is continuous, uniformly over $t\in(0,T]$, $\e>0$, over bounded balls in $\cD^{\sol}$, initial conditions in $\CC^\eta_\e$, and compatible models
%\eqref{e:assump-model} uniformly bounded.
satisfying Assumption~\ref{as:models} for a fixed $r>0$.
(We implicitly used the
additional bound $\|\Psi_i(t)\|_{L^\infty_\e}\lesssim \e^{-\kappa}$ from~\eqref{e:assump-Psi-1} to obtain~\eqref{eq:A_to_RA}.)
Lemma~\ref{lem:eps-improve-reg}\ref{pt:eps_f_vanish} thus implies $\|\CR\CA^\e(t)\|_{L^\infty_\e} \lesssim \e^{-\kappa}$.
Hence, for $\e<\e_0$ sufficiently small, which ensures $\e\CR\CA^\e \in V$
and thus $r_h ( \e \CR\CA^\e)$ and $r_l(\e \CR\CA^\e)$ are well-defined,
we obtain by analyticity of $r_h$, $r_l$ and Remark~\ref{rem:power_of_eps},
that the map
\begin{equ}
\cD^\sol\ni\CA^\e \mapsto \e^{-4+4\kappa} \big( r_h ( \e \CR\CA^\e) +  r_l(\e \CR\CA^\e)\big)(t)\in L^\infty_\e
\end{equ}
is locally Lipschitz, uniformly over the same data as above,
and so $\e^{-3} r_h ( \e \CR\CA^\e)
+  \e^{-3}  r_l(\e \CR\CA^\e) $ can be
lifted to polynomial regularity structure of order $0$ with $\cD^{0,0}$ norm bounded by $\eps^{1-4\kappa}$ uniformly in $\eps < \e_0$.
The bound $\$\CP^\e \bone_+  \boldR_i\$_{\gamma,2\eta+1}^{(\e)} \lesssim \eps^{1-4\kappa}$ follows from short-time convolution estimates
and the definition of $\boldR_i$ in~\eqref{e:def-boldR}.
%Note that we have not yet used the bounds~\eqref{e:assump-model_2}.

For $\boldH_i$, we obtain $\$\CP^\e \bone_+  \boldH_i\$_{\gamma,2\eta+1}^{(\e)} \lesssim \eps^{1-2\kappa}$ in the same way as for $\boldR_i$ upon using the bound $|c_j^\e[\<r_z_large>]|\lesssim\e^{-2\kappa}$ from~\eqref{eq:C_bounds_large_cherries}
and remarking that $h^\e_1 (A)=O(\e A^2)$, $h^\e_2 (A)=O(\e A^2)$, and $h^\e_3 (A)=O(\e)$.

Turning to the term $\widetilde{\boldR}_i$, recall
$\|\CR\CV^\e\|_{\eta;\K_\e;z;\e}\lesssim\$\CV^\e\$_{\gamma,\eta;\K_\e;\e}$ by Lemma~\ref{lem:reconstruct}.
One has, for any $\beta\in[0,1]$,
$\CR\CV^\e \in \CC^{\beta+\eta}_{-\beta,\e}$ (with blow-up due to initial condition).
Applying Lemma~\ref{lem:eps-improve-reg}\ref{pt:eps_d_f_vanish} with $\alpha=\beta+\eta$, $\beta\in[-\eta,1]$, therein,
\begin{equ}[eq:bounds_on_V]
\|\e^{1-\alpha}\partial\CR\CV^\e(t)\|_{L^\infty_\e}\lesssim t^{-\beta/2}\;,\qquad
\|\e^{2-\alpha}\partial^2\CR\CV^\e(t)\|_{L^\infty_\e}\lesssim t^{-\beta/2}\;.
\end{equ}
Taking $\beta = \frac12-\eta+\kappa$, so $1-\alpha=\frac12-\kappa$, we obtain a bound of order $\e^{2\kappa}$
in $\CC^{0,T}_{-2\beta,\e}=\CC^{0,T}_{-1+2\eta-2\kappa,\e}$ on the final two terms in the parenthesis $\{\ldots\}$ in~\eqref{e:def-tilde-boldR},
which gives a bound of order $\e^{2\kappa}$ in $\cD^{\gamma,3\eta+1}_{0,\e}$ after convolution with the heat kernel.

The rest of the terms in $\{\dots\}$ in \eqref{e:def-tilde-boldR} are of order $\e^\theta$
in $\CC^{-2\kappa,T}_{-1,\e}$ for $\e<\e_0$ by our choice of $\tilde c^\e_\star$ in Definition~\ref{def:tildec} and by Lemmas~\ref{lem:I3} and
\ref{lem:grad-R-hat}
together with Remark~\ref{rem:eA_small}
and the fact that $\CR\CV^\e$ satisfies~\eqref{eq:v_bound} (for a possibly larger $r$).
After convolution with the heat kernel, these terms, and thus $\widetilde{\boldR}_i$, are of order $\e^\theta$ in $\cD^{\gamma,1}_{0,\e}$.

Recalling $\delta_i$ from~\eqref{eq:delta_A_def}, by using the first bound in~\eqref{eq:bounds_on_V} with $\alpha=2\kappa$, $\beta = 2\kappa-\eta$ (and its obvious extension to derivatives of the form $\e^{-1}\{f(\cdot+\be+\be')-f(\cdot)\}$)
and $|\hat C^\e_{j,\be}|
+|C_{\bs,\pm}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]|
+|C_{\bs}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]| \lesssim \e^{-\kappa}$ (Lemma~\ref{lem:thick-trig-bahave}), we obtain
\begin{equ}[eq:delta_term_vanish]
\|(\delta_i \CR\CV^\e_i)(t)\|_{L^\infty_{\e}} \lesssim \e^{\kappa}t^{-\kappa+\eta/2}\;.
\end{equ}
After convolution with the heat kernel, this term is of order $\e^\kappa$ in $\cD^{\gamma,1}_{0,\e}$.

Finally, we recall $\bar\delta_i$ from~\eqref{eq:bar_delta_def}, and $|C_{\be,\bw}[\<IXiI'Xibar>]|,|C_{\be}[\<I'XiIXibar>]|\lesssim \eps^{-\kappa}$ by~\eqref{eq:C_bounds_large_cherries}.
Note the elementary identity
$\sum_{i=1}^n\lambda_iy_i = \sum_{i=1}^{n-1} (\sum_{j=1}^i\lambda_j)(y_{i}-y_{i+1})$
%\sum_{i=1}^n\lambda_iy_i = \sum_{i=1}^{n-1} (\sum_{j=1}^i\lambda_j)(y_{i}-y_{i+1}) \quad \mbox{whenever}
%\quad \sum_{i=1}^n\lambda_i=0
Using this identity with $n=4$,
and recalling~\eqref{eq:C_be_vanish},
the first term  in~\eqref{eq:bar_delta_def}
$\e^{\kappa} \sum_{\be\in\CE_\times} \ad_{C_{\be}[\<I'XiIXibar>]}A(e+\be)$
can be written as linear combinations of terms of the form $\e\d A$.
Using the above identity with $n=2$
and recalling~\eqref{eq:C_bs_be_vanish},
the second term  in~\eqref{eq:bar_delta_def}
\sum_{\bw\in\{\Northwest,\Southwest\}} \big( \sum_{\be\in\CE_\times} \ad_{C_{\be,\bw} [\<IXiI'Xibar>]}\big)A(e^\bw)$
can be also written in the form $\e\d A$.
We have $\|\CR\CA^\e(t)\|_{L^\infty_\e}\lesssim t^{\eta/2}\e^{-\kappa}$ by~\eqref{eq:A_to_RA},
which implies by Lemma~\ref{lem:eps-improve-reg}\ref{pt:eps_f_vanish}
\|\bar\delta_i\CR\CA_j^\e\|_{\CC^{-2\kappa}_\e}\lesssim \|\e\d \CR\CA_j^\e(t)\|_{\CC^{-2\kappa}_\e}\lesssim t^{\eta/2}\e^\kappa$.
After convolution with the heat kernel, this term is of order $\e^\theta$ in $\cD^{\gamma,1}_{0,\e}$.
\end{proof}

\subsection{Renormalised equation}
\label{subsec:renorm_eq}

Let $(Π^,Γ^)$ be the canonical model from Section~\ref{sec:canonical} and
let $(Π̂^,Γ̂^)$ be as in Definition~\ref{def:renorm_model}.
We call $(Π̂^,Γ̂^)$ the \textit{renormalised model}.
Let $c̃^_⋆$ be as in Definition~\ref{def:tildec} for $Ψ_i = K^i;*_(i)ξ^_i$ (which exists due to Lemma~\ref{lem:Psi_Wick}).
By Propositions~\ref{prop:canonical_model} and~\ref{prop:renormalised_model}, $(Π^,Γ^)$ and $(Π̂^,Γ̂^)$ are both compatible models.
Let $$ be the reconstruction operator for $(Π̂^,Γ̂^)$.
Recall $V⊂^$ from Section~\ref{sec:Notation}.

\begin{lemma}\label{lem:renormalised-equ}
There exists $c^\eps_\star\in L(\mfg^2,\mfg^2)$
such that $c^\eps_\star=O(1)$ and, if $\CA^\eps$ solves the
fixed point problem \eqref{e:fix-pt-Ae},
then $\hat\CR\CA^\eps$ solves the  discrete SPDE \eqref{e:Aeps}
up to the exit time of $\hat\CR\CA^\eps$ from the set $V$.
\end{lemma}
An important outcome of the following proof is that
in the equation for $A^$, the renormalisation map only produces
1)  vanishing terms corresponding to \eqref{eq:delta_A_def}-\eqref{e:def-h123A}
and 2) linear terms in $A^$.
Both of these are eventually cancelled thanks to our formulation of the fixed point problem in Section~\ref{subsec:FPP}.

%An important outcome of the following proof is that
%in the equation for $A^\e (e)$ where $e\in\obonds_i$ the renormalisation map only produces
%1) linear terms  evaluated at the same bond $e$  or the neighboring bonds of $e$ in $\obonds_i$, but {\it not} $\obonds_j$
%(with only one  exception which leads to a ``$\bar\delta$ term'' \eqref{eq:bar_delta_def} but this vanishes as we saw below \eqref{e:element-identity}); 2) vanishing terms corresponding to \eqref{e:def-h123A}.
%These terms produced by renormalisation are eventually all cancelled thanks to our formulation of the fixed point problem in Section~\ref{subsec:FPP}, so that our discrete Langevin dynamic  \eqref{e:Aeps} remains untouched.

\begin{proof}
We do not have the general results as in \cite{BCCH17} or \cite[Prop.~5.68]{CCHS_2D}   so we proceed by hand. Throughout the proof we assume $j\neq i$.
In view of the fixed point problem \eqref{e:fix-pt-Ae},
the solution $\CA^\e$ has the following expansion up to degree $1$:
\begin{equs}[e:CA-expand]
\CA^\e_i
\mcb{I} \bXi_i + a_i \bone^{(i)}
\mcb{I} \big[\cS_{a}\bPsi_j +\cS_{a} (a_j \bone^{(j)}),
2\CD_j \bPsi_i \big]
\\
& \quad
+\mcb{I}  \big[\cS_{a}\bPsi_j +\cS_{a} (a_j \bone^{(j)}),
\bar{\CD}_i \bPsi_j \big]
+\mcb{I} \big[ \bPsi_i+ a_i \bone^{(i)} , \CD_i \bPsi_i \big]
\\
\frac{\e^\kappa}{2}  \mcb{I} [  \bPsi_i +a_i \bone^{(i)},   \bar\bXi_i ]
- \frac{\eps^\kappa}{2} \mcb{I}  [(\CD_j^+ +\CD_j^-) \tilde\bPsi_i, \bar{\CD}_j \tilde\bPsi_j]
\\
+ \frac{\e^\kappa}{4}\mcb{I} [\CD_j^+\cS_{\southeast} \tilde\bPsi_j, \CD_j^+ \cS_{\southwest}\tilde\bPsi_j]
\langle \nabla a_i, {\bf X}^{(i)}\rangle\;.
\end{equs}
Here, we write $\bPsi_i =\mcb{I} \bXi_i$, $\bar\bPsi_i =\mcb{I} \bar\bXi_i$, $\tilde\bPsi_i =\mcb{I} \tilde\bXi_i$
as in Section~\ref{subsec:FPP}
(with the terms involving $\tilde\bPsi_i$ understood as in Remark~\ref{rem:quad-tildeXi}),
and $\nabla a_i = (\nabla_1 a_i,\nabla_2 a_i)$
is a vector-valued coefficient
which is not necessarily the derivative of $a_i$.
Note that % the terms such as $\CD_j \bPsi_i$ are the same as $\mcb{D}_j \bPsi_i$ since $\bPsi$ is constant modelled distribution. On the other hand
$\cS$ in \eqref{e:CA-expand} is a nonlocal operator which we treat carefully below (see e.g. \eqref{e:calc-double-shifts}).

Remark that the nonlocal parts of differentiation operators in  \eqref{e:fix-pt-Ae}
do not contribute to \eqref{e:CA-expand} at degree $\le 1$: for instance, for the terms of the form
$\CA^\e \mcb{D}\CA^\e = \CA^\e \CD \CA^\e+ \CA^\e (H \CA^\e \bone)$,
the last term  upon hitting by $\mcb{I}$ has degree $>1$.

By the definition \eqref{e:shift-Dga} of $\cS$,
and the definition of  $\CS_{\be}$ on abstract polynomials,
\begin{equ}[e:Sa1]
\cS_{a} (a_j \bone^{(j)}) (e) =
\frac14\sum_{\be\in \CE_\times} a_j (e+\be) \bone^{(i)} \;,
\qquad
e\in \obonds_i\;.
\end{equ}
We  remark that all the terms in \eqref{e:CA-expand} take values in $\CT^{(i)}\otimes \mfg$ and the Lie brackets act on the $\mfg$ component,
where $\CT^{(i)}$ is the $i$th component in our splitting
\eqref{e:T1T2}.

We now go through all the terms in~\eqref{e:fix-pt-Ae} that require renormalisation, expand them up to degree $0$,
and apply the map $M^\e$ to them in order to compute the
right-hand side of our equation.
The cubic term is expanded as
\begin{equs}[e:cubic-exp]
{} &
[\bPsi_j^{(a)}, [\bPsi_j^{(a)}, \bPsi_i]]
+ [a_j^{(a)}, [\bPsi_j^{(a)}, \bPsi_i]]
+ [\bPsi_j^{(a)}, [a_j^{(a)}, \bPsi_i]]
+ [\bPsi_j^{(a)}, [\bPsi_j^{(a)}, a_i]]
\\
+ [a_j^{(a)}, [a_j^{(a)}, \bPsi_i]]
+ [a_j^{(a)}, [\bPsi_j^{(a)}, a_i]]
+ [\bPsi_j^{(a)}, [a_j^{(a)}, a_i]]
+ [a_j^{(a)}, [a_j^{(a)}, a_i]]
\end{equs}
where $a_j^{(a)}$ is a shorthand for \eqref{e:Sa1},
and $a_i=a_i\bone^{(i)}$, $\bPsi_j^{(a)} =\cS_{a} \bPsi_j$ as shorthands.
Upon applying the renormalisation map $M^\e$, only the 1st and 4th terms are affected, and the expression transforms as
\begin{equ}[eq:cubic_ren_eq]
\eqref{e:cubic-exp} \mapsto
\eqref{e:cubic-exp}
%- C[\<IXi^2>] \,\Cas
\,(\bPsi_i+a_i \bone^{(i)})\;,
\end{equ}
where we recall the definition of $\bar C^\e$~\eqref{e:def-C16} and of $\ad_{X\otimes Y}=[X,[Y,\cdot]]$ for $X,Y\in\mfg$ following~\eqref{eq:delta_A_def}
and the minus sign in~\eqref{eq:M_tau_def}.
Here, $\bPsi_i+a_i \bone^{(i)}$ is just $\CA^\e_i$ truncated at degree $0$ and one has $\hat\CR \CA^\e_i =\hat\CR(\bPsi_i+a_i \bone^{(i)})$.

