall~ k' < k
$$
and the linear combination ${\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k})$ has rational coefficients.

Using the usual ``boundary conditions'' for ${\alpha}$ and
the inequality $m \ge 3$, we evaluate
\begin{equation}
\label{al-al-t-m-k}
\mu ({\alpha} {\mathbf{s}^{-1}\,} \otimes {\alpha} {\mathbf{s}^{-1}\,} ) {\cal D} ({\mathsf{t}}^{{\mathfrak{o}}}_{m, k}) =
\end{equation}
$$
{\partial}^{{\mathrm {Hoch} }}{\alpha} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k}) +
(-1)^{k} \sum_{\tau \in {\rm S h }_{2, n-2} } (\tau, {\rm  i d }) \big( {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1, k}) \circ_{1, {\mathfrak{c}}} {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} \big)
$$
$$
- \sum_{r = 1}^{m-1} \sum_{{\sigma} \in {\rm S h }_{r, m-r}} \sum_{0 \le p \le q  \le k}  (-1)^{p + (k-q)(q-p)}
 ({\sigma}, {\rm  i d }) \big( {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{r, p+k-q+1}) \circ_{p+1, {\mathfrak{o}}}
  {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-r, q-p})
\big)\,.
$$

Since ${\alpha}$ is a MC element of  ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$,
the left hand side of \eqref{al-al-t-m-k} is zero. So we conclude that
 the vector ${\alpha} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k})$ satisfies the following
cohomological equation
\begin{equation}
\label{al-at-m}
{\partial}^{{\mathrm {Hoch} }}{\alpha}  ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k}) =
\end{equation}
$$
 \sum_{r = 1}^{m-1} \sum_{{\sigma} \in {\rm S h }_{r, m-r}} \sum_{0 \le p \le q  \le k}  (-1)^{p + (k-q)(q-p)}
 ({\sigma}, {\rm  i d }) \big( {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{r, p+k-q+1}) \circ_{p+1, {\mathfrak{o}}}
  {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-r, q-p})
$$
$$
- (-1)^{k} \sum_{\tau \in {\rm S h }_{2, n-2} }
(\tau, {\rm  i d }) \big( {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1, k}) \circ_{1, {\mathfrak{c}}} {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} \big)\,.
$$

Since the subspace of vectors in ${\mathsf {KGra} }(m,k)$ of a fixed degree is
finite dimensional, equation \eqref{al-at-m} can be viewed as an inhomogeneous
linear system with the right hand side defined over rationals.
Thus, since the system has a solution over reals it must also have
a solution over rationals.

Let us denote by $\beta_{m,k}$ a linear combination of degree $2-2m-k$
in ${\mathsf {KGra} }(m,k)$ with all rational coefficients and such that
\begin{equation}
\label{beta-m-k}
{\partial}^{{\mathrm {Hoch} }}\beta_{m,k}  =
\end{equation}
$$
 \sum_{r = 1}^{m-1} \sum_{{\sigma} \in {\rm S h }_{r, m-r}} \sum_{0 \le p \le q  \le k}  (-1)^{p + (k-q)(q-p)}
 ({\sigma}, {\rm  i d }) \big( {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{r, p+k-q+1}) \circ_{p+1, {\mathfrak{o}}}
  {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-r, q-p})
$$
$$
- (-1)^{k} \sum_{\tau \in {\rm S h }_{2, n-2} }
(\tau, {\rm  i d }) \big( {\alpha} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1, k}) \circ_{1, {\mathfrak{c}}} {\Gamma}_{{\bullet\hspace{-0.05cm}-\hspace{-0.05cm}\bullet}} \big)\,.
$$

