For $i=1,2$ let $K_{ i+1,i}\in \cC^\infty(E^{(i+1)}\hotimes (E^{(i)})^* \setminus\triangle)$ be supported on $\{(p,q)\in (M\times M)\setminus \triangle\   | \ d_\fraks(p,q)<1\}$ and of order $\zeta_i$.
If $\zeta_{1}\wedge\zeta_2> -|\fraks|$, we write

$$K_{3,2} * K_{2,1} (p,q):= \int_M   K_{3,2}(p,z) K_{2,1}(z,q) \dVol_z \ ,$$

where $K_{3,2}(p,z) K_{2,1}(z,q) \in \left( E^{(3)}\hotimes (E^{(1)})^* \right)|_{p,q}$ is obtained
by contracting the factors in $(E^{(2)})^*$ and $E^{(2)}$.
$K_{3,2} * K_{2,1} \in \cC^\infty\left( E^{(3)}\hotimes (E^{(1)})^* \setminus\triangle \right)$.
If furthermore $\bar \zeta :=  \zeta_{1}+ \zeta_2 + |\fraks| < 0$, then

$$\vertiii{K_{3,2} * K_{2,1}}_{\bar{\zeta}; m}\leq C\vertiii{K_1}_{\zeta_1; m}\vertiii{K_1}_{\zeta_2; m}\ .$$

If $\bar{\zeta}\in \mathbb{R}_+\setminus \mathbb{N}$, then
$K_{3,2} * K_{2,1} \in \cC^{[\bar \zeta ]}(E^{(3)}\hotimes (E^{(1)})^* \setminus\triangle)$ and

$$K_{3,1}(p,q):=  K_{3,2} * K_{2,1}(p,q) - R \left(   j^{[\bar \zeta ]}_q K_{3,1}(\cdot, q ) \right) (p)$$

$$\vertiii{K_{3,1}}_{\bar{\zeta}; m}\leq C\vertiii{K_1}_{\zeta_1; \bar{m}}\vertiii{K_1}_{\zeta_2; \bar{m}}\ ,$$

for $\bar{m}= m \vee ([\bar{\zeta}] + \max_{i} \fraks_i)$ for any admissible realisation $R$.

Let $-|\fraks|-1<\zeta \leq |\fraks|$ and $K\in \cC^\infty(F\hotimes E^* \setminus \triangle)$ of order $\zeta$

such that $\supp(K(p,\cdot) )\subset B_{r_p}(p)$.
We define the renormalised operator $\mathcal{R}K: \cC^\infty(E) \to \cC^\infty(F)$ by

$$\mathcal{R}K (\varphi)= \int_M K(p,z) \left(\phi(z)- \tau_{z,p}\phi(p)\right) \dVol_z \ .$$

For $i=1,2$ let $K_{ i+1,i}\in \cC^\infty(E^{(i+1)}\hotimes (E^{(i)})^* \setminus\triangle)$ be supported on $\{(p,q)\in (M\times M)\setminus \triangle\   | \ d_\fraks(p,q)<1\}$ and of order $\zeta_i$.
If $-|\fraks|-1<\zeta_{2}\leq -|\fraks|$ and  $-2|\fraks|-\zeta_2 <\zeta_1<0$, then

$$\mathcal{R}K_{3,2} * K_{2,1} \in \cC^\infty\left( E^{(3)}\hotimes (E^{(1)})^* \setminus \triangle \right)\ ,$$

satisfies for $\bar{\zeta}= 0\wedge (\zeta_1+\zeta_2+|\fraks|)$,

$$\vertiii{\mathcal{R}K_{3,2} * K_{2,1}}_{\bar{\zeta}; m}\leq C\vertiii{K_1}_{\zeta_1; m}\vertiii{K_1}_{\zeta_2; \bar{m}}\ ,$$

for every $m \geq 0$ and
$\bar{m} = m + \max_i \fraks_i$.