For $[\CA^\e_i, \mcb{D}_i \CA^\e_i ]$, we have $\mcb{D}_i \CA^\e_i = \CD_i \CA^\e_i + b$, where $b\in \CT[\bone^{(i)}]\otimes \mfg$.
The product of $\CA^\e_i$ and $b$ is not renormalised, so it suffices to consider $[\CA^\e_i, \CD_i \CA^\e_i ]$,
which is expanded as,  up to degree $0$,
\begin{equs}[e:quad-exp-1]
[\bPsi_i + a_i  , \CD_i \bPsi_i + \nabla_i a_i \bone^{(i)}]
+ [\langle \nabla a_i,{\bf X}^{(i)} \rangle, \CD_i \bPsi_i]
\\
\mcb{I} [a_j^{(a)} +\bPsi_j^{(a)} ,  2 \CD_j \bPsi_i ]
+ \mcb{I} [a_j^{(a)} +\bPsi_j^{(a)} , - \bar{\CD}_i \bPsi_j  ]
\\
+ \mcb{I} [a_i  +\bPsi_i ,\CD_i \bPsi_i  ] \; , \;\; \CD_i \bPsi_i
\Big]
\\
&+ \Big[ \bPsi_i +a_i \; ,  \;\;
\CD_i \mcb{I} [a_j^{(a)} +\bPsi_j^{(a)} ,  2 \CD_j \bPsi_i ]
\\
+\CD_i  \mcb{I} [a_j^{(a)} +\bPsi_j^{(a)} , - \bar{\CD}_i \bPsi_j  ]
+\CD_i \mcb{I} [a_i + \bPsi_i , \CD_i \bPsi_i]
\Big]
+ \mcH^{(1)}\;.
% [\bPsi_i+a_i , \CD_i\mcb{I} [\bPsi_i+a_i  , \bar\bXi_i]]
%[\mcb{I} [\bPsi_i+a_i , \bar\bXi_i], \CD_i \bPsi_i]
% \Big[\bPsi_i+a_i  , \CD_i\mcb{I}  [(\CD_j^+ +\CD_j^-) \tilde\bPsi_i, \bar{\CD}_j \tilde\bPsi_j]
% -\frac12 \CD_i\mcb{I}  [\CD_j^+ \cS_{\southeast} \tilde\bPsi_j, \CD_j^+\cS_{\southwest} \tilde\bPsi_j]\Big]
% \\
%\Big[\mcb{I}  [(\CD_j^+ +\CD_j^-) \tilde\bPsi_i, \bar{\CD}_j \tilde\bPsi_j]
% -\frac12  \mcb{I}  [\CD_j^+ \cS_{\southeast} \tilde\bPsi_j, \CD_j^+ \cS_{\southwest} \tilde\bPsi_j], \CD_i \bPsi_i\Big] \;.
\end{equs}
The final term $\mcH^{(1)}$ involves trees
$\<IXiI'Xibar>$, $\<supercherry1>$, $\<I'XiIXibar>$, $\<supercherry2>$
in the group \hyperlink{(1-3)}{(1-3)}
and trees $\<I'(R2-1)new> $,
$ \<I'XiI(R2-1)new>$ in the group \hyperlink{(1-2)}{(1-2)}, all of which are not renormalised (for trees in \hyperlink{(1-3)}{(1-3)} this is because they are not of the form in~\eqref{eq:square_cherry_consts}).
The other terms only involve $\mfT^{\YM}_-$.
Upon applying the renormalisation map, only $[\mcb{I} [a_i  +\bPsi_i ,\CD_i \bPsi_i  ] , \;\CD_i \bPsi_i ]$ and $[\bPsi_i,\CD_i \mcb{I} [a_i + \bPsi_i , \CD_i \bPsi_i]]$ are affected, and the expression transforms as
\begin{equ}[eq:quad1_ren_eq]
\eqref{e:quad-exp-1} \;
\mapsto\;
\eqref{e:quad-exp-1}
%-  \left( C[ (\mcb{I} \CD_i \Psi_i ) \CD_i \Psi_i ]
%- C[ \Psi_i  \CD_i \mcb{I} (\CD_i \Psi_i)]\right)\, \Cas
\, (\bPsi_i+a_i \bone^{(i)})\;.
\end{equ}

For the term
$[ \cS_{a} \CA_j^\e , 2\mcb{D}_j \CA^\e_i  ] $, as before, it suffices to consider $[ \cS_{a} \CA_j^\e , 2\CD_j \CA^\e_i  ] $, which expands as,  up to degree $0$,
\begin{equs}[e:quad-exp-2]
[\bPsi_j^{(a)} + a_j^{(a)} , 2\CD_j \bPsi_i + 2\nabla_j a_i \bone^{(i)}]
+ [\cS_a \langle \nabla a_j,{\bf X}^{(j)} \rangle, 2\CD_j \bPsi_i]
\\
\cS_a \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} ,  2 \CD_i \bPsi_j ]
+ \cS_a \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} , - \bar{\CD}_j \bPsi_i ]
\\
&\qquad\qquad\qquad \qquad\qquad\qquad
+  \cS_a \mcb{I} [a_j  +\bPsi_j ,\CD_j \bPsi_j  ] \;, \;\; 2 \CD_j \bPsi_i
\Big]
\\
&+ \Big[ \bPsi_j^{(a)} +a_j^{(a)} \;,\;\;
2 \CD_j \mcb{I} [a_j^{(a)} +\bPsi_j^{(a)} ,  2 \CD_j \bPsi_i ]
\\
+2\CD_j  \mcb{I} [a_j^{(a)} +\bPsi_j^{(a)} , - \bar{\CD}_i \bPsi_j  ]
+2\CD_j \mcb{I} [a_i + \bPsi_i , \CD_i \bPsi_i]
\Big] + \mcH^{(2)}\;.
% [\bPsi_j^{(a)}+a_j^{(a)}  , \CD_j\mcb{I} [\bPsi_i+a_i , \bar\bXi_i]]
%[\cS_a\mcb{I} [\bPsi_j+a_j , \bar\bXi_j], \CD_j \bPsi_i]
% \Big[\bPsi_j^{(a)}+a_j^{(a)} , \CD_j\mcb{I}  [(\CD_j^+ +\CD_j^-) \tilde\bPsi_i, \bar{\CD}_j \tilde\bPsi_j]
% -\frac12 \CD_j\mcb{I}  [\CD_j^+\cS_{\southeast} \tilde\bPsi_j, \CD_j^+\cS_{\southwest} \tilde\bPsi_j]\Big]
% \\
%\Big[\cS_a \mcb{I}  [(\CD_i^+ +\CD_i^-) \tilde\bPsi_j, \bar{\CD}_i \tilde\bPsi_i]
% -\frac12 \cS_a \mcb{I}  [\CD_i^+\cS_{\southeast} \tilde\bPsi_i, \CD_i^+ \cS_{\southwest} \tilde\bPsi_i], \CD_j \bPsi_i\Big] \;.
\end{equs}
The final term $\mcH^{(2)}$ again denotes a term involving %the same trees
trees of the same form
as $\mcH^{(1)}$ in \eqref{e:quad-exp-1} -- again not renormalised (the trees in \hyperlink{(1-3)}{(1-3)} are again not of the form in \eqref{eq:square_cherry_consts}).
One can check that $M^\e$ affects only
$[\cS_a \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} , - \bar{\CD}_j \bPsi_i ], 2\CD_j\bPsi_i]$,
which at $e\in \obonds_i$ is equal to
(recalling \eqref{e:Sa1}, the definition of $\cS_a$, and the fact that $\bPsi_i$ is a constant modelled distribution)
%taking the constant value $\Psi_i\eqdef \id\in \CT[\CI\bXi_i]\otimes \mfg \simeq \mfg^*\otimes \mfg$, note that
\begin{equs}[e:calc-double-shifts]
%{} &\Big[ \cS_a \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} , - \bar{\CD}_j \bPsi_i ]
%, 2 \CD_j \bPsi_i \Big] (e)
{}&\frac{1}{16} \sum_{\be,\be'\in \CE_\times} \!\!
\Big(a_i(e+\be+\be') \Big[ \CS_{\be} \mcb{I} [\CS_{\be'} \bone^{(i)}, - \bar{\CD}_j \Psi_i ]
, 2 \CD_j \Psi_i \Big]
\\
&\qquad \qquad\qquad\qquad\qquad
+\Big[ \CS_{\be} \mcb{I} [\CS_{\be'} \Psi_i, - \bar{\CD}_j \Psi_i ]
, 2 \CD_j \Psi_i \Big]\Big)\;.
\end{equs}
Upon applying the renormalisation map  (in particular see Example~\ref{ex:M_tau}),
\begin{equs}[eq:C_1_ren_eq]
\eqref{e:quad-exp-2}
\mapsto
\eqref{e:quad-exp-2}
&+2 \ad_{\hat{C}^\e_1}
\,(\bPsi_i+a_i \bone^{(i)})
\\
&+  \frac{1}{16}\!\! \sum_{\be,\be'\in \CE_\times} \!\!  2 \ad_{\hat{C}^\e_{1,\be}} (a_i(\cdot+\be+\be')-a_i)\bone^{(i)}
\;.
\end{equs}
Note that $a_i(e+\be+\be')-a_i(e)=(\hat\CR \CV_i)(e+\be+\be')-(\hat\CR \CV_i)(e)$.

For the term
$[ \cS_{a} \CA_j^\e ,  - \bar{\mcb{D}}_i\bPsi_j -\hat{\mcb{D}}_i\CV^\e_j] $,
where we recall that $\hat{\mcb{D}}_i=\frac12
\mcb{D}_i^+ (\cS_{\northwest}+\cS_{\southwest})$,
we again  only need to consider $[ \cS_{a} \CA_j^\e ,  - \bar{\CD}_i \bPsi_j -\frac12
\CD_i^+ (\cS_{\northwest}+\cS_{\southwest}) \CV^\e_j] $, which expands as, up to degree $0$ and
with $b$ denoting a generic polynomial term that does not enter any terms that are renormalised,
\begin{equs}[e:quad-exp-3]
[\bPsi_j^{(a)} + a_j^{(a)} , -\bar{\CD}_i \bPsi_j +b]
+ [b, \bar \CD_i \bPsi_j]
\\
\cS_a \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} ,  2 \CD_i \bPsi_j ]
+ \cS_a \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} , - \bar{\CD}_j \bPsi_i ]
\\
&\qquad\qquad\qquad \qquad
+  \cS_a \mcb{I} [a_j+\bPsi_j ,\CD_j \bPsi_j  ] \;, \;\;  -\bar{\CD}_i \bPsi_j
\Big]
\\
&+ \Big[ \bPsi_j^{(a)} +a_j^{(a)} \;,\;\;
\CD_i^+ (\cS_{\northwest}+\cS_{\southwest}) \Big\{
\mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} ,  2 \CD_i \bPsi_j ]
\\
+  \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} , - \bar{\CD}_j \bPsi_i ]
+ \mcb{I} [a_j + \bPsi_j , \CD_j \bPsi_j]
\Big\}
\Big]
\\
[\bPsi_j^{(a)}+a_j^{(a)}  , \frac12
\CD_i^+ (\cS_{\northwest}+\cS_{\southwest})
\mcb{I} [\bPsi_j+a_j  , \bar\bXi_j]]
\\
&\qquad -\frac{\e^\kappa}{2}
[\cS_a\mcb{I} [\bPsi_j+a_j , \bar\bXi_j], \bar{\CD}_i \bPsi_j] + \mcH^{(3)}\;,
% \Big[\bPsi_j^{(a)}+a_j^{(a)} , \hat{\CD}_i\mcb{I}  [(\CD_i^+ +\CD_i^-) \tilde\bPsi_j, \bar{\CD}_i \tilde\bPsi_i]
% + \hat{\CD}_i\mcb{I}  [\CD_i^+ \cS_{\southeast} \tilde\bPsi_i, \CD_i^+ \cS_{\southwest} \tilde\bPsi_i]\Big]
% \\
%\Big[\cS_a \mcb{I}  [(\CD_i^+ +\CD_i^-) \tilde\bPsi_j, \bar{\CD}_i \tilde\bPsi_i]
% + \cS_a \mcb{I}  [\CD_i^+ \cS_{\southeast} \tilde\bPsi_i, \CD_i^+ \cS_{\southwest} \tilde\bPsi_i], \bar{\CD}_i \bPsi_j\Big] \;,
\end{equs}
where $\mcH^{(3)}$ denotes a term involving trees of the form $\<I'(R2-1)new> $,
$ \<I'XiI(R2-1)new>$ in the group \hyperlink{(1-2)}{(1-2)}, which are not renormalised.
%we used the shorthand $\hat\CD_i \eqdef \frac12 \CD_i^+(\cS_{\northwest}+\cS_{\southwest})$ in terms that are not renormalised.
The other terms in the final two lines involve trees $\<IXiI'Xibar>$, $\<supercherry1>$, $\<I'XiIXibar>$, $\<supercherry2>$ from the group \hyperlink{(1-3)}{(1-3)},
which {\it are} now renormalised according to~\eqref{eq:square_cherry_consts}.
The only terms from the first five lines that the renormalisation map affects are $[\bPsi_j^{(a)} , -\bar{\CD}_i \bPsi_j ]$,
$[\cS_a \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} ,  2 \CD_i \bPsi_j ]  , -\bar{\CD}_i \bPsi_j]$,
\CD_i^+ (\cS_{\northwest}+\cS_{\southwest}) \mcb{I} [a_i^{(a)} +\bPsi_i^{(a)} ,  2 \CD_i \bPsi_j ]  ]$.
Recalling~\eqref{eq:tilde_C_renorm}, the first term does not contribute anything to the renormalised equation
thanks to the Lie bracket.
upon applying the renormalisation map, \eqref{e:quad-exp-3} is mapped to itself plus
\begin{equs}[eq:non_local_A_j]{}
&\frac{1}{16}\sum_{\be,\be'\in \CE_\times}
\!\!
\Big\{
\,(\bPsi_i+a_i(\cdot+\be+\be'))
\!\!\!\!
\sum_{\bw\in\{\Northwest,\Southwest\}}
\!\!\!\!
\ad_{\hat{C}^\e_{3,\be,\bw}}(\bPsi_i+a_i(\cdot^\bw+\be'))
\Big\}
\\
&+ \frac{\e^{\kappa}}{4} \sum_{\be\in\CE_\times}
\!\!
\Big\{
2\ad_{C_{\be}[\<I'XiIXibar>]}(\CS_{\be}\Psi_j + a_j(\cdot+\be))
\!\!\!\!
\sum_{\bw\in\{\Northwest,\Southwest\}}
\!\!\!\!
\ad_{C_{\be,\bw}[\<IXiI'Xibar>]}(\CS_{\bw}\Psi_j + a_j(\cdot^\bw))
\Big\}\;.
\end{equs}
By a similar calculation as in \eqref{e:calc-double-shifts}-\eqref{eq:C_1_ren_eq},
the first line in~\eqref{eq:non_local_A_j} can be written as a local term plus a difference as
\begin{equs}{}
\,(\bPsi_i+a_i )
\label{eq:C_2_C_3_ren_eq}
\\
&+  \frac{1}{16} \!\! \sum_{\be,\be'\in \CE_\times} \!\!
\Big\{
2\ad_{\hat{C}^\e_{2,\be}} (a_i(\cdot+\be+\be')-a_i)
\!\!\!
\sum_{\bw\in\{\Northwest,\Southwest\}}
\!\!\!
\ad_{\hat C^\e_{3,\be,\bw}}(a_i(\cdot^\bw+\be')-a_i)\Big\}\;.
\end{equs}
Again, all the terms above take values in $\CT^{(i)}\otimes \mfg$.
For instance, for a term $\mcb{I} [a_j + \bPsi_j , \CD_j \bPsi_j] $  in \eqref{e:quad-exp-3}
it take values in $\CT^{(j)}\otimes \mfg$,
but then $\CD_i^+(\cS_{\northwest}+\cS_{\southwest})$ maps it into $\CT^{(i)}\otimes \mfg$
according to Section~\ref{sec:tr-reg}.

By \eqref{e:CA-expand} and the definition of $\cE$ in \eqref{eq:B_expand}, one has, up to degree $1$,
\begin{equs}
\cE \CA^\e_i
= (\e a_i + \e \Pi \bPsi_i ) \bone^{(i)}
+\frac{\e^{1+\kappa}}{2}  \mcb{I} [  \bPsi_i +a_i \bone^{(i)},   \bar\bXi_i ]
+\e\langle \nabla a_i, {\bf X}^{(i)}\rangle\;.
\end{equs}
Below we write $\e A_i = \e a_i + \e \Pi \bPsi_i = \e \hat\CR \CA_i^\e$
(the second equality follows from Lemma~\ref{lem:Rec_Mult_eps}).
So, up to degree $1$,
\begin{equs}
F(\cE \CA^\e_i)
&= F(\e A_i) \bone^{(i)}
\\
&\quad + F'(\e A_i)
\Big(\e^\kappa\bar\bPsi_i
+\frac{\e^{1+\kappa}}{2}  \mcb{I} [  \bPsi_i +a_i \bone^{(i)},   \bar\bXi_i ]
+\e\langle \nabla a_i, {\bf X}^{(i)}\rangle \Big)\;.
\end{equs}
Here, recalling $F$ from \eqref{eq:def_F},
$F'(\e A_i)$   takes values in $L(\mfg\otimes \mfg,\mfg)$ and
the right-hand side of the above identity takes values in $\CT^{(i)}\otimes L(\mfg,\mfg)$.
Moreover the term $ [\CA^\e_i,\bar\bXi_i]$
is expanded as,  up to degree $0$,
\begin{equs}[e:multi-noise-exp]
{}&   \Big[
\bPsi_i + a_i
\mcb{I} \big[\bPsi_j^{(a)} + a_j^{(a)},
2\CD_j \bPsi_i +\bar{\CD}_i \bPsi_j \big]
\mcb{I} \big[ \bPsi_i +a_i  , \CD_i \bPsi_i \big]
\;, \;\;
\bar\bXi_i \Big]
\\
\Big[
\frac{\e^\kappa}{2}  \mcb{I} [  \bPsi_i +a_i ,   \bar\bXi_i ] \;, \;\; \bar\bXi_i \Big]
\\
&+\Big[ - \frac{\eps^\kappa}{2} \mcb{I}  [(\CD_j^+ +\CD_j^-) \tilde\bPsi_i, \bar{\CD}_j \tilde\bPsi_j]
+ \frac{\e^\kappa}{4}\mcb{I} [\CD_j^+ \cS_{\southeast} \tilde\bPsi_j, \CD_j^+ \cS_{\southwest} \tilde\bPsi_j]\;,
\;\;
\bar\bXi_i \Big]\;.
\end{equs}
The first line involves the trees $\<Xi1>\,$,
$  \<PsiXi1>$,
$\<cherryY>$, and
$\<cherryYless>$ from group \hyperlink{(1-3)}{(1-3)}.
They are not renormalised,
except for $\<PsiXi1>\,$, but this does not contribute to the renormalised equation due to the Lie bracket
$[  \bPsi_i , \bar\bXi_i ] $.
The second line
$\<I(PsiXi1)Xi1>$ and $\<I(PsiXi1)Xi1less>$ from group \hyperlink{(3-3)}{(3-3)} which require renormalisation by  \eqref{e:M-XiIXi}.
The last line involves trees of the form $\<I(R2-1)Xi1new>$ from group \hyperlink{(2-3)}{(2-3)} which do not require renormalisation.

Therefore,  up to degree $0$,
\begin{equs}\label{e:F-multi-noise-exp}
\e^\kappa & F(\cE \CA^\e_i)[\CA^\e_i,\bar\bXi_i]
\e^\kappa  F(\e A_i)  \eqref{e:multi-noise-exp}
\\
& + \e^\kappa
F'(\e A_i)
\Big(
\e^\kappa\bar\bPsi_i
+\frac{\e^{1+\kappa}}{2}  \mcb{I} [  \bPsi_i
+a_i ,   \bar\bXi_i ]
+\e\langle \nabla a_i, {\bf X}^{(i)}\rangle \Big)
[ \bPsi_i + a_i,
\bar\bXi_i ] \;.
\end{equs}
the right-hand side takes values in $\CT^{(i)}\otimes \mfg$.
The trees involved in the first term on the right-hand side are explained below \eqref{e:multi-noise-exp}.
The second line
involves the trees  $\<I(PsiXi1)Xi1>$,
$\<I(PsiXi1)Xi1less>$, $\<r_z_large>$, $\<PsiPsibarXibar>$ from group \hyperlink{(3-3)}{(3-3)}
$ \X^{(i)} \bar\Xi_i$,
$\X^{(i)} \<PsiXi1>$  from group \hyperlink{(1-3)}{(1-3)}.\footnote{The second line
should be understood as $\e^\kappa$ times the derivative
of $F$ at $\e A_i$ in the direction given by the expression in the parenthesis, which gives us an element of $\CT\otimes L(\mfg,\mfg)$, and then we apply this to $[ \bPsi_i+a_i,\bar\bXi_i ] \in \CT\otimes \mfg$
which yields an element of $\CT\otimes \mfg$.}

To find the action of renormalisation map on \eqref{e:F-multi-noise-exp},
we note that
the contractions of  $\<PsiXi1>$ specified  in \eqref{e:M-XiIXi}
in the trees $\<r_z_large>$, $\<PsiPsibarXibar>$ and
$\X^{(i)} \<PsiXi1>$
do not have any contribution to the renormalised equation thanks to the last Lie bracket in
\eqref{e:F-multi-noise-exp}.
Only the contractions of  $\<I(PsiXi1)Xi1less>$  in \eqref{e:M-XiIXi} and
of  $\<r_z_large>$
in \eqref{e:M-four} need to be considered.