Equations \eqref{al-at-m} and \eqref{beta-m-k} imply that the
difference ${\alpha} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k})- \beta_{m,k}$ is ${\partial}^{{\mathrm {Hoch} }}$-closed.
Therefore, by Claim \ref{cl:Pi-Hoch}, there exist vectors
\begin{equation}
\label{xi-m-1k}
\xi_{m, k-1} \in \Big( {\mathsf {KGra} }(m,k-1)  \Big)^{S_m}\,, \qquad | \xi_{m, k-1}  | = 2-2m-k\,, \qquad k \ge 2
\end{equation}
such that
\begin{equation}
\label{al-beta-mk}
{\alpha} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k})- \beta_{m,k} = {\partial}^{{\mathrm {Hoch} }} \xi_{m, k-1} + {\gamma}_{m,k}
\end{equation}
where
\begin{equation}
\label{ga-m-k}
{\gamma}_{m,k} : = \Pi ( {\alpha} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k})- \beta_{m,k} ) \in
 \Big( \Pi {\mathsf {KGra} }(m,k)  \Big)^{S_m}\,.
\end{equation}

Let $\xi_{(k)}$ be a degree vector in  ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ defined by
the equations
\begin{equation}
\label{xi-m}
\begin{array}{c}
\xi_{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k-1}) = \xi_{m, k-1}\,, \qquad \forall~~ k \ge 1\,, \\[0.3cm]
\xi_{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k'}) = 0\,, \qquad \forall~~ n \neq m, ~~  k' \ge 0\,, \\[0.3cm]
\xi_{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{k'}) = 0\,, \qquad \forall ~~ k' \ge 2\,, \\[0.3cm]
\xi_{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{c}}}_{n}) = 0\,, \qquad \forall ~~ n \ge 2\,.
\end{array}
\end{equation}

We consider an SFQ corresponding to the MC element
\begin{equation}
\label{al-1m-wt}
\wt{{\alpha}} : = \exp({\mathrm{a d}}_{\xi_{(k)}}) \, {\alpha} \,.
\end{equation}

It is easy to see that
\begin{equation}
\label{wt-al-al}
(\wt{{\alpha}}  - {\alpha}) \big({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k} \big) = 0  \qquad \forall ~ n \le m-1, ~ k \ge 0\,.
\end{equation}

Furthermore,
\begin{equation}
\label{wt-al-al-at-m}
(\wt{{\alpha}}  - {\alpha}) \big({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,q} \big) =
\begin{cases}
- {\partial}^{{\mathrm {Hoch} }} \, \xi_{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k-1})  \qquad {\rm if} ~~ q = k  \\[0.3cm]
  0 \qquad {\rm otherwise}\,.
\end{cases}
\end{equation}
In particular,
\begin{equation}
\label{wt-al-4less-than-k}
\wt{{\alpha}}({\mathbf{s}^{-1}\,}  {\mathsf{t}}^{{\mathfrak{o}}}_{m,k'}) = {\alpha} ({\mathbf{s}^{-1}\,}  {\mathsf{t}}^{{\mathfrak{o}}}_{m,k'}) \qquad \forall~~ k' < k
\end{equation}
and
\begin{equation}
\label{wt-al-beta-mk}
\wt{{\alpha}} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k})= {\gamma}_{m,k} + \beta_{m,k} \,.
\end{equation}

Since $\wt{{\alpha}}$ is a MC element in  ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$
\begin{equation}
\label{MC-wt-al-m1-1k}
\mu \circ \wt{{\alpha}} {\mathbf{s}^{-1}\,}  \otimes \wt{{\alpha}} {\mathbf{s}^{-1}\,} ({\cal D}  {\mathsf{t}}^{{\mathfrak{o}}}_{m+1, k-1} ) = 0\,.
\end{equation}