Let $K\in \cC^\infty(F\hotimes E^* \setminus \triangle)$ be supported on $\{(p,q)\in (M\times M)\setminus \triangle\   | \ d_\fraks(p,q)<1\}$
and of order $\zeta$.
If $M=\bar{M}$ has constant scaling, set $\rho_\epsilon= G_{\epsilon^2}$, while if $M=\RR\times \bar{M}$ has scaling $\fraks=(2, 1)$ let $\rho_\epsilon$ be given as in (<ref>).
$K_\epsilon:= K * \rho_\epsilon$ has bounded derivatives of all orders and

$$|j_{p,q} K_\epsilon|_k\leq C (d_\fraks(p,q)+\epsilon)^{\zeta-k} \vertiii{K}_{\zeta, k} \ .$$

Finally, for $\bar{\zeta}\in [\zeta-1, {\zeta})$ and $m\geq 0$, one has the bound

$$\vertiii{K-K_\epsilon}_{\bar{\zeta}, m}\lesssim \epsilon^{\zeta-\bar{\zeta}} \vertiii{K}_{\zeta,\bar{m}} \ ,$$

where $\bar{m}= n + \max_i \fraks_i$.

Let $K\in \cC^\infty(F\hotimes E^* \setminus \triangle)$ be supported on $\{(p,q)\in (M\times M)\setminus \triangle\   | \ d_\fraks(p,q)<1\}$
and of order $\zeta\leq 0$. Then, for every $\alpha\in [0,1]$, one has the bound

$$|K( z_1,z)-\tau_{z_1, z_2}K(z_2,z)| \lesssim d(z_1,z_2)^\alpha ( d(z,z_1)^{\zeta-\alpha}+ d(z,z_2)^{\zeta-\alpha} ) \vertiii{K}_{\zeta,m}, $$

where $m= \sup_i \fraks_i$.

§.§ A Hairer–Quastel type criterion

In this section we state a non-translation invariant weaker version of <cit.> which is sufficient to estimate the terms in (<ref>) and (<ref>).
Let us first recall the setting: We are given a finite directed multigraph $\CCG= (\CCV, \CCE)$ with edges $e \in \CCE$ labelled by $(a_e, r_e)\in \mathbb{R}_+\times \{0,1\}$, and kernels $K_e: M\times M\setminus \triangle \to \mathbb{R}$ compactly supported on $\{(p,q)\in (M\times M)\setminus \triangle\   | \ d_\fraks(p,q)<1\}$ and satisfying $\vertiii{K_e}_{a_e; 2}<\infty .$
We assume $\CCG$ contains $M\geq 0$ distinguished edges[The green arrows in Section <ref>.] $e_{\star,1},\ldots,e_{\star,M}$ connecting $\star \in \CCV$ distinguished vertices $v_{\star,1},\ldots,v_{\star,M}$. We write $\CCV_\star= \{ \star,  v_{\star,1},\ldots,v_{\star,M}\}$
$\CCV_0= \CCV\setminus \{\star \}$.
For $e\in E$, we write $(e_-,e_+)$ for the pair of vertices such that that $e$ is directed from the vertex $e_-$ to $e_+$.
For simplicity[This is not needed for the argument but generally satisfied for diagrams appearing in regularity structures.]
assume that for any pair of vertices
$v_1, v_2$ there exists at most one edge connecting them such that $r_e=1$.

Then, we define a new fully connected directed graph $\hat \CCG =(\CCV , \hat \CCE)$ where for each edge $e\in \hat \CCE$ the label
$(\hat a_e, r_e)$ is the sum of the labels over all edges directed from $e_-$ to $e_+$ (with the empty sum being understood to be $0$ and $K_e=1$ for all such newly added edges).

For a subset $\bar\CCV \subset \CCV$ we call

* $\CCE^\uparrow (\bar \CCV)  =
\{ e\in \CCE : e\cap \bar\CCV = e_-\}$ outgoing edges,

* $\CCE^\downarrow (\bar \CCV)  =   \{ e\in \CCE : e\cap \bar\CCV = e_+\}$ incoming edges,

* $\CCE_0 (\bar \CCV)  =   \{ e\in \CCE : e\cap \bar\CCV = e\}$ internal edges,

* $\CCE (\bar \CCV)  =   \{ e\in \CCE : e\cap \bar\CCV \neq \emptyset\}$ incident edges

and write
$\CCE_+ (\bar \CCV)  =   \{ e\in \CCE(\bar \CCV) :  r_e>0\}$, $\CCE_+^\uparrow = \CCE_+ \cap \CCE^\uparrow$ and $\CCE_+^\downarrow = \CCE_+ \cap \CCE^\downarrow$.