Recall from \eqref{eq:def_F} and \eqref{eq:Phi_inverse} that
\begin{equ}\label{eq:def_F-rep}
F(X) Z %= \frac{\Phi(X)^{-1}-1}{\ad_X}
= \sum_{k=0}^\infty c_k  \ad_X^k Z
= \frac12 Z + \frac{1}{12}\ad_X Z %+ r_F(X)\;, \qquad  r_F(X) = O(X^2)
\end{equ}
for some coefficients $c_k\in\R$. So,
\begin{equ}\label{eq:F-prime}
=\sum_{k=1}^\infty c_k \sum_{m=0}^{k-1} \ad_X^m \, \ad_Y \, \ad_X^{k-m-1} Z
= \frac{1}{12}\ad_Y Z + O(XY)Z\;. %+ r'_F(X)(Y)\;.
\end{equ}
Consider the following term from the first line on the right-hand side of \eqref{e:F-multi-noise-exp}:
\begin{equ}[e:F-multi-noise-1]
\frac{\e^{2\kappa}}{2}  F(\e A_i)
[\mcb{I} [  \bPsi_i +a_i \bone^{(i)},   \bar\bXi_i ],  \bar\bXi_i ] \;.
\end{equ}
Upon applying by $M^\e$,  by  \eqref{e:M-XiIXi} and the exact value of $C^\e[ \<I(PsiXi1)Xi1less> ]= \e^{-2\kappa}\Cas$ given in \eqref{e:C-XiIXi},\footnote{Heuristically,
the renormalisation $-\frac{1}{4} \ad_{\Cas}(\bPsi_i+a_i \bone^{(i)})$ for \eqref{e:F-multi-noise-1} could be understood
as obtained by ``substituting $A$ in $\frac\eps2[A,\xi]$ in
\eqref{e:Z-terms} by $\frac\eps2 \mcb{I}[A,\xi]$''.}
\begin{equs}[e:F-multi-noise-1-result]
\eqref{e:F-multi-noise-1} & \mapsto
\eqref{e:F-multi-noise-1}
- \frac{1}{2} F(\e A_i)
\ad_{\Cas}(\bPsi_i+a_i \bone^{(i)})
\\
\eqref{e:F-multi-noise-1}
- \frac{1}{4}
\ad_{\Cas}(\bPsi_i+a_i \bone^{(i)})
%- \frac{1}{2} (\frac{1}{12}\ad_{\e A_i} + r_F(\e A_i))
- \frac{1}{2} (F(\e A_i)-\frac12)\,
\ad_{\Cas}(\bPsi_i+a_i \bone^{(i)}) \;.
\end{equs}
It is important to note that
the last term
will not contribute to the renormalised equation.
Indeed, this follows from  $\e A_i = \e\hat\CR \CA^\e_i = \e a_i + \e \Pi \bPsi_i $,
%and, recalling our shorthand  $\e A_i = \e a_i + \e \Pi \bPsi_i $,
%one has
%$\e A_i = \e \hat\CR \CA^\e_i$
and the fact that, in \eqref{eq:def_F-rep}, the summand for every $k\ge 1$ vanishes if $X=Z$.

writing again $(e_l)_l$ for an orthonormal basis of $\mfg$,
consider the following term from the second line of \eqref{e:F-multi-noise-exp},
corresponding to the trees $\<PsiPsibarXibar>$ and $\<I(PsiXi1)Xi1less>\,$:
\begin{equs}\label{e:F-multi-noise-2}
\e^{2\kappa} &
F'(\e A_i)
[ \bPsi_i + a_i \bone^{(i)},
\bar\bXi_i ]
\\
\e^{2\kappa}
\sum_{j,n}
e_j^*\otimes e_n^* \otimes
F'(\e A_i)
[\bPsi_i + a_i\bone^{(i)},
\in ( \CT[\<PsiPsibarXibar>] \oplus \CT[\<I(PsiXi1)Xi1less>] )  \otimes\mfg\;.
\end{equs}
Here and below, for $f\colon \R\times\T^2\to\CT\otimes \mfg$ and $\tilde\CT\subset \CT$, we write $f\in \tilde\CT  \otimes\mfg$ to mean $f$ takes values in $\tilde\CT  \otimes\mfg$.
We remark that the $\CT[\<PsiPsibarXibar>]   \otimes\mfg$ component can be alternatively written as
\sum_{j,m,n}
e_j^*\otimes e_m^* \otimes e_n^*\otimes
F'(\e A_i) (e_j)
[e_m,e_n]  $.
Contracting $\<I(PsiXi1)Xi1less>$ we have
\begin{equs}
\e^{2\kappa}
\langle
e_j^* \otimes e_n^* , C^\e[ \<I(PsiXi1)Xi1less> ]\,
\rangle
F'(\e A_i)
[ \bPsi_i + a_i \bone^{(i)},
\\
\sum_{j}
F'(\e A_i)
[ \bPsi_i + a_i \bone^{(i)},
e_j]  \;\;\in ( \CT[\<IXi>] \oplus \CT[\bone^{(i)}] )  \otimes\mfg\;,
\end{equs}
where we used again the value of $C^\e[ \<I(PsiXi1)Xi1less> ]$ in \eqref{e:C-XiIXi}.
upon applying $M^\e$,\footnote{Again heuristically, the renormalisation $\frac{1}{12} \ad_{\Cas}(\bPsi_i+a_i \bone^{(i)})$  for \eqref{e:F-multi-noise-2} can be seen as contracting
the first $A$ and $\xi$ in $\frac{\eps^2}{12}[A,[A,\xi]]$ in \eqref{e:Z-terms}.}
\begin{equs}[e:F-multi-noise-2-result]
\eqref{e:F-multi-noise-2}
\eqref{e:F-multi-noise-2}
- \sum_{j} F'(\e A_i)(e_j)[\bPsi_i + a_i \bone^{(i)},e_j]
\\
\eqref{e:F-multi-noise-2}
+ \frac{1}{12}
\ad_{\Cas}(\bPsi_i+a_i \bone^{(i)})
\\
- \sum_{j} %r'_F(\e A_i) (e_j)
(F'(\e A_i)(e_j) - \frac{1}{12}  \ad_{e_j})
[\bPsi_i + a_i \bone^{(i)},e_j]\;.
\end{equs}
Next, consider the following terms from the second line of \eqref{e:F-multi-noise-exp},
\begin{equs}\label{e:F-multi-noise-3}
{}& \frac{\e^{1+2\kappa}}{2}  F'(\e A_i ) ( \mcb{I} [  \bPsi_i + a_i \bone^{(i)} ,  \bar\bXi_i ] )
[ \bPsi_i + a_i \bone^{(i)},
\bar\bXi_i ]
\\
%&= \frac{\e^{1+2\kappa}}{2}  \sum_{j,n}  e_j^*\otimes  e_n^* F'(\e A_i)([\bPsi_i+a_i \bone^{(i)},e_j])[\bPsi_i+a_i \bone^{(i)},e_n]
&\qquad\qquad \in (\CT[\<r_z_large>]\oplus \CT[\<PsiPsibarXibar>] \oplus \CT[\<I(PsiXi1)Xi1>] \oplus \CT[\<I(PsiXi1)Xi1less>] )\otimes\mfg\;.
\end{equs}
Contracting $\<I(PsiXi1)Xi1less>$ and $\<r_z_large>$ with the basis of $\mfg$ as above  using the definitions of  $C^\e[\<I(PsiXi1)Xi1less>]$  in \eqref{e:C-XiIXi}, and $C^\e[\<r_z_large>]$
in \eqref{e:def-C-four},
one has that, upon applying $M^\e$,
\begin{equs}[e:F-multi-noise-3-result]
\eqref{e:F-multi-noise-3}
\mapsto
\eqref{e:F-multi-noise-3}
& -  \frac{\e}{2} \sum_{j}  F'(\e A_i)([\bPsi_i+a_i \bone^{(i)},e_j])[\bPsi_i+a_i \bone^{(i)},e_j]
\\
& - \frac{\e^{1+2\kappa}}{2} ( c_1^\e[\<r_z_large>]-c_2^\e[\<r_z_large>]) \sum_{j,k}  F'(\e A_i) ([e_j,e_k]) [e_j,e_k]\;.
\end{equs}
Here the second term on the right-hand side belongs to
$( \CT[ \<IXi^2>]\oplus \CT[ \<IXi>] \oplus \CT[ \bone^{(i)}] )\otimes\mfg$,
and the third term belongs to $\CT[ \bone^{(i)}] \otimes\mfg$.

Combining~\eqref{e:F-multi-noise-1-result},~\eqref{e:F-multi-noise-2-result}, and~\eqref{e:F-multi-noise-3-result},
we find that upon applying $M^\e$,
\begin{equ}[eq:F_ren_eq]
\eqref{e:F-multi-noise-exp} \mapsto \eqref{e:F-multi-noise-exp}
- \frac{1}{6}
\ad_{\Cas}(\bPsi_i+a_i \bone^{(i)})
+ (\cdots) \;,
\end{equ}
where $- \frac{1}{6} =- \frac{1}{4}+ \frac{1}{12}$ and
$ (\cdots)$ denotes the sum of
the last term in \eqref{e:F-multi-noise-1-result} (which will not contribute
to the renormalised equation upon reconstruction by the remark following~\eqref{e:F-multi-noise-1-result}),
the last term in \eqref{e:F-multi-noise-2-result}, and the last two terms  in \eqref{e:F-multi-noise-3-result}.
Since $\e A_i = \e \hat\CR \CA^\e_i$, the contribution of $(\cdots)$
to the renormalised equation (upon reconstruction) is precisely
-h_1^\e(\hat\CR \CA^\e_i)
-h_2^\e(\hat\CR \CA^\e_i)
-h_3^\e(\hat\CR \CA^\e_i)
for $h^\e_\ell$ defined by~\eqref{e:def-h123A},
which is cancelled by $\boldH_i$ in~\eqref{e:fix-pt-Ae}.

Finally, for $ \CQ^{\e}_i$ defined by~\eqref{e:def-tildeCR2},
we have $\mcb{D} \CV = \CD\CV + b$ for the various differentiations, where $b\in \CT[\bone]\otimes \mfg$.
Since the product of $\mcb{D}\bar\bPsi$ and $b$ is not renormalised,
it suffices to expand $ \CQ^{\e}_i$ with all $\mcb{D}$ replaced by $\CD$,
which gives, up to degree $0$,
\begin{equs}[e:tildeR2-exp]
%  - \frac{\eps^\kappa}{2}  [\CD_j^+ +\CD_j^-) \tilde\bPsi_i, \bar{\CD}_j \tilde\bPsi_j]
%   + \frac{\e^\kappa}{4} [\CD_j^+\cS_{\southeast} \tilde\bPsi_j, \CD_j^+\cS_{\southwest} \tilde\bPsi_j]
% \\
% &
\mcH^{(4)} -\frac{\eps^\kappa}{4}
\Big[(\CD_j^+ +\CD_j^-) \bar\bPsi_i,
{\CD}_j^+ (\cS_{\southeast}+\cS_{\southwest})\CI_j
\CH_i\{\<IXiI'Xi_notriangle>\}
\Big]
\\
\Big[
(\CD_j^+ +\CD_j^-) \CI_i
\CH_j\{\<IXiI'Xi_notriangle>\},
\bar{\CD}_j \bar\bPsi_j
\Big]
\\
+ \frac{\e^\kappa}{4}
\Big[
\CD_j^+ \cS_{\southeast}\bar\bPsi_j, \CD_j^+\cS_{\southwest}\CI_j
\CH_i\{\<IXiI'Xi_notriangle>\}
\Big]
+ \frac{\e^\kappa}{4}
\Big[
\CD_j^+\cS_{\southeast}\CI_j
\CH_i\{\<IXiI'Xi_notriangle>\}
,  \CD_j^+ \cS_{\southwest}\bar\bPsi_j
\Big]\;,
% \\
% &
% -\frac{\eps^\kappa}{2}
% \Big[(\CD_j^+ +\CD_j^-) \bar\bPsi_i,
%\mcb{I}_j [\bPsi_j+a_j,\bar\bXi_j]
% \Big]
% \\
% &
%  -\frac{\eps^\kappa}{2}
% \Big[
% \frac{\eps^\kappa}{2}
% (\CD_j^+ +\CD_j^-)
%\mcb{I} [\bPsi_i+a_i,\bar\bXi_i],
% \bar{\CD}_j \bar\bPsi_j
% \Big]
% \\
%+ \frac{\e^\kappa}{4}
%\CD_j^+ \cS_{\southeast}\bar\bPsi_j,
% \CD_j^+ \cS_{\southwest}
%\mcb{I} [\bPsi_j+a_j,\bar\bXi_j]
% \Big]
% \\
% &
%+ \frac{\e^\kappa}{4}
%\CD_j^+  \cS_{\southeast}
%\mcb{I} [\bPsi_j+a_j,\bar\bXi_j]
%  ,  \CD_j^+ \cS_{\southwest} \bar\bPsi_j
%+ \CH_i\{\<I'Xi1I'(R2-1)new>\}
\end{equs}
\begin{equ}[e:def-CHi]
\CH_i \{\<IXiI'Xi_notriangle>\}\eqdef
\big[\bPsi_i^{(a)} + a_i^{(a)},
2\CD_i \bPsi_j +\bar{\CD}_j \bPsi_i \big]
\big[ \bPsi_j +a_j \bone^{(j)} , \CD_j \bPsi_j \big]\;,
\quad j\eqdef 3-i
\end{equ}
and $\CH_j \{\<IXiI'Xi_notriangle>\}$
is defined in the same way with $i,j$ exchanged.
The first term
involves trees of the form $\<R2-1new>$ in \eqref{e:trees-deg-1-rem}, $\<cherry231>$ and
$\<cherry232>$ in group \hyperlink{(2-3)}{(2-3)}, and $ \<I'Xi1I'(R2-1)new>$
in group \hyperlink{(2-2)}{(2-2)},
which are not renormalised.
The remaining terms are all of the form
$\<I'XiI'[IXiI'Xi]>$ and $\<I'XiI'[IXiI'Xi]less>$ in group \hyperlink{(1-2)}{(1-2)}.

By definition of $M^\e$ in Section~\ref{subsec:renorm_group},
especially in view of \eqref{e:div-R2}, only $6$ terms in \eqref{e:tildeR2-exp}
will be renormalised:
they are precisely the $6$ terms containing $\CH_i \{\<IXiI'Xi_notriangle>\}$
(but not $\CH_j \{\<IXiI'Xi_notriangle>\}$).
Upon applying $M^\e$ \eqref{e:tildeR2-exp} is mapped to itself plus
\begin{equs}{}
&- \frac{\e^\kappa}{4}
\ad_{C_{\southeast,+}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
\ad_{C_{\southeast,-}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
+2\ad_{C_\southeast^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
(\bPsi_i + a_i^{(a)}(\cdot^{\southeast}))
\\
&- \frac{\e^\kappa}{4}
(\ad_{C_{\southwest,+}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
+\ad_{C_{\southwest,-}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
-2\ad_{C_\southwest^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
(\bPsi_i + a_i^{(a)}(\cdot^{\southwest}))\;.
\end{equs}
Note that the first line above
can be written as a local term plus a difference as
\begin{equs}[e:C_tree_ren_eq]{}
&- \frac{\e^\kappa}{4}
\ad_{C_{\southeast,+}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
\ad_{C_{\southeast,-}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
+2\ad_{C_\southeast^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
(\bPsi_i + a_i)
\\
\frac14\sum_{\be\in\CE_\times} \frac{\e^\kappa}{4} (
\ad_{C_{\southeast,+}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
\ad_{C_{\southeast,-}^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
+2\ad_{C_\southeast^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}
(a_i(\cdot^{\southeast}+\be) - a_i)\;,
\end{equs}
and likewise for the second line with $\Southeast$ and $2\ad_{C_\southeast^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}$ replaced by $\Southwest$ and $-2\ad_{C_\southwest^\e [\, \<I'XiI'[IXiI'Xi]less>\,]}$.

We now observe that $\delta_i\hat\CR\CV_i^\e$ in~\eqref{e:fix-pt-Ae}, by its definition~\eqref{eq:delta_A_def}, upon reconstruction cancels the
`non-local' renormalisations above involving $a_i$, i.e. the second lines of~\eqref{eq:C_1_ren_eq},~\eqref{eq:C_2_C_3_ren_eq}, and~\eqref{e:C_tree_ren_eq} (and its analogue for $\Southwest$).
Likewise, $\bar\delta_i\hat\CR\CA_j^\e$ in~\eqref{e:fix-pt-Ae}, by its definition~\eqref{eq:bar_delta_def},
cancels the second line of~\eqref{eq:non_local_A_j}.
Furthermore, recalling
$C^\e_{\sym}$ defined in \eqref{e:def-CSYM},
the reconstruction of the sum of the renormalisation terms in~\eqref{eq:cubic_ren_eq},~\eqref{eq:quad1_ren_eq},
and the first lines of~\eqref{eq:C_1_ren_eq} and~\eqref{eq:C_2_C_3_ren_eq}
is $\ad_{C^\e_{\sym}} \hat\CR\CA_i^\e$.
Finally, recalling $C^\e_{\rem}$ defined in~\eqref{e:def-CREM},
the reconstruction of the sum of the renormalisation terms in the first lines of~\eqref{eq:F_ren_eq} and~\eqref{e:C_tree_ren_eq} (and its analogue for $\Southwest$) is
$\ad_{C^\e_{\rem}} \hat\CR\CA_i^\e$.