Unfolding this equation, we get
\begin{equation}
\label{iden-ga-m-k}
{\mathfrak{d}} ({\gamma}_{m,k}) =
{\partial}^{{\mathrm {Hoch} }} \big( \wt{{\alpha}} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m+1, k-2} )  \big)
\end{equation}
$$
+ \sum_{p=1}^{k} \sum_{i=1}^{m+1}  (-1)^p (\tau_{m+1, i}, {\rm  i d })  \big( \beta_{m,k} \circ_{p, {\mathfrak{o}}} {\Gamma}^{{\mathrm{br}}}_0 \big)
+ \dots\,,
$$
where $\tau_{m+1, i}$ is the cycle $(i, i+1, i+2, \dots, m, m+1)$ and
$\dots$ is a sum of terms involving only $\wt{{\alpha}} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k'})$ for $n < m$
or  $\wt{{\alpha}} ( {\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,  k'})$ for $k' < k$\,. In particular, all coefficients in the terms
$\dots$ and in the sum
$$
\sum_{p=1}^{k} \sum_{i=1}^{m+1}  (-1)^p (\tau_{m+1, i}, {\rm  i d })  \big( \beta_{m,k} \circ_{p, {\mathfrak{o}}} {\Gamma}^{{\mathrm{br}}}_0 \big)
$$
are rational.

Applying the projection $\Pi$ \eqref{Pi}  to both sides of \eqref{iden-ga-m-k} and
using Claim \ref{cl:Pi-Hoch}, we conclude that coefficients of ${\gamma}_{m,k}$ solve
the following finite size linear system:
\begin{equation}
\label{Pi-iden-ga-m-k}
{\mathfrak{d}} ({\gamma}_{m,k}) =
\sum_{p=1}^{k} \sum_{i=1}^{m+1}  (-1)^p \Pi \Big( (\tau_{m+1, i}, {\rm  i d })
\big( \beta_{m,k} \circ_{p, {\mathfrak{o}}} {\Gamma}^{{\mathrm{br}}}_0 \big) \Big) + \Pi(\dots)
\end{equation}
with the right hand side defined over rationals.

Thus, we conclude that, there exists a degree $2-2m -k$ vector
\begin{equation}
\label{wtga-m-k}
\wt{{\gamma}}_{m,k} \in \big( \Pi {\mathsf {KGra} }(m,k)^{{\mathfrak{o}}} \big)^{S_m}({\mathbb Q})
\end{equation}
such that
\begin{equation}
\label{md-wtga-ga}
{\mathfrak{d}}(\wt{{\gamma}}_{m,k} - {\gamma}_{m,k}) = 0\,.
\end{equation}

Let us denote by ${\kappa}^r_{m,k}$, $(r \ge 0)$ the linear combination which is obtained from
$\wt{{\gamma}}_{m,k} - {\gamma}_{m,k}$ by retaining only graphs with exactly $r$ pikes. (In fact,
since ${\mathfrak{d}}(\wt{{\gamma}}_{m,k} - {\gamma}_{m,k}) = 0$, the component  ${\kappa}^0_{m,k} = 0$, i.e. each graph
in the difference $\wt{{\gamma}}_{m,k} - {\gamma}_{m,k}$ has a pike.)

Then equation \eqref{md-md-star} implies that
\begin{equation}
\label{wtga-ga}
\wt{{\gamma}}_{m,k} - {\gamma}_{m,k} = {\mathfrak{d}} ({\kappa}_{m,k}),
\end{equation}
where
$$
{\kappa}_{m,k} = \sum_{r\ge 0} \frac{1}{k+r} {\mathfrak{d}}^* {\kappa}^r_{m,k}\,.
$$

Using the vector ${\kappa}_{m,k}$, we define the degree zero vector
$\psi_{(k)} \in {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ by setting
\begin{equation}
\label{psi-k}
\psi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1, k+1} ) =  {\kappa}_{m,k}\,,
\qquad
\psi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n', k'} ) =
\psi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{c}}}_{n_1}) =
\psi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{k_1} ) = 0
\end{equation}
for all pairs $(n',k') \neq (m-1, k+1)$ and for all $n_1 \ge 2$
and $k_1\ge 2$\,.