The resulting directed graph $(\CCV,\hat\CCE)$ with labels $(\hat a_e,r_e)$ satisfies
$r_e = 0$ for all edges connected to $\star$ or connecting two elements of $\CCV_\star$. Furthermore assume the following.

* For every subset $\bar \CCV \subset \CCV_0$ of cardinality at least $2$,

\begin{equation}\label{e:assEdges}
\sum_{e  \in \hat\CCE_0 (\bar \CCV)} \hat a_e < (|\bar \CCV| - 1)|\fraks|\;;
\end{equation}

* For every subset $\bar \CCV \subset \CCV$ containing $0$ of cardinality at least $2$,

\begin{equation}\label{e:assEdges2}
\sum_{e  \in \hat \CCE_0 (\bar \CCV) } \hat a_e
+\sum_{e  \in \hat\CCE^{\uparrow}_+ (\bar \CCV)  } (\hat a_e+ r_e -1)  - \sum_{e  \in \hat\CCE^{\downarrow}_+ (\bar \CCV)  }  r_e< (|\bar \CCV| - 1)|\fraks|\;;
\end{equation}

* For every non-empty subset $\bar \CCV \subset \CCV\setminus \CCV_\star$,

\begin{equation}\label{e:assEdges3}
\sum_{e\in \hat\CCE(\bar\CCV)\setminus \hat\CCE^{\downarrow}_+(\bar \CCV) }  \hat a_e
+\sum_{e\in \hat \CCE^{\uparrow}_+(\bar\CCV)}  r_e
- \sum_{e \in \hat \CCE^\downarrow_+(\bar \CCV)} (r_e-1)
> |\bar \CCV||\fraks|\;.
\end{equation}

Next, for edges $e\in \CCE$ such that $r_e=0$ set $\hat{K}_e(p,q)={K}_e(p,q)$ while for edges $e\in \CCE$ such that $r_e=1$ set
$\hat{K}_e(p,q)={K}_e(p,q)-K_e(\star. q)$ and let

$$\mathcal{I}^{\CCG}(\phi^\lambda_\star, K):= \int_{M^{\CCV_0}}  \prod_{e\in \CCE} \hat{K}_e(x_{e_+},x_{e_-})\prod_{i=1}^M \phi^\lambda ({x_{v_{\star,i}}}) \dVol_x \ .$$

Let $\CCG=(\CCV,\CCE)$ be a finite directed multigraph with labels $\{a_e,r_e\}_{e\in \CCE}$ and kernels
$\{K_e\}_{e\in \CCE}$ with  the resulting graph satisfying Assumption <ref> and its preamble.
Then, there exist  $C<\infty$ depending only on the
structure of the graph $(\CCV,\CCE)$ and the labels $r_e$ such that

\begin{equation}\label{genconvBound}
\le C\lambda^{\tilde \alpha}\prod_{e\in \CCE} \|K_e\|_{a_e;2}\;,
\end{equation}

for $0<\lambda\le 1$, where
\begin{equation}
\tilde \alpha = |\fraks||\CCV\setminus \CCV_\star| - \sum_{e\in \CCE} a_e.
\end{equation}

This theorem is a simpler version of the one in <cit.>, so we shall only outline the (minor) changes necessary to adapt it to manifolds.
One first introduces $\phi: \mathbb{R}\to [0,1]$ supported on $[3/8,1]$ such that for $\Phi^{(n)}(s)= \varphi(2^n s)$ one has for every $s>0$

$$\sum_{n\in \mathbb{Z}} \varphi^{(n)} (s)=1\ .$$

Then, for $\mathbf{n}\in \mathbb{N}^3$ and an edges $e\in \CCE$ such that $r_e=0$ set

$$\hat{K}^{k,l,m}_e (p,q) =
\begin{cases}
\varphi^{(k)}(d(p,q))  K_e (p,q)  & \text{if }  l=m=0 \ , \\
0 & \text{else. }
\end{cases}

One observes that the analogue of <cit.> holds.
If $r_e=1$, set

$$\hat{K}^{k,l,m}_e (p,q) = \varphi^{(k)}(d(p,q)) \varphi^{(l)}(d(\star, p)) \varphi^{(m)} (d(\star, q)) \left( K_e (p,q)- K(p, \star)\right) \ . $$