To finally find the equation that $\hat\CR\CA^\e$ solves,
note that $(\hat\Pi^\e,\hat\Gamma^\e)$ is admissible, and thus $\hat\CR \CP^{i;\e} = P^{i;\e} *_{(i)} \hat\CR$.
It therefore suffices to find $\hat\CR f$ as a function of $\hat\CR\CA^\e$, where $f$ is any one of the terms in the parentheses $(\cdots)$ in~\eqref{e:fix-pt-Ae}.
To this end, for all such $f$ and all $x\in\R\times\obonds_i$,
\begin{equ}
\hat \CR f(x) = (\hat \Pi_x f(x))(x) = (\Pi_x M^\e f(x))(x) = \CR (M^\e f)(x)\;,
\end{equ}
where $\CR$ is the reconstruction operator associated to the canonical model $(\Pi^\e,\Gamma^\e)$
(this identity holds as written if $\hat \CR f$ is continuous in time
and needs to be understood in the distributional sense otherwise, i.e.
for $f=\bXi_i$ or $f=\e^\kappa F(\cE\CA_i^\e)[\CA^\e_i, \bar\bXi_i]$).
Moreover $\CR$ commutes with derivatives (Lemma~\ref{lem:DRcommute}), shiftings (Lemma~\ref{lem:RecShft}), products (Remark~\ref{rem:canonical_R_mult}), and with composition of smooth functions,
and we have
\begin{equ}
\CR\CA_i^\e(x)=(\Pi_x \CA_i^\e(x))(x) = (\hat \Pi_x \CA_i^\e(x))(x) = \hat\CR \CA_i^\e(x)\;,
\end{equ}
and likewise for $\CV_i^\e, \bXi_i, \bar \bXi_i$ and for $\mcb{D}\cS\CA^\e$ for any derivative and shift operators $\mcb{D},\cS$.
We conclude from the above computations that $\hat\CR\CA^\eps$ solves the discrete SPDE~\eqref{e:Aeps} plus the following terms on the right-hand side:
\begin{equ}[e:Ae-renormalised-mass]
(\ad_{C^\e_{\sym}}+\ad_{C^\e_{\rem}})A_i^\e +  (c^\eps_{\star,i} -  \tilde{c}^\e_{\star,i} ) A^\e\;.
\end{equ}
Note that $C^\e_{\sym}=O(1)$ and $C^\e_{\rem}=O(1)$ by Lemmas~\ref{lem:C_sym-finite} and~\ref{lem:thick-trig-bahave} respectively,
while $\tilde c^\e_\star = O(1)$ by
Definition~\ref{def:tildec} and since each $c^\e$ in Lemmas~\ref{lem:I3} and \ref{lem:grad-R-hat} is $O(1)$ due to Lemma~\ref{lem:Psi_Wick}.
The proof is therefore done with the choice
\begin{equ}[e:specify-c_star]
c^\e_\star \eqdef - \ad_{C^\e_{\sym}} - \ad_{C^\e_{\rem}} + \tilde{c}_\star^\e = O(1)\;.
\end{equ}
\end{proof}

\subsection{Uniform bounds on renormalised discrete models}
\label{subsec:moments_discrete_models}

In this section we derive uniform moment bounds on the renormalised models
$Ẑ^= (Π̂^,Γ̂^)$ from Section~\ref{subsec:renorm_eq}.
\begin{proposition}\label{prop:Ze_moments}
For all $p\geq 1$, $\gamma>0$, and compact $\K\subset \R\times\T^2$, one has $\sup_{\e>0} \E |\$\hat Z^\e\$^{(\e)}_{\gamma;\K}|^p < \infty$.
\end{proposition}
\begin{proof}
The proof follows from Lemmas~\ref{lem:mom-YM},~\ref{lem:mom-xi},~\ref{lem:mom-12},~\ref{lem:mom-13} and~\ref{lem:mom-222333}
and a Kolmogorov-type criterion (see~\cite[Thm.~6.1]{HM18} or~\cite[Thm.~10.7]{Hairer14}).
\end{proof}
The main ingredient in the above proof is the bounds
\minilab{e:mom-bounds}
\begin{equs}
\E[(\hat\Pi_{z_*}^\e\tau)(\phi_{z_*}^{\lambda})^2]
\lambda^{2(|\tau|+\kappa)},			\label{e:mom-bounds1}
\\
\E\Big[\Big(\int (\hat\Pi_{z_*}^\e\tau)(s,x_*)(\CS_{2,t_*}^{\lambda}\phi)(s)\mrd s\Big)^2\Big]
&\lesssim \lambda^{2(|\tau|+\kappa)}\;,		\label{e:mom-bounds2}
\end{equs}
for each  $τ∈^(i)_-∩{^⊕^}$
(recall Remark~\ref{rem:model_bounds_sector}), uniformly in $z_*=(t_*,x_*)∈×_i$,
and $> 0$. Here the second moment bounds suffice since our noise is Gaussian.
The first bound is also uniform in $λ∈(,1]$
and all space-time test functions $ϕ$ as in~\eqref{eq:Pi},
and the second bound is uniform in $λ∈(0,]$ and all temporal  functions $ϕ$ as in~\eqref{eq:seminorm}.
The rest of this subsection is devoted to the proof of these bounds.

We will need to verify \eqref{e:mom-bounds} for each of our trees in Section~\ref{sec:tr-reg}
{\it one by one} for several reasons.
First, in our discrete setting, we do not have the general ``black-box'' results for probabilistic bounds as in \cite{CH16,HS23_BPHZ,LOTT21} (see also~\cite{BB23,BN22_discrete,BN22,KPZJeremy}).
Also we do not  use the general discrete bounds in the appendix of \cite{EH21},
since in our setting, verifying the conditions therein would  not be easier.
Another, more important reason
is that
since we introduced the noises such as $ξ̅=^1-κ ξ$, the models
will yield functions possibly containing factors of $$ which are crucial for us to obtain  \eqref{e:mom-bounds}.
we need to carefully examine parity, i.e. whether
certain functions in the integrands are exactly or only approximately odd,
due to discrete derivatives (see Remark~\ref{rem:loss-odd}) and the shifting operators.
%and this will further relate with the verification that the equation for $A_i$
%does not involve any mass renormalisation term of the form $cA_j$ for $j\neq i$
%(so in a sense the general argument as in \cite[Section~4]{CCHS_3D} does not directly apply to the discrete setting here).
Fortunately, in 2D  the number of  trees is  relatively small
so it is manageable to obtain the bounds by hand.

Recall  the kernel norm $·_ζ;m^()$
and the notation $z_$ from \eqref{e:DSingKer}. For the rest of this subsection we drop the $$-dependence in our kernel notation, e.g. $K=K^$.
%The following lemma states that slightly shifting a kernel will essentially preserve its norm.
\begin{lemma}\label{lem:shift-K}
Let $\zeta<0$. %For $\be\in \CE_+\cup \CE_\times$, $\tilde K(z) = K(z+\be)$,
For two kernels $\tilde K$ and $K$ such that $\tilde K(z) = K(z+a)$ where $a=a_1\e_1+a_2\e_2$ with $|a_1|+|a_2|<10$,
one has
$\VERT \tilde K \VERT_{\zeta;m}^{(\eps)}
\asymp
\VERT K \VERT_{\zeta;m}^{(\eps)}$.
\end{lemma}

\begin{proof}
This is immediate from the definition \eqref{e:DSingKer} and $\|z+a \|_\e \asymp \|z\|_\e$.
\end{proof}
We write $K_1_K_2$  for any  product of the form $K_1(z)K_2(z+a)$,
and $K_1*_K_2$ for any  convolution  of the form $K_1(·) * K_2(·+a)$,
%where $\be$ is a generic element of $ \CE_+\cup \CE_\times$.
where $a$ is as in Lemma~\ref{lem:shift-K}.
Here $*$ is the (semi-discrete) convolution over space-time.
We will also write $K^*_2 K*_K̂$,
where $K̂$ the reflection of $K$ as in Section~\ref{subsubsec:behaviour}.
Recall  $=(2,1,1)$ and $||=4$.

\begin{lemma}\label{lem:OpDSingKer}
Let $K_1$ and $K_2$ be functions of order $\zeta_1$ and $\zeta_2$ respectively.
Let $m\in\N$, $\bar{\zeta}\eqdef \zeta_1+\zeta_2+|\s|$.
Then one has, uniformly in $\eps$ for $\e$ sufficiently small,
\begin{enumerate}
\item
(Multiplication bound):
$\VERT K_1 \bigcdot_\e K_2 \VERT^{(\eps)}_{\zeta_1+\zeta_2;m}
\lesssim
\VERT K_1 \VERT^{(\eps)}_{\zeta_1;m}
\VERT K_2 \VERT^{(\eps)}_{\zeta_2;m}$.
\item 			\label{item:convolution}
(Convolution bound): $\VERT K_1 \ast_\eps K_2 \VERT^{(\eps)}_{\bar{\zeta};m}
\lesssim
\VERT K_1\VERT^{(\eps)}_{\zeta_1;m}
\VERT K_2 \VERT^{(\eps)}_{\zeta_2;m}$ provided that $\zeta_1\wedge \zeta_2>-|\s|$ and $\bar{\zeta}<0$.
%\item 			\label{item:positive}
%If $\bar{\zeta}\in (0,2) \setminus\N$, then
%$\VERT \bar{K}^\eps\VERT^{(\eps)}_{\bar{\zeta};m}\lesssim \VERT K_1^\eps\VERT^{(\eps)}_{\zeta_1;\bar{m}}\VERT K_2^\eps\VERT^{(\eps)}_{\zeta_2;\bar{m}}$ where
%$\bar{m}=m\vee(\lfloor\bar{\zeta}\rfloor +2)$ and
%\eqdef K_1^\eps\ast_\eps K_2^\eps(z)
%	-\sum_{|k|_\s<\bar{\zeta}}
%	\frac{z^k}{k!} D_\eps^k (K_1^\eps\ast_\eps K_2^\eps) (0)\;.
%%\quad (t,x)_{k,\eps} \eqdef t^{k_0} \prod_{i\neq 0}\prod_{0\le j<k_i} (x_i - \eps j)\;.
\item				\label{item:renormalized}
(Renormalised kernel):
$\zeta_1 \in \bigl(-|\s|-1, -|\s|\bigr]$ and $\zeta_2 \in \bigl(-2 |\s|-\zeta_1, 0\bigr]$,
\begin{equ} [e:RenorConv]
\VERT{\bigl(\mathscr{R} K_1\bigr) \ast_\eps K_2}\VERT_{\bar{\zeta}; m} \lesssim \VERT{K_1}\VERT_{\zeta_1; m} \VERT{K_2}\VERT_{\zeta_2; m + 2}
\end{equ}
$\left(\mathscr{R} K\right)(\varphi) \eqdef \int_{\R \times (\e\Z)^2} K(z) \left( \varphi(z) - \varphi(0) \right) dz$
for any $\varphi \in \CC^\infty_0(\R^{3})$.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows from Lemma~\ref{lem:shift-K} and \cite[Lemmas~7.3, 7.5]{HM18}.
\end{proof}
%Our first step in the proof of the bounds \eqref{e:mom-bounds} is the same as that in~\cite{Hairer14,HM18}. Namely,
As in~\cite{Hairer14,HM18},
for each $τ$ we have a Wiener chaos decomposition for
$Π̂^_z_* τ(ϕ) = ∑_k I_k^((^(; k)τ)(ϕ))$
where $I_k^$ is the $k$th order Wiener integral with respect to $ξ^$,
and $^(; k)τ$ are kernels such that
$(^(; k)τ)(z) ∈H_^⊗k$
with $H_$ being the Hilbert space for $ξ^$.
Similarly, for a temporal test function $ϕ$,
\[
\int_{\R} (\hat\Pi_{z_*}^\e\tau)(s,x_*)\phi(s)\mrd s =
\sum_k I_k^\e
\int_{\R}
(\CW^{(\eps; k)}\tau)(s,x_*) \phi(s)\mrd s\;.
\]
Then we consider their ``symmetric pairings''
\begin{equs}[e:CovDef]
\bigl(\CK^{(\eps; k)}\tau\bigr)(z_1, z_2) \eqdef \langle \bigl(\CW^{(\eps; k)}\tau\bigr)(z_1), \bigl(\CW^{(\eps; k)}\tau\bigr)(z_2) \rangle_{H_\eps^{\otimes k}}\;.
\end{equs}
Below we will always assume $z_1=(t_1,x_1)$ and $z_2=(t_2,x_2)$.
We will also assume $z_*=(t_*,x_*)=(0,0)$ since our model is stationary, but  we often still write
$z_*=(t_*,x_*)$ as an intuitive placeholder.
%In \cite[Prop.~10.11]{Hairer14}  the uniform bounds on models amounts to finding the bounds
Consider the bound
\begin{equ}[e:CovarianceBounds]
|\big(\CK^{(\eps; k)}\tau\big)\big(z_1, z_2\big)|
\lesssim
\sum \e^{n_\e}
\big( \|z_1\|_\s + \|z_2\|_\s\big)^\zeta
\|z_1- z_2\|_\e^\alpha   %{2|\tau| +\kappa - \zeta}
\,\delta_\e (z_1-z_2)^{n_\delta}\;,
\end{equ}
where the sum is over finitely many values of $(ζ,α,n_δ,n_)$
such that
$α∈(-||,0]$, $ζ≥0$, $n_δ∈{0,1}$, $n_≥0$,
\[
\alpha+n_\e \in [\, |\s|n_\delta-|\s|, n_\delta d \,) \;, \qquad
\alpha+\zeta-|\s|n_\delta + n_\e = 2|\tau|+\kappa\;.
\]
Here, $δ_(z_1-z_2)  δ(t_1-t_2) ^-d (x_1-x_2)$  where $$ is the indicator of the origin.
We claim that \eqref{e:CovarianceBounds} suffices
to prove both bounds in \eqref{e:mom-bounds}.
Indeed, \eqref{e:CovarianceBounds} $⇒$ \eqref{e:mom-bounds1} in the regime $λ> $
follows as in \cite[Prop.~10.11]{Hairer14}.
(Note that when $n_δ=1$, $^n_ z_1- z_2_^αδ_(z_1-z_2)$ is integrable uniformly in $$ thanks to $α+n_≥0$.)
Regarding  \eqref{e:mom-bounds2} in the regime  $λ≤$,
the 2nd moment of the $k$th chaos  $I_k^∫_ (^(; k)τ)(t,x_*)(_2,t_*^λϕ)(t)t$ is bounded by
\begin{equs}
%\E & \Big[\Big( I_k^\e \int_{\R} (\CW^{(\eps; k)}\tau)(t,x_*)(\CS_{2,t_*}^{\lambda}\phi)(t)\mrd t\Big)^2\Big]
\int_{\R^2}  \langle (\CW^{(\eps; k)}\tau)(t_1,x_*), (\CW^{(\eps; k)}\tau)(t_2,x_*)\rangle
\mrd t_1\mrd t_2
\\
\sum%_{\zeta \geq 0}
\int_{\R^2} \e^{n_\e}  \big( \|(t_1,x_*)\|_\s + \|(t_2,x_*)\|_\s\big)^\zeta
(|t_1- t_2|^{\frac12}\vee \e)^{\alpha}
\\
\cdot  (\e^{-d} \delta (t_1-t_2))^{n_\delta}   (\CS_{2,t_*}^{\lambda}\phi)(t_1)(\CS_{2,t_*}^{\lambda}\phi)(t_2)
\mrd t_1\mrd t_2
% \sum %_{\zeta \geq 0}
%  \lambda^\zeta
% \int_{\R^2}
%(|t_1- t_2|^{\frac12}\vee \e)^{2|\tau|+\kappa - \zeta}
%\mrd t_1\mrd t_2
% \sum %_{\zeta \geq 0}
% \lambda^\zeta
% \lambda^{2|\tau| +\kappa - \zeta}
% \lambda^{2 |\tau|+\kappa}\;,
\\
\begin{cases}
\int_{\R^2}  \lambda^\zeta (|t_1- t_2|^{\frac12}\vee \e)^{\alpha+n_\e}   (\CS_{2,t_*}^{\lambda}\phi)(t_1)(\CS_{2,t_*}^{\lambda}\phi)(t_2)\mrd t_1\mrd t_2 \lesssim \lambda^{\zeta+\alpha+n_\e}
\\
\lambda^\zeta \e^{n_\e} \int_\R \e^\alpha \e^{-d} (\CS_{2,t_*}^{\lambda}\phi)(t)^2 \mrd t
\lesssim \lambda^{\zeta-2} \e^{\alpha-d+n_\e} \lesssim \lambda^{\zeta+\alpha-|\s|+n_\e}
\end{cases}
\end{equs}
where  the first bound is for $n_δ=0$ where we used $α+n_<0$ in the last step;
the second bound is for $n_δ=1$ where we used $α+n_<d$ in the last step.
In both cases we have the bound $λ^2|τ|+κ$ as required, which proves the claim.
%we will write  (the contribution of a certain chaos to) the left-hand sides of
% \eqref{e:mom-bounds1} and  \eqref{e:mom-bounds2}
%as  ``L.S.'' and ``S.S.''
% respectively,
%which stand for ``large scale'' and ``small scale'' bounds.
%We always assume $\lambda > \e$ when bounding L.S. and $\lambda\le \e$  when bounding S.S.

In the next lemmas, we keep in mind that $i≠j$, and that $κ>0$ is an arbitrarily small parameter.
We will introduce graphic notation:
$[baseline=-3](0,0)–(.8,0);$ stands for the truncated heat kernel $K$,\footnote{understood
as $K(z-w)$ where $z$ is a point lower than $w$ in the graph; same for other kernels.}
$[baseline=-3][very thick] (0,0)–(.8,0);$ stands for a generic derivative of $K$,
$at (0,0) [var] ; $ stands for $ξ^$,
$at (0,0) [var1] ; $ stands for $^1-κξ^$.
Joining these lines correspond to multiplications or convolutions  possibly with shifts (i.e.
they can be
and $K_1*_K_2$ described above), but we do not explicitly show these shifts in the graphs.
$[baseline=-3][kernel1] (0,0) –(.8,0);$ stands for the ``positively renormalised'' truncated heat kernel
$K(z-w)-K(-w)$.% where $z$ is a point lower than $w$ in the graph.