Next, we consider the new MC element
\begin{equation}
\label{al-pr}
{\alpha}' : = \exp({\mathrm{a d}}_{\psi_{(k)} }) (\wt{{\alpha}})\,.
\end{equation}

It is obvious that for every $n < m-1$ and $k' \ge 0$
$$
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k'}) =
\wt{{\alpha}}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k'})\,.
$$

Furthermore, since ${\partial}^{{\mathrm {Hoch} }}({\kappa}_{m,k}) = 0$, we also have
\begin{equation}
\label{al-pr-1m-q}
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1, q}) =
\wt{{\alpha}}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1, q})\,, \qquad \forall~~ q \ge 0\,.
\end{equation}

It is not hard to see that, for $q < k$,
\begin{equation}
\label{al-pr-m-less-k}
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, q}) =
\wt{{\alpha}}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, q})\,,
\end{equation}
and finally, for ${\mathsf{t}}^{{\mathfrak{o}}}_{m,k}$, we have
\begin{equation}
\label{al-pr-m-k}
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k}) =
\wt{{\alpha}}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k}) +
{\mathfrak{d}} \big( \psi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m-1, k+1} ) \big) =
\end{equation}
$$
\wt{{\alpha}}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, k}) + {\mathfrak{d}} ({\kappa}_{m,k})\,.
$$

Combining the above observations with
equations \eqref{wt-al-beta-mk} and \eqref{wtga-ga},  we conclude
that
$$
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k_1}) = {\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k_1}) \qquad \forall~ n \le m-1, ~k_1\ge 0
$$
$$
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k'}) = {\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k'}) \qquad \forall~ k' < k
$$
and
$$
{\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k}) =  \wt{{\gamma}}_{m,k} + \beta_{m,k},
$$
i.e. all coefficients in ${\alpha}'({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,k})$ are rational.

Thus we proved the following statement
\begin{prop}
\label{prop:level-m}
Let $m$ be an integer $\ge 3$ and ${\alpha}$ be
a MC element in ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ satisfying
``boundary conditions'' \eqref{al-Schouten},
\eqref{al-mult}, \eqref{al-HKR} and such that
\begin{itemize}

\item all coefficients in ${\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,k'})$ are rational for $n \le m-1$ and all $k' \ge 0$

\item all coefficients in ${\alpha}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$ are rational as well.

\end{itemize}

Then there exist sequences of degree zero vectors
\begin{equation}
\label{xi-and-psi}
\{\xi_{(k)}\}_{k \ge 1} \qquad \textrm{and} \qquad
\{\psi_{(k)}\}_{k \ge 1}
\end{equation}
in ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ such that  for every $k \ge 1$
\begin{equation}
\label{bound-cond-xi-psi}
\psi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{c}}}_n)=
\xi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{c}}}_n) = \psi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{1, q}) = \xi_{(k)} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{1, q}) = 0 \qquad
\forall~~ n \ge 2, ~~ q \ge 0\,,
\end{equation}
\begin{equation}
\label{xi-psi-deeper}
\xi_{(k)}  \in {\cal F}_{m+k-2} {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })
\qquad \textrm{and} \qquad
\psi_{(k)}  \in {\cal F}_{m+k-1} {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })\,,
\end{equation}
and the MC elements
\begin{multline}
\label{new-MC-k}
{\alpha}^{(k)} : =
\exp({\mathrm{a d}}_{\psi_{(k)}}) \exp({\mathrm{a d}}_{\xi_{(k)}})
\exp({\mathrm{a d}}_{\psi_{(k-1)}}) \exp({\mathrm{a d}}_{\xi_{(k-1)}})
\dots \\
\dots
\exp({\mathrm{a d}}_{\psi_{(2)}}) \exp({\mathrm{a d}}_{\xi_{(2)}}) \exp({\mathrm{a d}}_{\psi_{(1)}}) \exp({\mathrm{a d}}_{\xi_{(1)}}) ({\alpha})
\end{multline}
enjoy the following properties

\begin{itemize}

\item for every $k \ge 0$,  the element ${\alpha}^{(k)}$ satisfies the
required boundary conditions \eqref{al-Schouten}, \eqref{al-mult},
and \eqref{al-HKR},

\item for every $k \ge 0$,
$$
{\alpha}^{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,q}) =  {\alpha} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,q})\,,
\qquad \forall ~~ n \le m-1, \quad q \ge 0\,,
$$

\item for every $k \ge 1$ and $q < k$
$$
{\alpha}^{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,q}) = {\alpha}^{(k-1)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,q})\,,
$$

\item for every $q \le k$, all coefficients in ${\alpha}^{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, q})$ are rational.