After possibly enhancing the edge set $\CCE$ to include for any $(v,w)\in \CCV^{2}$ an edge $e$ from $v$ to $w$ and setting $K_e=1$ for such edges, define for $\mathbf{n}: \CCE \to \mathbb{N}^3$

$$ \hat{K}^{(n)}(x)= \prod_{e\in \CCE} \hat{K}^{(n_e)}_e (x_{e_-}, x_{e_+}) \ .$$
One observes
that if we set $\mathcal{N}_\lambda=\big\{\mathbf{n}: \CCE \to \mathbb{N}^3 \ : \ 2^{-{\mathbf{n}_{e_{\star, i}} }}  \leq \lambda\ \big\} $
it suffices to show that the absolute value of

I^\mathbb{G}_{\lambda}(K) = \sum_{\mathbf{n}\in \mathcal{N}_\lambda} \int_{M^{\CCV_0}} \hat{K}^{(\mathbf{n})}(x) \dVol_x$$

is bounded by the right-hand side of (<ref>), see <cit.>.

At this point, we can apply the multiscale clustering argument in <cit.> essentially ad verbatim, even with the further simplification that in our case the naive bound of <cit.> is sufficient. Let us therefore only recall the main cornerstones of that argument.
To every configuration $x\in M^{\CCV}$ one associates a labelled binary rooted tree $(T,l)$ where
the leave set $L_T$ of $T$ is given by $\CCV$ and
the inner nodes $T^\circ= N_T\setminus L_T$ have a labels

$$l:T^\circ \to \mathbb{N}$$

satisfying the following properties.

* The labels are order preserving, i.e. for inner nodes $\nu, \omega\in T^\circ$,

$$\nu\geq \omega \Rightarrow l(\nu)\geq l(\omega) \ .$$

* For every $v,w\in \CCV$, one has

$$2^{-l(v\wedge w)}\leq d_\fraks(x_v,x_w) \leq |\CCV| 2^{-l(v \wedge w)} \ ,$$

where $v\wedge w\in T^\circ$ denotes the largest common ancestor of $v$ and $w$.

After defining $\mathcal{N}(T,l)$ and $\CCT_\lambda(\CCV)$ as in <cit.> and right thereafter,
it follows by the same argument as for <cit.> that

\begin{equation}
|{I}_\lambda^\CCG (K)| \lesssim \sum_{(T,\ell) \in \CCT_\lambda(\CCV)} \sum_{n \in \mathcal{N}(T,\ell)} \Bigl|  \int_{M^{\CCV}} \hat K^{(\mathbf{n} )}(x)\,\dVol_x \Bigr|\;.
\end{equation}

Next, it follows as in <cit.>
that for  $\eta\colon T^\circ \to \RR$ given by
$\eta(v) =|\fraks|+ \sum_{e \in \hat\CCE} \eta_e(v)$, where

\begin{align*}
\eta_e(v)   &= - \hat{a}_e \mathbf{1}_{e_\uparrow} (v)
+ r_e  (\mathbf{1}_{e_+\wedge 0} (v)
- \mathbf{1}_{e_\uparrow} (v)\bigr) \mathbf{1}_{r_e >0, e_+\wedge 0 > e_\uparrow} \\
&\quad + (1-r_e -  \hat{a}_e) (\mathbf{1}_{e_-\wedge 0} (v)
- \mathbf{1}_{e_\uparrow} (v)\bigr) \mathbf{1}_{r_e >0, e_-\wedge 0 > e_\uparrow}\;,
\end{align*}

with $\mathbf{1}_v(v) = 1$ and $\mathbf{1}_v(w) = 0$ for $w \neq v$, one has the bound

\begin{equation}\label{naive bound}
\Bigl(\prod_{v \in T^\circ} 2^{-\ell_v |\fraks|}\Bigr)\sup_x |\hat K^{(\mathbf{n})}(x)| \lesssim \prod_{v \in T^\circ} 2^{-\ell_v \eta(v)}\; .
\end{equation}

Finally, combining (<ref>) with <cit.> concludes the argument. Note that we do not consider any negative renormalisation here and therefore we can omit the considerations of <cit.>.