%The following fact will be useful.
%Let $F(w)$ be a function of $w \in\R\times \obonds_i$.
%Let $J(z,w)$ be a function with arguments
%$z\in  \R\times \obonds$ and
%$w\in \R\times \obonds_j$, which only depends on the distance $\|z-w\|_\s$ (recall that we identify the bonds $z,w$ with their midpoints).
%Assume that $j\neq i$. We can then write, for any $z\in \R\times \obonds$,
%\frac14 \sum_{\be\in \CE_\times}(J * _{(j)} F(\cdot + \be))(z)
%=\frac14 \sum_{\be\in \CE_\times} \int_{\R\times \obonds_j} J(z,w) F(w + \be) \mrd w \;.
%If we define $J^\times$ by
%$J^\times (z,w) \eqdef \frac14 \sum_{\be\in \CE_\times} J(z,w-\be)$ where
%$w\in \R\times \obonds_i$, then
%\frac14 \sum_{\be\in \CE_\times}(J * _{(j)} F(\cdot + \be))(z)
%(J^\times * _{(i)} F)(z) \;.

\begin{remark}
In Lemmas~\ref{lem:mom-YM}--\ref{lem:mom-222333} below
we will frequently apply the multiplication and convolution bounds in Lemma~\ref{lem:OpDSingKer}.
For instance to obtain \eqref{e:mom-cherry},
the convolution of the two kernels of degree $-3$ (resp. $-2$) yields a new kernel of degree $-2$ (resp. $-\kappa$),
%and the convolution of the two kernels of degree $-2$ yields a new kernel of degree $-\kappa$,
and the multiplication of the two new kernels is of degree $-2-\kappa$.
This procedure is elementary and standard, so we will directly give the resulting bound
without repeatedly referring to Lemma~\ref{lem:OpDSingKer}.
Also, we will slightly abuse notation
and write $\Psi,\bar\Psi,\tilde\Psi$ for the trees $\mcb{I}\Xi$,  $\mcb{I}\bar\Xi$,  $\mcb{I}\tilde\Xi$,
as well as the Gaussian processes they realise;
the meaning is always clear from the context.
\end{remark}

\begin{lemma}
\label{lem:mom-YM}
For every $\tau\in \CT^{\YM}_-$,  the bounds \eqref{e:mom-bounds} hold.
\end{lemma}

\begin{proof}
Consider $\tau=\Xi$ and  recall $\xi^\e= \e^{-1}\dt \BM$ as in Section~\ref{sec:canonical}.
%recall that $\xi^\e = \e^{-1}\dt \BM$ where $\BM$ is a collection of i.i.d. Brownian motions.
One has $\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big) = \delta_\e(z_1-z_2)$.
%\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big) = \delta_\e(z_1-z_2)
% \eqdef \delta(t_1-t_2) \e^{-2} \bone(x_1-x_2)
%where $\bone$ equals $1$ at the origin and $0$ otherwise.
%This is not exactly the case as in \eqref{e:CovarianceBounds} due to $\delta(t_1-t_2) $.
%However one still has the desired bounds
%\mbox{L.S.} &= \int (\phi^\lambda_{z_*}(z))^2 \mrd z \lesssim \lambda^{-4}\;,
%%\E \Big[\Big(\int_\R \xi^\e (t,x) \CS^\lambda_2 \phi(t) \mrd t \Big)^2 \Big]
%%&\le \int_\R \e^{-2}  \delta(t_1-t_2)  \CS^\lambda_{2,t_*} \phi(t_1)\CS^\lambda_{2,t_*} \phi(t_2) \mrd t_1\mrd t_2
%&= \int_{\R^2} \e^{-2} \CS^\lambda_{2,t_*} \phi(t)^2 \mrd t
%\lesssim \e^{-2} \lambda^{-2} \lesssim \lambda^{-4}\;, \quad (\mbox{by }\lambda\le \e)\;.
%%where in the last step we used $\lambda\le \e$.
So \eqref{e:CovarianceBounds} applies with $(\zeta,\alpha,n_\delta,n_\e)=(0,0,1,0)$.

%Consider $\tau=\<XXi>$. One has
%\big|\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big)\big|
% \lesssim \|z_1\|_\s \|z_2\|_\s   \delta_\e(z_1-z_2)\;.
%Similarly as \eqref{e:xiS22} we have $\mbox{L.S.} \lesssim \lambda^{-2} $,
% and
%$\mbox{S.S.} \lesssim \e^{-2} \le \lambda^{-2}$,
%where we bound $\|z_1\|_\s \|z_2\|_\s \le \lambda^2$ as in
%The same argument shows that
%for $\tau=X^2 \Xi$
%we have $\mbox{L.S.} \lesssim \lambda^{-\kappa} $,
% and
%$\mbox{S.S.} \lesssim \e^{-\kappa} \le \lambda^{-\kappa}$.

Consider $\tau=\<IXiI'Xi_notriangle>$
which includes
$\CS_{\be} \Psi_j \CD_j \Psi_i $,
$\CS_{\be} \Psi_j \bar\CD_i \Psi_j$, and
$\Psi_i \CD_i \Psi_i $ for  $\be\in \CE_\times$.  %\CS \in \{\CS_a,\CS_{\be},\be\in \CE_\times\}
The $0$th chaos of $(\hat\Pi_z^\e\tau)(\phi_z^{\lambda})$ for each of them is $0$. Indeed,
this holds for $\CS_{\be} \Psi_j \CD_j \Psi_i$ since $i\neq j$,
for $\Psi_i \CD_i \Psi_i $ by Lemma~\ref{lem:ePsiDPsi} thanks to
the symmetrised derivative $\partial_i$,
and for $\CS_{\be} \Psi_j \bar\CD_i \Psi_j$ by the choice of $\tilde C_{\be}$. %~\eqref{e:def-C67etc}.
For the 2nd chaos,
\footnote{Below the dotted lines represent
%$\delta_\e$ as in \eqref{e:CovarianceBounds}.
the parings, i.e. replacing the pair of noises by their covariances.} \vspace{-5pt}
\begin{equ}[e:mom-cherry]
\begin{tikzpicture}  [scale=0.4,baseline=5]
\node at (.6,1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\draw[very thick] (0,0) -- (a);
\draw (0,0) -- (b);
\node at (2.3+.6,1)  [var] (c)  {};
\node at (2.3-.6,1) [var] (d) {};
\draw[very thick] (2.3,0) -- (d);
\draw (2.3,0) -- (c);

\draw[bend left =40, delta]  (a) to (d) ;
\draw[bend left =40, delta]  (b) to (c) ;
\end{tikzpicture}
\qquad
\big(\CK^{(\eps; 2)}\tau\big)\big(z_1, z_2\big) \lesssim \|z_1-z_2\|_{\e}^{-2-\kappa}
\end{equ}
%one has
%$\mbox{L.S.} \lesssim \lambda^{-2} $ and
%$\mbox{S.S.} \lesssim \e^{-2} \le \lambda^{-2}$.
% where for S.S. we brutally bound $ \|z\|_{\e}^{-2}$ by $\e^{-2}$.
as required by \eqref{e:CovarianceBounds}.
For $\tau=\<I'Xi_notriangle>$, one has essentially the same bound \eqref{e:mom-cherry}.

Consider $\tau = \<IXi^3_notriangle>$,
namely %$(\CS_a \Psi_j)(\CS_a \Psi_j)\Psi_i$ and
$(\CS_{\be} \Psi_j)(\CS_{\be'} \Psi_j)\Psi_i$.
The 1st chaos vanishes by our choice of renormalisation
%$\bar C^\e$ and
$\bar C^\e_{\be\be'}$.
For the 3rd chaos, \vspace{-5pt}
\begin{equ}[e:cubic-pairing]
\begin{tikzpicture}  [scale=0.45,baseline=5]
\node at (.7, .8)  [var] (a)  {};
\node at (-.7,.8) [var] (b) {};
\node at (0,1) [var] (x) {};
\draw (0,0) -- (a);
\draw (0,0) -- (b);
\draw (0,0) -- (x);
\node at (2.3+.7,.8)  [var] (c)  {};
\node at (2.3-.7,.8) [var] (d) {};
\node at (2.3,1)  [var] (y)  {};
\draw (2.3,0) -- (d);
\draw (2.3,0) -- (c);
\draw (2.3,0) -- (y);

\draw[bend left =50, delta]  (a) to (d) ;
\draw[bend left =60, delta]  (b) to (c) ;
\draw[bend left =40, delta]  (x) to (y) ;
\end{tikzpicture}
\qquad
\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big) \lesssim \|z_1-z_2\|_{\e}^{-\kappa}
\end{equ}
as required by \eqref{e:CovarianceBounds}.
The bounds for $\<IXi^2>$ and  $\<IXi>$ are similar and simpler.
%Consider $\tau=\<IXi^2>$.
%The 0th chaos  vanishes by the choice of renormalisation.
%For the 2nd chaos,
%\begin{tikzpicture}  [scale=0.6,baseline=10]
%\node at (.6,1)  [var] (a)  {};
%\node at (-.6,1) [var] (b) {};
%\draw (0,0) -- (a);
%\draw (0,0) -- (b);
%\node at (2.6,1)  [var] (c)  {};
%\node at (2-.6,1) [var] (d) {};
%\draw (2,0) -- (d);
%\draw (2,0) -- (c);
%\draw[bend left =40, dotted]  (a) to (d) ;
%\draw[bend left =40, dotted]  (b) to (c) ;
%\big(\CK^{(\eps; 2)}\tau\big)\big(z_1, z_2\big) \lesssim \|z_1-z_2\|_{\e}^{-\kappa}
%as required by \eqref{e:CovarianceBounds}.
%The case $\<IXi>$ is even simpler.

Consider $\<IXiI'[IXiI'Xi]_notriangle>$, which consists of the following trees
\begin{gather*}
{} {\bf(1)}\;\; \Psi_i \CD_i \mcb{I} (\CS_{\be} \Psi_j \CD_j \Psi_i)\;,
\quad
{\bf(2)}\;\; \Psi_i \CD_i \mcb{I} (\CS_{\be} \Psi_j \bar\CD_i \Psi_j)\;,
\quad
{\bf(3)}\;\; \Psi_i \CD_i \mcb{I} ( \Psi_i \CD_i \Psi_i)\;,
\\
{\bf(4)}\;  \CS_{\be} \Psi_j \CD_j \mcb{I} (\CS_{\be'} \Psi_j \CD_j \Psi_i),
\;
{\bf(5)}\; \CS_{\be} \Psi_j \CD_j \mcb{I} (\CS_{\be'} \Psi_j \bar\CD_i \Psi_j),
\;
{\bf(6)}\; \CS_{\be} \Psi_j \CD_j \mcb{I} (\Psi_i\CD_i \Psi_i),
\\
{\bf(7)}\; \CS_{\be} \Psi_j \CD_i^+ \CS_{\bw} \mcb{I} (\CS_{\be'} \Psi_i \CD_i \Psi_j),
\;
{\bf(8)}\; \CS_{\be} \Psi_j \CD_i^+\CS_{\bw} \mcb{I} (\CS_{\be'} \Psi_i \bar\CD_j \Psi_i),
\;
{\bf(9)}\; \CS_{\be} \Psi_j \CD_i^+ \CS_{\bw} \mcb{I} (\Psi_j \CD_j \Psi_j).
\end{gather*}
Here $\be,\be'\in \CE_\times$ and $\bw\in\{\Northwest,\Southwest\}$.
For the 3rd chaos, as well as one term of the 1st chaos of the form \tikz[scale=0.25,baseline=-5]{\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (-1.2,0) [var] (x) {};
\draw[very thick] (0,0) -- (a);
\draw (0,0) -- (b);
\draw[very thick] (0,0) -- (-0.6,-1);
\draw (x) -- (-0.6,-1); \draw[bend right =50, delta,thick]  (b) to (x);},
%obtained by contracting
%the $\Psi$ outside $\mcb{I}$ and the $\Psi$ (without derivative) inside $\mcb{I}$,
%their second moments are represented by the following graphs;
proceeding as above
we again get a bound by $\|z_1-z_2\|_{\e}^{-\kappa}$.
%\begin{tikzpicture}  [scale=0.35,baseline=10]
%\node at (.6, 1)  [var] (a)  {};
%\node at (-.6,1) [var] (b) {};
%\node at (1.2,0) [var] (x) {};
%\draw[very thick] (0,0) -- (a);
%\draw (0,0) -- (b);
%\draw[very thick] (0,0) -- (0.6,-1);
%\draw (x) -- (0.6,-1);
%\node at (4.6,1)  [var] (c)  {};
%\node at (4-.6,1) [var] (d) {};
%\node at (4-1.2,0) [var] (y) {};
%\draw[very thick] (4,0) -- (d);
%\draw (4,0) -- (c);
%\draw[very thick] (4,0) -- (4-.6,-1);
%\draw (y) -- (4-0.6,-1);
%\draw[bend left =40, dotted]  (b) to (c) ;
%\draw[bend left =40, dotted]  (x) to (y) ;
%\draw[bend left =40, dotted]  (a) to (d) ;
%\begin{tikzpicture}  [scale=0.35,baseline=10]
%\node at (.6, 1)  [var] (a)  {};
%\node at (-.6,1) [var] (b) {};
%\node at (-1.2,0) [var] (x) {};
%\draw[very thick] (0,0) -- (a);
%\draw (0,0) -- (b);
%\draw[very thick] (0,0) -- (-0.6,-1);
%\draw (x) -- (-0.6,-1);
%\node at (3.6,1)  [var] (c)  {};
%\node at (3-.6,1) [var] (d) {};
%\node at (3+1.2,0) [var] (y) {};
%\draw[very thick] (3,0) -- (d);
%\draw (3,0) -- (c);
%\draw[very thick] (3,0) -- (3.6,-1);
%\draw (y) -- (3+0.6,-1);
%\draw[bend right =50, dotted]  (b) to (x) ;
%\draw[bend left =50, dotted]  (c) to (y) ;
%\draw[bend left =40, dotted]  (a) to (d) ;
For each of these trees there is another term in the 1st chaos component of $\hat\Pi_{z_*}^\e\tau (z)$, denoted by $I_1^\e(\bar\CW^{(\eps; 1)}\tau)(z)$,
% obtained by contracting
%the $\Psi$ outside $\mcb{I}$ with the derivative of $\Psi$ inside $\mcb{I}$,
of the form
\tikz[scale=0.25,baseline=-5]{
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (-1.2,0) [var] (x) {};
\draw (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[very thick] (0,0) -- (-0.6,-1);
\draw (x) -- (-0.6,-1); \draw[bend right =50, delta,thick]  (b) to (x);},
which we focus on now.\footnote{The term in the 1st chaos
obtained by contracting the two noises on the top is zero, since the kernel $\d K$
annihilates constants. We will often use this fact below, together with the fact that the positively renormalised kernel $K(z-w)-K(-w)$ also kills constants, without explicitly mentioning.}

For (2)(4)(6)(8): $I_1^\e(\bar\CW^{(\eps; 1)}\tau)=0$ by  independence
of $\Psi_i$ and $\Psi_j$.

For (1)(5)(9):
$I_1^\e(\bar\CW^{(\eps; 1)}\tau)(z)$ for $z\in \R\times \obonds_i$ is given by
(J *_{(j)} \Psi_j) (z) = \int_{\R\times \obonds_j} J(z-w) \Psi_j (w) \mrd w
where for $z\in \R\times \obonds_i$ and $w\in \R\times \obonds_j$
\begin{equ}[e:def-J1]
J(z-w) \eqdef
\begin{cases}
\E [\Psi_i(z) \partial_i K (z+\be-w) \partial_j \Psi_i(w-\be)] \;,\\
\E [\Psi_j(z+\be) \partial_j K (z+\be'-w) \bar\partial_i \Psi_j(w-\be')]\;,\\
\E [\Psi_j(z+\be) \partial_i^+ K (z^{\bw}-w) \partial_j \Psi_j(w)]
\end{cases}
\end{equ}
respectively for the three trees.
Now one has
\begin{equ}[e:RJ-convolve-Psi]
%I_1^\e (\bar\CW^{(\eps; 1)}\tau)(z)  =
(J *_{(j)} \Psi_j) (z) - \mathrm{Int}[J] \Psi_j (z)
= (\mathscr{R} J *_{(j)} \Psi_j) (z)\;.
\end{equ}
Here we write  $\mathrm{Int}[J]$ for $\int_{\R\times \obonds_j} J(z-w) \mrd w$
which is independent of $z\in \R\times \obonds_i$.
$J$ has degree $-4$,
by the bound for renormalised kernels in Lemma~\ref{lem:OpDSingKer} one  can
bound the second moment of $\mathscr{R} J *_{(j)} \Psi_j$ by  \vspace{-5pt}
\begin{equ}[e:mom-2424]
\begin{tikzpicture}  [scale=0.32,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\draw (0,0) -- (a);
\draw[kernels] (0,0) -- (0.6,-1);

\node at (3-.6, 1)  [var] (b)  {};
\draw (3,0) -- (b);
\draw[kernels] (3,0) -- (3-.6,-1);

\draw[bend right =50, delta]  (b) to (a) ;
\end{tikzpicture}
\qquad
\qquad
\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big) \lesssim \|z_1-z_2\|_{\e}^{-\kappa}
\end{equ}
satisfying  \eqref{e:CovarianceBounds}.
Here $\tikz[baseline=-3]\draw[kernels] (0,0) --(.8,0);$ stands for $\mathscr{R}  J$.
Now it suffices to show that $\mathrm{Int}[J]$ is finite (recalling $i\neq j$).
Indeed, in the first case of \eqref{e:def-J1},
$\mathrm{Int}[J]$ can be written as
\int_{\R\times \obonds_i} \E [\Psi_i(z) \partial_i K (z-w) \partial_j \Psi_i(w)] \mrd w
by parity, namely,
$\partial_i K(w)$ is odd and  $\E[\Psi_i(z) \partial_j \Psi_i(z-w)]$ is even (in $w$)
in the $i$th spatial coordinate, noting that $\partial_i$ is a symmetrised derivative.
In the second case, we write $\mathrm{Int}[J]$ as
\begin{equ}
\frac12\sum\nolimits_{\sigma\in \{0,1\}}\int_{\R\times \obonds_i}
\E [(\Psi_j(z+\be)+ (-1)^{\sigma}\Psi_j(z-\be)) \partial_j K (z-w) \bar\partial_i \Psi_j(w)]
%+\E [(\Psi_j(z+\be)-\Psi_j(z-\be)) \partial_j K (z-w) \bar\partial_i \Psi_j(w)]
\mrd w\;.
\end{equ}
The $\sigma=0$ term vanishes by parity; the $\sigma=1$ term
$\lesssim\int \e \|w\|_\e^{-5} \mrd w$
which is finite uniformly in $\e$.
The third case of \eqref{e:def-J1} follows in the same way.