\end{itemize}

\end{prop}
$\qed$

\section{The proof of Theorem \ref{thm:main}}
\label{sec:proof}
We will now assemble all the statements presented in
Section \ref{sec:step-by-step} into a proof of Theorem \ref{thm:main}.

The desired MC element ${\alpha}_2$ is constructed in Section \ref{sec:al-2}.
So let us assume that, for $m \ge 3$, the desired MC element ${\alpha}_{m-1}$ is constructed.

According to Claim \ref{cl:kill-pikes}, there exists a MC element ${\alpha}'_{m-1}$ in
${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ such that
\begin{itemize}

\item it satisfies the required boundary conditions \eqref{al-Schouten}, \eqref{al-mult},
and \eqref{al-HKR},

\item the linear combination ${\alpha}'_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,0})$ does not
involve graphs with pikes,

\item for every $n \le m-1$ and every $k \ge 0$
$$
{\alpha}'_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k}) = {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k})\,;
$$
in particular, all coefficients in ${\alpha}'_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k})$ are rational.

\end{itemize}

Next, replacing ${\alpha}_{m-1}$ by ${\alpha}'_{m-1}$ and using Claim \ref{cl:adj-by-dfGC},
we conclude that there exists a MC element $\wt{{\alpha}}_{m-1}$ in
${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ such that
\begin{itemize}

\item it satisfies the required boundary conditions \eqref{al-Schouten}, \eqref{al-mult},
and \eqref{al-HKR},

\item for every $n \le m-1$ and every $k \ge 0$
$$
\wt{{\alpha}}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k}) = {\alpha}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k})\,;
$$
in particular, all coefficients in $\wt{{\alpha}}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n, k})$ are rational,

\item all coefficients in the linear combination $\wt{{\alpha}}_{m-1}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, 0})$
are also rational.

\end{itemize}

Setting ${\alpha}= \wt{{\alpha}}_{m-1}$ and using Proposition \ref{prop:level-m}, we conclude that
there exists a sequence of MC elements
\begin{equation}
\label{al-k}
\big\{ {\alpha}^{(k)}  \big\}_{k \ge 0} \in {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })
\end{equation}
such that
\begin{itemize}

\item $ {\alpha}^{(0)} = \wt{{\alpha}}_{m-1}$\,,

\item for every $k \ge 0$,  the element ${\alpha}^{(k)}$ satisfies the
required boundary conditions \eqref{al-Schouten}, \eqref{al-mult},
and \eqref{al-HKR},

\item for every $k \ge 0$,
$$
{\alpha}^{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,q}) =  \wt{{\alpha}}_{m-1} ({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{n,q})\,,
\qquad \forall ~~ n \le m-1, \quad q \ge 0\,,
$$

\item for every $k \ge 1$ and $q < k$
$$
{\alpha}^{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,q}) = {\alpha}^{(k-1)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m,q})\,,
$$

\item for every $q \le k$, all coefficients in ${\alpha}^{(k)}({\mathbf{s}^{-1}\,} {\mathsf{t}}^{{\mathfrak{o}}}_{m, q})$ are
rational.

\end{itemize}

Thus, the desired MC element ${\alpha}_{m}$ in \eqref{al-sequence} is obtained
as the limiting element\footnote{Recall that the dg Lie algebra ${\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$ is complete
with respect to the filtration ${\cal F}^{{\mathfrak{o}}}_{{\bullet}}  {\mathrm{Conv}}({\mathbf{s}^{-1}\,} {\mathfrak{o} \mathfrak{c}} , {\mathsf {KGra} })$\,.}
of the sequence \eqref{al-k}.

Theorem \ref{thm:main} is proved.  $\qed$

redpajama_set_name: RedPajamaArXiv