For (3)(7):
by \eqref{e:M-on-YM},  %\eqref{e:def-C-main},
$\hat\Pi_{z_*}^\e\tau (z)$ for $z\in \R\times \obonds_i$ is equal to
\begin{equs}
{} &\Psi_i (z) \,(\partial_i K*_{(i)}( \Psi_i \partial_i \Psi_i))(z)
- \E [\Psi_i(z) (\partial_i K*_{(i)} \partial_i \Psi_i)(z)]
\,\Psi_i (z)\;,
\\
& \Psi_j (z+\be)\, (\partial_i^+ K*_{(j)}( \Psi_i(\cdot+\be') \partial_i \Psi_j))(z^\bw)
\\
&\qquad\qquad\qquad \qquad
- \E [ \Psi_j(z+\be) (\partial_i^+ K *_{(j)} \partial_i \Psi_j) (z^\bw) ]
\, \Psi_i (z)
\end{equs}
respectively, recalling $\hat{C}^{\e}_5$, $\hat C^\e_{3,\be,\bw}$ in Section \ref{sec:renorm_constants}.
So with
\[
J(z-w) \eqdef
\begin{cases}
\E [\Psi_i(z) \partial_i K (z-w) \partial_i \Psi_i(w)] \;,\\
\E [\Psi_j(z+\be) \partial_i^+ K (z^\bw+\be'-w) \partial_i \Psi_j(w-\be')]
\end{cases}
\]
where $z,w\in \R\times \obonds_i$,
one has
\begin{equ}[e:37RJ]
I_1^\e (\bar\CW^{(\eps; 1)}\tau)(z)
= (J *_{(i)} \Psi_i) (z) - \mathrm{Int}[J] \Psi_i (z)
= (\mathscr{R} J *_{(i)} \Psi_i) (z)
\end{equ}
and one again has the bound \eqref{e:mom-2424}.

Consider $\<I[IXiI'Xi]I'Xi_notriangle>$, which consists of the trees
\begin{gather*}
{}{\bf(1)}\;\;\mcb{I} (\CS_{\be}\Psi_j\CD_j \Psi_i) \CD_i \Psi_i\;,
\quad
{\bf(2)}\;\;\mcb{I} (\CS_{\be}\Psi_j\bar\CD_i \Psi_j) \CD_i \Psi_i\;,
\quad
{\bf(3)}\;\; \mcb{I} (\Psi_i\CD_i \Psi_i) \CD_i \Psi_i\;,
\\
{\bf(4)} \;\CS_{\be} \mcb{I} (\CS_{\be'}\Psi_i\CD_i \Psi_j) \CD_j \Psi_i,
\;\;
{\bf(5)} \; \CS_{\be} \mcb{I} (\CS_{\be'}\Psi_i \bar\CD_j \Psi_i) \CD_j \Psi_i,
\;\;
{\bf(6)} \;\CS_{\be} \mcb{I} (\Psi_j\CD_j \Psi_j) \CD_j \Psi_i,
\\
{\bf(7)} \;\CS_{\be} \mcb{I} (\CS_{\be'}\Psi_i\CD_i \Psi_j) \bar\CD_i \Psi_j,
\;\;
{\bf(8)}\;\CS_{\be} \mcb{I} (\CS_{\be'}\Psi_i \bar\CD_j \Psi_i) \bar\CD_i \Psi_j,
\;\;
{\bf(9)}\;\CS_{\be} \mcb{I} (\Psi_j\CD_j \Psi_j) \bar\CD_i \Psi_j.
\end{gather*}
For the 3rd chaos, we have \vspace{-5pt}
\begin{equ}[e:K3-01]
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (1.2,0) [var] (x) {};
\draw[very thick] (0,0) -- (a);
\draw (0,0) -- (b);
\draw[kernel1] (0,0) -- (0.6,-1);
\draw[very thick] (x) -- (0.6,-1);

\node at (4.6,1)  [var] (c)  {};
\node at (4-.6,1) [var] (d) {};
\node at (4-1.2,0) [var] (y) {};
\draw[very thick] (4,0) -- (d);
\draw (4,0) -- (c);
\draw[kernel1] (4,0) -- (4-.6,-1);
\draw[very thick] (y) -- (4-0.6,-1);

\draw[bend left =40, delta]  (b) to (c) ;
\draw[bend left =40, delta]  (x) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\quad
\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big) \lesssim
\|z_1-z_2\|_{\e}^{-2} (\|z_1\|_{\e} +\|z_2\|_{\e})^{2-2\kappa}
\end{equ}
%so that $\mbox{L.S.}\lesssim \lambda^{-\kappa}$
%and  $\mbox{S.S.}\lesssim \e^{-\kappa} + \e^{-1-\kappa}\lambda \lesssim \lambda^{-\kappa}$.
as required by \eqref{e:CovarianceBounds} with $\zeta=2-\kappa$ therein.
Here, the factor $\|z_1-z_2\|_{\e}^{-2} $
arises from pairing the two noises in the bottom. For the  kernel
$\tikz[baseline=-3]\draw[kernel1] (0,0) --(.8,0);$ we bound
\begin{equ}[e:Diff-K]
|K(z-w)-K(-w)| \lesssim \|z\|_\e^\alpha (\|z-w\|_\e^{-2-\alpha} + \|w\|_\e^{-2-\alpha})\;,
\quad
\forall \alpha\in [0,1]
\end{equ}
which can be proved in the same way as
\cite[Lem.~10.18]{Hairer14}, see also \cite[Lem.~7.4]{HM18}.
The above bound \eqref{e:K3-01} is then obtained by applying
\eqref{e:Diff-K} to both $\tikz[baseline=-3]\draw[kernel1] (0,0) --(.8,0);$ %of the  ``positively renormalised'' kernels
with $\alpha=1-\kappa$:
one has $ \|z_1\|_\e^{1-\kappa}  \|z_2\|_\e^{1-\kappa}\le
(\|z_1\|_{\e} +\|z_2\|_{\e})^{2-2\kappa}$,
and the integration over the other variables is bounded due to our choice of $\alpha$.

Consider the term in the 1st chaos not requiring renormalisation. Graphically:\vspace{-5pt}
\begin{equ}[e:K3-02]
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (-1.2,0) [var] (x) {};
\draw[very thick] (0,0) -- (a);
\draw (0,0) -- (b);
\draw[kernel1] (0,0) -- (-0.6,-1);
\draw[very thick] (x) -- (-0.6,-1);

\node at (3.6,1)  [var] (c)  {};
\node at (3-.6,1) [var] (d) {};
\node at (3+1.2,0) [var] (y) {};
\draw[very thick] (3,0) -- (d);
\draw (3,0) -- (c);
\draw[kernel1] (3,0) -- (3.6,-1);
\draw[very thick] (y) -- (3+0.6,-1);

\draw[bend right =50, delta]  (b) to (x) ;
\draw[bend left =50, delta]  (c) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\end{equ}
We bound the two terms in the ``positively renormalised'' kernel separately
(i.e. \eqref{e:Diff-K} with $\alpha=0$), which boils down to estimating the following two terms
\begin{equs}
\int
\|z_1-w_1\|_\e^{-3} \,& \|w_1-w_2\|_\e^{-2}
\|z_2-w_2\|_\e^{-3} \mrd w_1 \mrd w_2 \;,
\\
\int
\|w_1\|_\e^{-2}\,\|z_1-w_1\|_\e^{-1}\, & \|w_1-w_2\|_\e^{-2}
\|w_2\|_\e^{-2}\,\|z_2-w_2\|_\e^{-1} \mrd w_1 \mrd w_2\;.
\end{equs}
The first term is easily bounded by $\|z_1-z_2\|_\e^{-\kappa}$ using Lemma~\ref{lem:OpDSingKer}.
For the second term, by Young's inequality
$\|w_1\|_\e^{-2}\|z_1-w_1\|_\e^{-1} \lesssim \|w_1\|_\e^{-3}+\|z_1-w_1\|_\e^{-3}$.
For the integral with the term $\|w_1\|_\e^{-3}$, Lemma~\ref{lem:OpDSingKer} leads to
a bound by $\|z_2\|_\e^{-\kappa}$.
The integral with the term $\|z_1-w_1\|_\e^{-3}$ is bounded by
\int
\|z_1-w_2\|_\e^{-1}
\|w_2\|_\e^{-2}\,\|z_2-w_2\|_\e^{-1}  \mrd w_2
By Young's inequality as above with the subscript $1$ replaced by $2$, it is bounded by
$\|z_1-z_2\|_\e^{-\kappa}+ \|z_2\|_\e^{-\kappa}$.
Summarising, we obtain a bound
\begin{equ}[e:kappa3]
\|z_1-z_2\|_\e^{-\kappa}+\|z_1\|_\e^{-\kappa}+\|z_2\|_\e^{-\kappa}\;.
\end{equ}
The first term  has the form of  \eqref{e:CovarianceBounds}.
The  other two terms integrated against test functions are bounded by $\lambda^{-\kappa}$,
where in the regime $\lambda<\e$ one uses $\|z_i\|_\e^{-\kappa} \leq \e^{-\kappa}$.
%Regarding the other two terms,
%one has
%\int \phi^\lambda_{z_*}(z_1)\phi^\lambda_{z_*}(z_2) \|z_2\|_\e^{-\kappa} \mrd z_1\mrd z_2 \lesssim \lambda^{-\kappa}\;,
%\int_{\R^2}  \|z_1\|_\e^{-\kappa}
%\mrd t_1\mrd t_2
%where in the second bound we used $  \|z_2\|_\e^{-\kappa} \leq \e^{-\kappa}$.

We now focus on
the other term in the 1st chaos which has a potential divergence, again denoted by $I_1^\e\bar\CW^{(\eps; 1)}\tau$.

For (2)(4)(6)(8): $I_1^\e(\bar\CW^{(\eps; 1)}\tau)=0$ by  independence
of $\Psi_i$ and $\Psi_j$.

For (1)(9): we proceed similarly as for the terms (1)(5)(9) of $\<IXiI'[IXiI'Xi]_notriangle>$.
$I_1^\e (\bar\CW^{(\eps; 1)}\tau)(z)$ for $z\in \R\times \obonds_i$  is given by $(J *_{(j)} \Psi_j) (z)$ %as in  \eqref{e:J-convolve-Psi}
where for $z\in \R\times \obonds_i$ and $w\in \R\times \obonds_j$,
\begin{equ}[e:def-J2]
J(z-w) \eqdef
\begin{cases}
\E [\partial_i\Psi_i(z)  K (z+\be-w) \partial_j \Psi_i(w-\be)] \;,\\
\E [\bar\partial_i \Psi_j(z)  K (z+\be-w) \partial_j \Psi_j(w)]
\end{cases}
\end{equ}
respectively, plus a remainder term due to the %``positively renormalised''
$\tikz[baseline=-3]\draw[kernel1] (0,0) --(.8,0);$ kernels.
We again have \eqref{e:RJ-convolve-Psi}
and the bound \eqref{e:mom-2424} for $\mathscr{R} J *_{(j)} \Psi_j$.
One can then show that
$\mathrm{Int}[J]$ is bounded uniformly in $\e$ by similar arguments as for the terms (1)(5)(9) of $\<IXiI'[IXiI'Xi]_notriangle>$.
Now we only need to consider the remainder term,
which boils down to estimating
\begin{equ}[e:w-z-w-w-w]
\int
\|w_1\|_\e^{-2}\,\|z_1-w_1\|_\e^{-2}\,  \|w_1-w_2\|_\e^{-\kappa} \,
\|w_2\|_\e^{-2}\,\|z_2-w_2\|_\e^{-2} \mrd w_1 \mrd w_2\;.
\end{equ}
%Since $0=z_*,w_1,w_2$ form a triangle, $\|w_1\|_\e^{-2} \|w_1-w_2\|_\e^{-\kappa} \|w_2\|_\e^{-2}$ is bounded by
%+\|w_1\|_\e^{-2} \|w_1-w_2\|_\e^{-2-\kappa}
%+\|w_2\|_\e^{-2} \|w_1-w_2\|_\e^{-2-\kappa}\;.
By Young's inequality
$\|w_1\|_\e^{-2} \|w_1-w_2\|_\e^{-\kappa}
\lesssim \|w_1\|_\e^{-2-\kappa}+\|w_1-w_2\|_\e^{-2-\kappa}$.
If the first term is chosen, the integral is bounded by $\|z_1\|_\e^{-\kappa}+\|z_2\|_\e^{-\kappa}$.
If the other term is chosen, we again apply Lemma~\ref{lem:OpDSingKer}  and Young's inequality to obtain
\eqref{e:kappa3}. % and thus \eqref{e:kappa3LSSS}.

For (3)(5)(7): we proceed similarly as for the terms (3)(7) of $\<IXiI'[IXiI'Xi]_notriangle>$.
By \eqref{e:M-on-YM} and the choices of $\hat C^\e_{4}$, $\hat C^\e_{1,\be}$,
$\hat C^\e_{2,\be}$  in Section \ref{sec:renorm_constants},
% \eqref{e:def-C-main},
$\hat\Pi_{z_*}^\e\tau (z)$ for $z\in \R\times \obonds_i$ is equal to
\begin{equs}
\partial_i \Psi_i (z) (K*_{(i)}( \Psi_i \partial_i \Psi_i))(z)
&- \E [\partial_i\Psi_i(z) (K*_{(i)} \partial_i \Psi_i)(z)]
\Psi_i (z)\;,
\\
\partial_j \Psi_i (z) ( K*_{(j)}( \Psi_i(\cdot+\be') \bar\partial_j \Psi_i))(z+\be)
&- \E [\partial_j \Psi_i (z) ( K *_{(j)} \bar\partial_j \Psi_i) (z+\be) ]
\Psi_i (z)\;,
\\
\bar\partial_i \Psi_j (z) ( K*_{(j)}( \Psi_i(\cdot+\be') \partial_i \Psi_j))(z+\be)
&- \E [\bar\partial_i \Psi_j(z) ( K *_{(j)} \partial_i \Psi_j) (z+\be) ]
\Psi_i (z)
\end{equs}
respectively, plus remainder terms due to the ``positively renormalised'' kernels
which are bounded in exactly the same way as above.
So with
\[
J(z-w) \eqdef
\begin{cases}
\E [\partial_i  \Psi_i(z) K (z-w) \partial_i \Psi_i(w)] \;,\\
\E [\partial_j \Psi_i (z) K (z+\be+\be'-w) \bar\partial_j \Psi_i (w-\be') ]\;,\\
\E [\bar\partial_i \Psi_j(z)  K (z+\be+\be'-w) \partial_i \Psi_j(w-\be')]
\end{cases}
\]
where $z,w\in \R\times \obonds_i$,
the above three expressions can all be written as in \eqref{e:37RJ},
and one again has the bound \eqref{e:mom-2424}.

%Consider $\<I'[IXiI'Xi]_notriangle>$. It only has 2nd chaos and
%\begin{tikzpicture}  [scale=0.4,baseline=0]
%\node at (.6, 1)  [var] (a)  {};
%\node at (-.6,1) [var] (b) {};
%\draw[very thick] (0,0) -- (a);
%\draw (0,0) -- (b);
%\draw[very thick] (0,0) -- (0.6,-1);
%\node at (3.6,1)  [var] (c)  {};
%\node at (3-.6,1) [var] (d) {};
%\draw[very thick] (3,0) -- (d);
%\draw (3,0) -- (c);
%\draw[very thick] (3,0) -- (3-.6,-1);
%\draw[bend left =40, dotted]  (b) to (c) ;
%\draw[bend left =40, dotted]  (a) to (d) ;
%\big(\CK^{(\eps; 2)}\tau\big)\big(z_1, z_2\big) \lesssim \|z_1-z_2\|_{\e}^{-\kappa}
%%one has
%%$\mbox{L.S.} \lesssim \lambda^{-2} $ and
%%$\mbox{S.S.} \lesssim \e^{-2} \le \lambda^{-2}$.
%% where for S.S. we brutally bound $ \|z\|_{\e}^{-2}$ by $\e^{-2}$.
%as required by \eqref{e:CovarianceBounds}.
%The case $\<I'[I'Xi]_notriangle>$ satisfies the same bound.
The bounds for $\<I'[IXiI'Xi]_notriangle>$ and  $\<I'[I'Xi]_notriangle>$
follow from the bounds for $\<IXiI'Xi_notriangle>$ and $\<I'Xi_notriangle>$.
%Consider $ \<IXiI'[I'Xi]_notriangle>$. This is controlled in the same way as $\<IXiI'[IXiI'Xi]_notriangle>$.
%In particular, the two trees that need renormalisation are
%$ \Psi_i \CD_i \mcb{I} ( \bone^{(i)} \CD_i \Psi_i)$
%$\CS_a \Psi_j \bar\CD_i \mcb{I} (\CS_a \bone^{(i)} \CD_i \Psi_j)$,
%and for each of them
%the $0$th chaos exactly vanishes  by the choice of renormalisation.
%This is also the case for the other trees of this form
% either by independence
%of $\Psi_i$ and $\Psi_j$, or by exact spatial parity due to
%symmetrised derivatives.
%The case $\<I[I'Xi]I'Xi_notriangle>$ is also treated in the same way as $\<I[IXiI'Xi]I'Xi_notriangle>$.
The trees  $\<IXiI'[I'Xi]_notriangle>$ and $\<I[I'Xi]I'Xi_notriangle>$
are treated in the same way as
$\<IXiI'[IXiI'Xi]_notriangle>$ and
$\<I[IXiI'Xi]I'Xi_notriangle>$, and we omit the details.
\end{proof}

\begin{lemma}
\label{lem:mom-xi}
%For every $\tau\in \CT^{\xi}_-$,
For every $\tau$ in \eqref{e:trees-deg-1-rem},
the bounds \eqref{e:mom-bounds} hold.
\end{lemma}

\begin{proof}
For $\tau=\<Xi1>=\bar\Xi$ such that $\hat\Pi^\e\tau = \e^{1-\kappa} \xi^\e$,
similarly as $\Xi$, \eqref{e:CovarianceBounds} applies with $(\zeta,\alpha,n_\delta,n_\e)=(0,0,1,2-2\kappa)$.
%the same calculation as in \eqref{e:xiS22} yields the desired bounds:
%\mbox{L.S.} \lesssim (\e^{1-\kappa})^2 \lambda^{-4} \le \lambda^{-2-2\kappa}\;,
%\mbox{S.S.} \lesssim (\e^{1-\kappa})^2 \e^{-2} \lambda^{-2} \lesssim \lambda^{-2-2\kappa}\;.

Let $\tau= \<PsiXi1>$.
The $0$th chaos vanishes by our choice of renormalisation. For the 2nd chaos,
$\bigl(  \CK^{(\eps; 2)}\tau \bigr)(z_1,z_2)  = \e^{2-2\kappa} \delta_\e(z_1-z_2)  K^{*2}(z_1-z_2)$,
so \eqref{e:CovarianceBounds} applies with $(\zeta,\alpha,n_\delta,n_\e)=(0,-\kappa,1,2-2\kappa)$.
%for which
%%\E [(\hat\Pi^\e \tau) (\phi^\lambda)^2]
%& \le  (\e^{1-\kappa})^2  \int \delta_\e(z_1-z_2)  \phi_{z_*}^\lambda(z_1)   \phi_{z_*}^\lambda(z_2)
%  K^{*2}(0) \mrd z_1 \mrd z_2
%& \lesssim \e^{2-2\kappa} \int \phi_{z_*}^\lambda(z)^2 \e^{-\kappa}\mrd z
% \lesssim  \e^{2-3\kappa} \lambda^{-4}
% \le \lambda^{-2-3\kappa}\;.
%%For $\lambda<\e$,
%Here we recall $\delta_\e$ in \eqref{e:def-delta-eps}. By similar calculation,
%%\E \Big[\Big(\int_\R \hat\Pi^\e \tau (t,x) \CS^\lambda_2 \phi(t) \mrd t \Big)^2 \Big]
%\le  \e^{2-2\kappa}\int_\R \e^{-2} \CS^\lambda_{2,t_*} \phi(t)^2 \e^{-\kappa}\mrd t
%\lesssim \e^{-3\kappa} \lambda^{-2}
%\le \lambda^{-2-3\kappa}\;.
%Here $ \e^{-2}$ inside the integral arises from the factor $\e^{-1}$ in $\e^{-1}dW$.
%In general, the bounds at small scales and large scales
%can be obtained in essentially the same way:
%as in the above calculation one simply needs to replace a factor $\lambda^{-2}$
%in $ \e^{2-3\kappa} \lambda^{-4} $ for the large scale bound
%by a factor $\e^{-2}$ which is simply the right small scale bound.

Let $\tau = \<R2-1new> = \big(\CD_j^+ \CS_{\southeast} \mcb{I}_j \tilde\Xi_j \big)
\big(\CD_j^+ \CS_{\southwest} \mcb{I}_j \tilde\Xi_j \big)$. For the 2nd chaos,
\begin{equs}
\bigl(  \CK^{(\eps; 2)}\tau \bigr)(z_1,z_2)
& = \e^{2-2\kappa} (\partial_j^+ K)^{*_\e 2} (z_1-z_2) (\partial_j^+ K)^{*_\e 2} (z_1-z_2)
\\
& \lesssim \e^{2-2\kappa} \|z_1-z_2\|_\e^{-4-\kappa}
\lesssim  \|z_1-z_2\|_\e^{-2-3\kappa}
\end{equs}
so we  have \eqref{e:CovarianceBounds}.
%and we get the bound $\lambda^{-2-3\kappa}$ on both large and small scales.
For the $0$th chaos, one has, for $z = (t,x)$ with $x \in \obonds_i$,
\[
\bigl(\CW^{(\eps; 0)}\tau \bigr)(z) =
\int_{\R} \e^2 \sum\nolimits_{y\in \obonds_j}
\partial_j^+ K_{t-s}(x^\southeast - y) \,
\partial_j^+ K_{t-s}(x^\southwest - y)\mrd s \;.
\]
By naive dimension counting, this appears divergent. However, it is actually bounded.
This is because replacing $K$ above by $P$ only causes  an error uniformly bounded in $\e$.
since $x^\southeast \neq x^\southwest$ and $P_0$
vanishes away from origin, by Lemma~\ref{lem:identity},
the above quantity with
$K$ replaced by $P$ vanishes.
\end{proof}

\begin{lemma}
\label{lem:mom-12}
For every $\tau$ in group \hyperlink{(1-3)}{(1-3)},  the bounds \eqref{e:mom-bounds} hold.
\end{lemma}

\begin{proof}
$\tau= \<cherryY>$, consisting of three trees,
%We will see that they do not contribute to any mass renormalisation.
with $\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big)$ all bounded as \vspace{-5pt}
\begin{equ}[e:mom-Y333]
\begin{tikzpicture}  [scale=0.35,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (.6,-1) [var1] (x) {};
\draw[very thick] (0,0) -- (a);
\draw (0,0) -- (b);
\draw[kernel1] (0,0) -- (x);

\node at (3.6,1)  [var] (c)  {};
\node at (3-.6,1) [var] (d) {};
\node at (3-.6,-1) [var1] (y) {};
\draw[very thick] (3,0) -- (d);
\draw (3,0) -- (c);
\draw[kernel1] (3,0) -- (y);

\draw[bend left =40, delta]  (b) to (c) ;
\draw[delta]  (x) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\qquad
\|z_1-z_2\|_{\e}^{-\kappa} (\|z_1\|_{\e}^2 +\|z_2\|_{\e}^2)
\delta_\e(z_1-z_2)  (\e^{1-\kappa})^2
\end{equ}
so \eqref{e:CovarianceBounds} applies with $(\zeta,\alpha,n_\delta,n_\e)=(2,-\kappa,1,2-2\kappa)$.
%Therefore, since $\|z_1\|_{\e}^2 +\|z_2\|_{\e}^2\lesssim \lambda^2$
%on the supports of the rescaled test functions,
%\e^{-\kappa} \lambda^2 (\e^{1-\kappa})^2
% \int \phi_{z_*}^\lambda(z)^2 \mrd z
% \lesssim
%\e^{-\kappa} \lambda^2 \e^{2-2\kappa}  \lambda^{-4}
%\le \lambda^{-3\kappa}\;,
%\e^{-\kappa} \lambda^2 (\e^{1-\kappa})^2
%\int_\R \e^{-2} \CS^\lambda_{2,t_*} \phi(t)^2 \mrd t
% \lesssim
% \e^{-\kappa} \lambda^2 \e^{2-2\kappa}\e^{-2}  \lambda^{-2}
%\le \lambda^{-3\kappa}\;.
For the 1st chaos,
one term is treated as follows. Note that the contraction of two noises within each tree
yields a product of kernels with total degree $-4$, but we can use the powers of $\e$ to improve it, e.g.
$\e^{1-\kappa} \|z\|_\e^{-1} \le \|z\|_\e^{-\kappa}$. Then $\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big)$ is bounded by
\begin{equ}[e:mom-Y11]
\begin{tikzpicture}  [scale=0.35,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (0,-1.2) [var1] (x) {};
\draw[very thick] (0,0) -- (a);
\draw (0,0) -- (b);
\draw[kernel1] (0,0) -- (x);

\node at (3.6,1)  [var] (c)  {};
\node at (3-.6,1) [var] (d) {};
\node at (3,-1.2) [var1] (y) {};
\draw[very thick] (3,0) -- (d);
\draw (3,0) -- (c);
\draw[kernel1] (3,0) -- (y);

\draw[bend right =40, delta]  (b) to (x) ;
\draw[bend left =40, delta]  (c) to (y) ;
\draw[delta]  (a) to (d) ;
\end{tikzpicture}
\qquad
\|z_1-z_2\|_{\e}^{-2-\kappa}
(\|z_1\|_{\e}^2 +\|z_2\|_{\e}^2+\|z_1-z_2\|_{\e}^2)
\end{equ}
so we get  \eqref{e:CovarianceBounds}.
There is then another term in the 1st chaos, with 2nd moment
\[
\begin{tikzpicture}  [scale=0.35,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (0,-1.2) [var1] (x) {};
\draw (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[kernel1] (0,0) -- (x);

\node at (3.6,1)  [var] (c)  {};
\node at (3-.6,1) [var] (d) {};
\node at (3,-1.2) [var1] (y) {};
\draw (3,0) -- (d);
\draw[very thick] (3,0) -- (c);
\draw[kernel1] (3,0) -- (y);

\draw[bend right =40, delta]  (b) to (x) ;
\draw[bend left =40, delta]  (c) to (y) ;
\draw[delta]  (a) to (d) ;
\end{tikzpicture}
%\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big) \lesssim
%(\|z_1\|_{\e}^2 +\|z_2\|_{\e}^2+\|z_1-z_2\|_{\e}^2)
\]
Define $\bar K\eqdef \e^{1-\kappa} (K \partial K)$ (where $\partial$ could be replaced by  $\bar\partial$ depending on the particular tree).
One has
$|\bar K(z)| \lesssim \|z\|_\e^{-4-\kappa}$
since the derivative is symmetrised,
$\bar K=\mathscr{R} \bar K$ and  $\bigl(\CK^{(\eps; 1)}\tau \bigr)(z_1,z_2)$ is given by  (recall $z_*=0$)
\begin{equ}
\int
\big(\mathscr{R} \bar K (z_1-w_1)
-\bar K (-w_1)\big)
K^{*2} (w_1- w_2)
\big(\mathscr{R} \bar K (z_2-w_2)
-\bar K (-w_2)\big)
\mrd w_1\mrd w_2\;.
\end{equ}
We can derive the same bound \eqref{e:mom-Y11} for this case,
so one again has \eqref{e:CovarianceBounds}.
The case $\<cherryYless>$ follows in the same way.

$\tau = \<IXiI'Xibar>$ (which includes trees added in Step 3 in Section \ref{subsubsec:Step_3}).
The $0$th chaos vanishes
thanks to our choice of renormalisation \eqref{eq:tilde_C_renorm}, \eqref{eq:square_cherry_consts}, \eqref{eq:C_square_cherries}.
%\bigl(\CK^{(\eps; 2)}\tau \bigr)(z,\bar z) = \e^{2-2\kappa} (K^\e)^{*2} (\partial K^\e)^{*2} (z-\bar z) \lesssim \e^{2-2\kappa} \|z-\bar z\|_\e^{-2-\kappa}\;.
%One gets a bound $\e^{2-2\kappa} \lambda^{-2-\kappa} \lesssim \lambda^{-3\kappa}$ for $\e<\lambda$.
The 2nd chaos is treated similarly as  $\<IXiI'Xi_notriangle>$,
and by absorbing powers of $\e$ into kernels $\e^{1-\kappa} \|z\|_\e^{-1} \le \|z\|_\e^{-\kappa}$.
The cases $\tau=\<I'Xibar>$  and $\tau=\<I'XiIXibar>$ follow similarly.
%This is similar as the last one, except that
%\bigl(\CK^{(\eps; 2)}\tau \bigr)(z,\bar z)
% \lesssim
%  \e^{2-2\kappa} \|z-\bar z\|_\e^{-2-\kappa}
% +  \e^{2-2\kappa} \|z-\bar z\|_\e^{-3-\kappa} (\|z\|_\e + \|\bar z\|_\e)\;.
%So we get the same bound.   The zeroth chaos also vanishes.

Consider $\tau\in \{ \<supercherry1>, \<supercherry2>\}$.  This includes the following trees
\begin{equs}[e:supercherry1-list]
\Psi_i \CD_i\mcb{I} (\Psi_i \bar\Xi_i)\;,
\mcb{I} ( \Psi_i \bar\Xi_i) \CD_i \Psi_i\;,
\\
\CS_{\be}\Psi_j  \CD_j\mcb{I} (\Psi_i  \bar\Xi_i)\;,
\CS_{\be}\mcb{I} (\Psi_j \bar\Xi_j) \CD_j \Psi_i\;,
\\
\CS_{\be} \Psi_j
\CD_i^+ \CS_{\bw}
\mcb{I} (\Psi_j \bar\Xi_j)\;,
\CS_{\be}\mcb{I} (\Psi_j \bar\Xi_j) \bar{\CD}_i \Psi_j\;,
\end{equs}
where $\be\in\CE_\times$ and $\bw\in\{\Northwest,\Southwest\}$.
%Remark that these trees showed up in \eqref{e:quad-exp-1}, \eqref{e:quad-exp-2}, \eqref{e:quad-exp-3}.
The 3rd chaos parts of the trees in the left column are all bounded as follows:
\begin{equ}[e:superch-3rd]
\begin{tikzpicture}  [scale=0.35,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (0,0) [var1] (b) {};
\node at (1.2,0) [var] (x) {};
\draw (b) -- (a);
\draw[very thick] (b) -- (0.6,-1);
\draw (x) -- (0.6,-1);

\node at (4,0)  [var1] (c)  {};
\node at (4-.6,1) [var] (d) {};
\node at (4-1.2,0) [var] (y) {};
\draw (c) -- (d);
\draw[very thick] (c) -- (4-.6,-1);
\draw (y) -- (4-0.6,-1);

\draw[bend left =30, delta]  (b) to (c) ;
\draw[bend left =10, delta]  (x) to (y) ;
\draw[bend left =10, delta]  (a) to (d) ;
\end{tikzpicture}
\qquad
\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big) \lesssim
\e^{2-3\kappa}
\|z_1- z_2\|_\e^{-2} \lesssim \|z_1- z_2\|_\e^{-3\kappa}
\end{equ}
where we bounded
$|K^{*_\e 2}(0)| \lesssim \e^{-\kappa}$ at the top
(it is evaluated at $0$ due to the contraction of the two $\bar\xi$'s).
For the 3rd chaos  of the trees in the right column:
\[
\begin{tikzpicture}  [scale=0.35,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (0,0) [var1] (b) {};
\node at (1.2,0) [var] (x) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (0.6,-1);
\draw[very thick] (x) -- (0.6,-1);

\node at (4,0)  [var1] (c)  {};
\node at (4-.6,1) [var] (d) {};
\node at (4-1.2,0) [var] (y) {};
\draw (c) -- (d);
\draw[kernel1] (c) -- (4-.6,-1);
\draw[very thick] (y) -- (4-0.6,-1);

\draw[bend left =30, delta]  (b) to (c) ;
\draw[bend left =10, delta]  (x) to (y) ;
\draw[bend left =10, delta]  (a) to (d) ;
\end{tikzpicture}
\quad
\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big)
\lesssim
\|z_1-z_2\|_\e^{-1-3\kappa}
\]
These two bounds are again in the scope of \eqref{e:CovarianceBounds}.
The 1st chaos of each of these trees consists of two terms,
one of which is bounded as follows
\begin{equ}[e:mom-supercheche]
\begin{tikzpicture}  [scale=0.35,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (1.2,0) [var1] (b) {};
\node at (0,0) [var] (x) {};
\draw (b) -- (a);
\draw[very thick] (b) -- (0.6,-1);
\draw (x) -- (0.6,-1);

\node at (4-.6,1) [var] (d) {};
\node at (4-1.2,0)  [var1] (c)  {};
\node at (4,0) [var] (y) {};
\draw (c) -- (d);
\draw[very thick] (c) -- (4-.6,-1);
\draw (y) -- (4-0.6,-1);

\draw[bend right =40, delta]  (a) to (x) ;
\draw[delta]  (b) to (c) ;
\draw[bend left =40, delta]  (d) to (y) ;
\end{tikzpicture}
\qquad
\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big)
\lesssim
\e^{2-2\kappa} \|z_1-z_2\|_\e^{-2-2\kappa}
\lesssim \|z_1- z_2\|_\e^{-4\kappa}
\end{equ}
\begin{equ}
\begin{tikzpicture}  [scale=0.4,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (1.2,0) [var1] (b) {};
\node at (0,0) [var] (x) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (0.6,-1);
\draw[very thick] (x) -- (0.6,-1);

\node at (3.7-.6,1) [var] (d) {};
\node at (3.7-1.2,0)  [var1] (c)  {};
\node at (3.7,0) [var] (y) {};
\draw (c) -- (d);
\draw[kernel1] (c) -- (3.7-.6,-1);
\draw[very thick] (y) -- (3.7-0.6,-1);

\draw[bend right =40, delta]  (a) to (x) ;
\draw[delta]  (b) to (c) ;
\draw[bend left =40, delta]  (d) to (y) ;
\end{tikzpicture}
\qquad
\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big)
\lesssim \|z_1- z_2\|_\e^{-4\kappa}
+ \|z_1\|_\e^{-3\kappa}+ \|z_2\|_\e^{-3\kappa}
\end{equ}
so we are again in the scope of \eqref{e:CovarianceBounds} and \eqref{e:kappa3}.
The other term in the 1st chaos is
%obtained by contracting the two noises in the bottom:
\tikz[scale=0.29,baseline=-5]
{\node at (.6, 1)  [var] (a)  {};
\node at (1.2,0) [var1] (b) {};
\node at (0,0) [var] (x) {};
\draw (b) -- (a);
\draw[very thick] (b) -- (0.6,-1);
\draw (x) -- (0.6,-1);
\draw[very thick, delta]  (b) to (x) ;
\tikz[scale=0.29,baseline=-5]
\node at (.6, 1)  [var] (a)  {};
\node at (1.2,0) [var1] (b) {};
\node at (0,0) [var] (x) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (0.6,-1);
\draw[very thick] (x) -- (0.6,-1);
\draw[very thick, delta]  (b) to (x) ;
} .
This term vanishes for trees in the second row of \eqref{e:supercherry1-list}, by independence of $\xi_i,\xi_j$.
For trees in the first row of \eqref{e:supercherry1-list},
this term is equal to $J*_{(i)}\Psi_i$ (up to a remainder due to the  kernel $\tikz[baseline=-3]\draw[kernel1] (0,0) --(.8,0);$ in the case of $\mcb{I} ( \Psi_i \bar\Xi_i) \CD_i \Psi_i$) with
$J(z-u)$  being
$\e^{1-\kappa}  \E [\Psi_i(z)\xi_i(u) \partial_i K(z-u)]$
$\e^{1-\kappa} \E [\partial_i \Psi_i(z)\xi_i(u)  K(z-u)]$
for $z,u\in \R\times \obonds_i$. Since each $J$  here is odd in the $i$th coordinate, its integral vanishes and thus $J=\mathscr{R} J $.
Since $J$ is of order $-4-\kappa$,
we therefore have the bound
\eqref{e:mom-2424}. The ``remainder'' is bounded exactly as in \eqref{e:w-z-w-w-w} for $\<I[IXiI'Xi]I'Xi_notriangle>$.

Consider now  the trees in last row of  \eqref{e:supercherry1-list},
which are renormalised as in  \eqref{eq:square_cherry_consts} and \eqref{eq:C_square_cherries}.
For $ \CS_{\be} \Psi_j\CD_i^+ \CS_{\bw}\mcb{I} (\Psi_j \bar\Xi_j)$,
the term of interest in the 1st chaos is
\begin{equs}
\e^{1-\kappa} &\int \E[\Psi_j (z+\be) \partial_i^+ K(z^\bw-u)  \xi_j(u)] \Psi_j(u) \mrd u
\\
\e^{1-\kappa} \E [\Psi_j(z+\be) \d_i^+\Psi_j(z^\bw) ] \Psi_j(z^\bw) = (\mathscr{R} J *_{(j)} \Psi_j)(z^\bw)
\end{equs}
where for $\bar z,u\in  \R\times \obonds_j$, writing $\hat z$ for the element of $\R\times \obonds_i$ such that $\hat z^{\bw}=\bar z$,
\[
J(\bar z-u) = \e^{1-\kappa} \E[\Psi_j (\hat z+\be) \partial_i^+ K(\bar z-u)  \xi_j(u)]\;.
\]
We then again obtain  the bound
\eqref{e:mom-2424}.  The other tree in the last row of  \eqref{e:supercherry1-list} follows in the same way,
with ``positively renormalised'' kernels treated as above.

For $\tau=\<supercherry1less>$ one only needs to bound the 2nd chaos and it is similar as \eqref{e:superch-3rd}.
%$\bigl(\CK^{(\eps; 1)}\tau\bigr)(z_1,z_2)$ equals
% \e^{2-2\kappa}
%(\partial K^\e)(K^\e)^{*2} (z_1-y_1) \,
%(\partial K^\e)(K^\e)^{*2} (z_2- y_2) \,
%\mrd y_1 \mrd y_2
%One then gets a large scale bound $ \e^{2-2\kappa}  \lambda^{-2-2\kappa} \lesssim \lambda^{-4\kappa}$.

Finally, for $\tau=\X^{(i)} \<PsiXi1>$, the 0th chaos vanishes by parity.
For the 2nd chaos the bound follows as in $\<PsiXi1>$ above except that $\zeta=1$.
%Regarding the 2nd chaos, writing $z_j=(t_j,x_j)$ for $j\in \{1,2\}$ one has
%\le  (\e^{1-\kappa})^2  \int \delta_\e(z_1-z_2) \, \| x_1^{(i)}\|_\e \| x_2^{(i)}\|_\e\, \phi_{z_*}^\lambda(z_1)   \phi_{z_*}^\lambda(z_2)
%  K^{*2}(0) \mrd z_1 \mrd z_2
%which is bounded by $\lambda^{-3\kappa}$
%similarly as \eqref{e:PsiXi1-mom} for $ \<PsiXi1>$, noting that $\|x_j^{(i)}\|_\e \le \lambda$ on the support of $\phi_{z_*}^\lambda$.
%The small scale bound is trivial since
%in \eqref{e:IkWk-2nd}
%$\CW^{(\eps; 2)}\tau$ is evaluated at $(t,x_*)$ which is zero.
The tree $\X^{(i)} \bar\Xi_i$ is even simpler.
\end{proof}

\begin{lemma}
\label{lem:mom-13}
For every $\tau$ in group \hyperlink{(1-2)}{(1-2)},  the bounds \eqref{e:mom-bounds} hold.
\end{lemma}

\begin{proof}
Let $\tau=\<I'XiI'[IXiI'Xi]>$. Recall (below \eqref{e:trees-deg-0-rem}) that trees of this form arise from the term $\tilde R_2$ in \eqref{e:tildeR2}, \eqref{eq:R_2_other_form},
with one of the factors $A$ there replaced by an integration of $A\partial A$. (See also \eqref{e:tildeR2-exp}.)
% where they appear on the level of modelled distributions when we expand the term $ \CQ^{\e}_i$ defined in \eqref{e:def-tildeCR2}.
These trees have different properties in terms of parity, noise indices and renormalisation;
to enumerate these trees more easily and conceptually,  we will count
the two trees  corresponding to $\CD^+ + \CD^-$ as ``one tree'',
the two trees corresponding to $\sum_{\bw} \CD^+ \cS_{\bw}$ as ``one tree'',
the two trees which only differ by simultaneously swapping all indices $i\to 3-i$ as ``one tree'',
and the four trees corresponding to $\cS_{a}$ defined in \eqref{e:shift-a} (see e.g. \eqref{e:def-CHi}) as ``one tree''.
Counting this way,
there are $2\times 2\times 3 = 12$ trees of this form (two terms in $\tilde R_2$, each of them is quadratic in $A$, and the equation for $A$ has $3$ terms of the form $A\partial A$).
The 3rd chaos can be all bounded as follows: \vspace{-5pt}
\[
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (1.2,0) [var1] (x) {};
\draw[very thick] (0,0) -- (a);
\draw (0,0) -- (b);
\draw[very thick] (0,0) -- (0.6,-1);
\draw[very thick] (x) -- (0.6,-1);

\node at (4.6,1)  [var] (c)  {};
\node at (4-.6,1) [var] (d) {};
\node at (4-1.2,0) [var1] (y) {};
\draw[very thick] (4,0) -- (d);
\draw (4,0) -- (c);
\draw[very thick] (4,0) -- (4-.6,-1);
\draw[very thick] (y) -- (4-0.6,-1);

\draw[bend left =40, delta]  (b) to (c) ;
\draw[bend left =40, delta]  (x) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\qquad
\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big) \lesssim
\e^{2-2\kappa}\|z_1-z_2\|_{\e}^{-2-\kappa}
\lesssim
\|z_1-z_2\|_{\e}^{-3\kappa}\;.
\]
The same bound holds for the term
\tikz[scale=0.27,baseline=-5]
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (-1.2,0) [var1] (x) {};
\draw[very thick] (0,0) -- (a);
\draw (0,0) -- (b);
\draw[very thick] (0,0) -- (-0.6,-1);
\draw[very thick] (x) -- (-0.6,-1);
\draw[bend right =40, delta,thick]  (b) to (x) ;
} in the 1st chaos
using $\e^{1-\kappa} \|z\|_\e^{-4} \le  \|z\|_\e^{-3-\kappa}$.
%For the 1st chaos, the following term is easy to bound: using $\e^{1-\kappa} \|z\|_\e^{-4} \le  \|z\|_\e^{-3-\kappa}$  \vspace{-5pt}
%\begin{tikzpicture}  [scale=0.33,baseline=0]
%\node at (.6, 1)  [var] (a)  {};
%\node at (-.6,1) [var] (b) {};
%\node at (-1.2,0) [var1] (x) {};
%\draw[very thick] (0,0) -- (a);
%\draw (0,0) -- (b);
%\draw[very thick] (0,0) -- (-0.6,-1);
%\draw[very thick] (x) -- (-0.6,-1);
%\node at (3.6,1)  [var] (c)  {};
%\node at (3-.6,1) [var] (d) {};
%\node at (3+1.2,0) [var1] (y) {};
%\draw[very thick] (3,0) -- (d);
%\draw (3,0) -- (c);
%\draw[very thick] (3,0) -- (3+.6,-1);
%\draw[very thick] (y) -- (3+0.6,-1);
%\draw[bend right =40, delta]  (b) to (x) ;
%\draw[bend left =40, delta]  (a) to (d) ;
%\draw[bend left =40, delta]  (c) to (y) ;
%\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big) \lesssim
Below we only need to focus on
the term
\tikz[scale=0.27,baseline=-5]
\node at (.6, 1)  [var] (a)  {};
\node at (-.6,1) [var] (b) {};
\node at (-1.2,0) [var1] (x) {};
\draw (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[very thick] (0,0) -- (-0.6,-1);
\draw[very thick] (x) -- (-0.6,-1);
\draw[bend right =40, delta,thick]  (b) to (x) ;
in the 1st chaos.
% obtained by contracting the two noises in the subtree $\<I'XiI'[IXiI'Xi]less>$,
%which form a potential divergent subgraph  with 3 derivatives.
By enumerating the $12$ trees, one finds that for $6$ of them
the two noises being contracted %in $\<I'XiI'[IXiI'Xi]less>$
have distinct spatial indices,
% (the expansion of $\CA_i^\e$~\eqref{e:CA-expand} is helpful for this),
for instance
$\big(\CD_j^+ \CS_{\southeast} \mcb{I} (\CS_{\be}\Psi_i \CD_j \Psi_i)\big)
\big( \CD_j^+ \CS_{\southwest} \bar\Psi_j\big)$.
By independence
of $\xi_i,\xi_j$,
we only need to consider the other $6$ trees.
%one might be able to use parity, but on lattice we must check if we have {\it exact} parity,
%so we look into each tree more explicitly.
First, consider
$\big(\CD_j^+ \CS_{\southeast} \mcb{I} (\Psi_j \CD_j \Psi_j)\big)
\big( \CD_j^+ \CS_{\southwest} \bar\Psi_j\big)$.
The term of interest  in the 1st chaos has a potential divergence, where $z\in \R\times \obonds_i$:  %the form $\bar K * \Psi_j$ where
\begin{equ}[e:mom-R2-DjK3]
\e^{1-\kappa} \!\!
\int_{(\R\times \obonds_j)^2} \!\!
\partial_j^+ K(z^\southwest - y)
\partial_j^+ K(z^\southeast - w)
\partial_j K( w-y) \mrd w \mrd y\;.
\end{equ}
By a change of variables $w_j \mapsto 2z_j-w_j$ and $y_j \mapsto 2z_j-y_j$,
and recalling that $K$ is even in each spatial direction and the definitions of $z^\southwest ,z^\southeast,\partial_j^+,\partial_j$,
we see that each of the three factors in the integrand flips the sign, so \eqref{e:mom-R2-DjK3} is zero. (Essentially
$\partial_j^+ K(z^\southwest) = \frac{1}{\e} (K (z^\northwest) - K (z^\southwest))$ is still a ``symmetrised'' derivative in the $j$th direction.)
%\hao{Probably don't need this sentence now. On the other hand,
%if this is $\e^{-\kappa}$ instead of $0$,
%having $c\CA_j^\e$ is fine due to your new argument;
%it's just that $\e^{-\kappa}$ isn't allowed here to bound the model...
%In principle
%  we could also treat it in the same way as the 3 trees listed in the displayed equation below, namely we renormalize it (and we have analog of Lemma \ref{lem:thick-trig-bahave}), but that doesn't necessarily shorten things}
Now with the kernel
$\bar K=\mathscr{R} \bar K $ of degree $-4-\kappa$
denoted by $\tikz[baseline=-3]\draw[kernels] (0,0) --(.8,0);$ which is defined to be the above integral without integrating $w$,
one  has the same bound as \eqref{e:mom-2424},
again satisfying  \eqref{e:CovarianceBounds}.
%\bigl(\CK^{(\eps; 1)}\tau \bigr)(z,\bar z) = \e^{2-2\kappa}\int \bar K^\e (z-w) (K^\e)^{*2} (w-\bar w) \bar K^\e (\bar z-\bar w)\mrd w\mrd \bar w
%and we get a bounded by $\lambda^{-2\kappa}$.
Exactly the same argument applies to
$\big( \CD_j^+ \CS_{\southeast} \bar\Psi_j\big)
\big(\CD_j^+ \CS_{\southwest} \mcb{I} (\Psi_j \CD_j \Psi_j)\big)$.
We then turn to the case
$ (\CD_j^{\pm} \mcb{I} (\CS_{\be} \Psi_j \bar\CD_i \Psi_j) )(\bar\CD_j \bar\Psi_j)$.
The potential divergence (which is independent of $\be$) is an integral of a function which is odd in the $i$th direction, so we again have a bound \eqref{e:mom-2424}.
It then remains to
\begin{equs}
(\CD_j^{\pm} \bar\Psi_i)  \CD_j^+ &\CS_{\bs}  \mcb{I} (\CS_{\be} \Psi_i \bar\CD_j \Psi_i))\;,
\\
\big( \CD_j^+ \CS_{\southeast} \bar\Psi_j\big)
\big(\CD_j^+ \CS_{\southwest} \mcb{I} (\CS_{\be}\Psi_i \CD_i \Psi_j)\big)
\;,
& \qquad
\big(\CD_j^+ \CS_{\southeast} \mcb{I} (\CS_{\be}\Psi_i \CD_i \Psi_j)\big)
\big( \CD_j^+ \CS_{\southwest} \bar\Psi_j\big)\;,
\end{equs}
where $\be\in\CE_\times$ and $\bs\in\{\Southeast,\Southwest\}$.
These are precisely the trees renormalised as in
\eqref{e:div-R2} and
\eqref{e:def-C-triangle} by
$C_{\bs,\pm}^\e[ \, \<I'XiI'[IXiI'Xi]less>\,]$,
%$C_{\southeast,\pm}^\e[ \, \<I'XiI'[IXiI'Xi]less>\,]$,
$C_{\southwest}^\e[ \, \<I'XiI'[IXiI'Xi]less>\,]$,
$C_{\southeast}^\e[ \, \<I'XiI'[IXiI'Xi]less>\,]$,
for which we can obtain bounds of the form \eqref{e:mom-2424}.
For instance if $\tau$ is the last tree,
we have (see \eqref{e:def-C-triangle3})
\begin{equs}[e:product-expectation]
\hat\Pi_{z_*}^\e\tau (z)
\partial_j^+\Psi_j(z^\southwest)
(\partial_j^+ K *( \Psi_i(\cdot+\be) \partial_i \Psi_j))(z^\southeast)
\\
&\qquad -  \E [ \e^{1-\kappa} \partial_j^+\Psi_j(z^\southwest)
(\partial_j^+ K * \partial_i \Psi_j)(z^\southeast)]
\Psi_i (z)\;,
\end{equs}
which is equal to $(\mathscr{R} J *_{(i)} \Psi_i)(z)$ with $J$ of degree $-4-\kappa$ as above.
The other trees follow similarly.
(These constants may not in general vanish, see Remark \ref{rem:C_non_vanish}.)\footnote{In these calculations, the 3 trees which do require renormalisation due to lack of exact parity
are renormalised by $c\Psi_i$ which will contribute a term $c A_i$ to the renormalised equation;
the 3 trees which would potentially renormalise the equation by a term $c A_j$ for $j\neq i$ all enjoy exact parity so
they are not renormalised. This is consistent with the fact that we show in Section \ref{sec:gauge-covar} (by other means) that we do not have a renormalisation term $c A_j$ in the $A_i$ equation.}
%Instead, we can replace the kernels $K^\e$ by the full heat kernel $P^\e$;
%the error of this replacement vanishes.
%Moreover, we can replace the $w$ integral in \eqref{e:mom-R2-DjK2DiK} by an integral over $z$.
%With these replacements, and
%applying Lemma~\ref{lem:identity}, \eqref{e:mom-R2-DjK2DiK} is then turned into
%\e^{1-\kappa} \!\!
%\int_{\R\times \obonds_j} \!\!
%P^\e(w - y - \e_i)
%\partial_i P^\e( w-y) \mrd y\;.
%By Lemma~\ref{lem:ePsiDPsi} this is $\e^{-\kappa} O(1)$.

The case  $\<I'XiI'[IXiI'Xi]less>$ is analogous to the above arguments for $\<I'XiI'[IXiI'Xi]>$.

The bound for $\<I'(R2-1)new>$ follows from that for $\<R2-1new>$.
%It only has the 2nd chaos  which is easily bounded:
%\begin{tikzpicture}  [scale=0.4,baseline=0]
%\node at (.6, 1)  [var2] (a)  {};
%\node at (-.6,1) [var2] (b) {};
%\draw[very thick] (0,0) -- (a);
%\draw[very thick] (0,0) -- (b);
%\draw[very thick] (0,0) -- (0,-1);
%\node at (3.6,1)  [var2] (c)  {};
%\node at (3-.6,1) [var2] (d) {};
%\draw[very thick] (3,0) -- (d);
%\draw[very thick] (3,0) -- (c);
%\draw[very thick] (3,0) -- (3,-1);
%\draw[bend left =40, dotted]  (b) to (c) ;
%\draw[bend left =40, dotted]  (a) to (d) ;
%\big(\CK^{(\eps; 2)}\tau\big)\big(z_1, z_2\big) \lesssim \|z_1-z_2\|_{\e}^{-\kappa}

Consider $\tau=\<IXiI'(R2-1)new>$ and  $\tau=\<I'XiI(R2-1)new>$.
For the 3rd and 1st chaoses we can obtain
%a bound of the form \eqref{e:momR2-Ydiamond}
the desired bounds
by observing the following graphs
and absorbing powers of $\e$ into kernels as before: \vspace{-5pt}
\[
\begin{tikzpicture}  [scale=0.33,baseline=10]
\node at (.6, 1)  [var2] (a)  {};
\node at (-.6,1) [var2] (b) {};
\node at (1.2,0) [var] (x) {};
\draw[very thick] (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[very thick] (0,0) -- (0.6,-1);
\draw (x) -- (0.6,-1);

\node at (4.6,1)  [var2] (c)  {};
\node at (4-.6,1) [var2] (d) {};
\node at (4-1.2,0) [var] (y) {};
\draw[very thick] (4,0) -- (d);
\draw[very thick] (4,0) -- (c);
\draw[very thick] (4,0) -- (4-.6,-1);
\draw (y) -- (4-0.6,-1);

\draw[bend left =40, dotted]  (b) to (c) ;
\draw[bend left =40, dotted]  (x) to (y) ;
\draw[bend left =40, dotted]  (a) to (d) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=10]
\node at (.6, 1)  [var2] (a)  {};
\node at (-.6,1) [var2] (b) {};
\node at (-1.2,0) [var] (x) {};
\draw[very thick] (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[very thick] (0,0) -- (-0.6,-1);
\draw (x) -- (-0.6,-1);

\node at (3.6,1)  [var2] (c)  {};
\node at (3-.6,1) [var2] (d) {};
\node at (3+1.2,0) [var] (y) {};
\draw[very thick] (3,0) -- (d);
\draw[very thick] (3,0) -- (c);
\draw[very thick] (3,0) -- (3.6,-1);
\draw (y) -- (3+0.6,-1);

\draw[bend right =50, delta]  (b) to (x) ;
\draw[bend left =50, delta]  (c) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\qquad
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=10]
\node at (.6, 1)  [var2] (a)  {};
\node at (-.6,1) [var2] (b) {};
\node at (1.2,0) [var] (x) {};
\draw[very thick] (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[kernel1] (0,0) -- (0.6,-1);
\draw[very thick] (x) -- (0.6,-1);

\node at (4.6,1)  [var2] (c)  {};
\node at (4-.6,1) [var2] (d) {};
\node at (4-1.2,0) [var] (y) {};
\draw[very thick] (4,0) -- (d);
\draw[very thick] (4,0) -- (c);
\draw[kernel1] (4,0) -- (4-.6,-1);
\draw[very thick] (y) -- (4-0.6,-1);

\draw[bend left =40, delta]  (b) to (c) ;
\draw[bend left =40, delta]  (x) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=10]
\node at (.6, 1)  [var2] (a)  {};
\node at (-.6,1) [var2] (b) {};
\node at (-1.2,0) [var] (x) {};
\draw[very thick] (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[kernel1] (0,0) -- (-0.6,-1);
\draw[very thick] (x) -- (-0.6,-1);

\node at (3.6,1)  [var2] (c)  {};
\node at (3-.6,1) [var2] (d) {};
\node at (3+1.2,0) [var] (y) {};
\draw[very thick] (3,0) -- (d);
\draw[very thick] (3,0) -- (c);
\draw[kernel1] (3,0) -- (3.6,-1);
\draw[very thick] (y) -- (3+0.6,-1);

\draw[bend right =50, delta]  (b) to (x) ;
\draw[bend left =50, delta]  (c) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\]
Here the 3rd graph has  ``positively renormalised'' kernels and is bounded in the same way as
\eqref{e:K3-01} using \eqref{e:Diff-K}.
The last graph is also treated as in \eqref{e:K3-02}.
%also satisfies the same bound \eqref{e:kappa3LSSS} as in the case \eqref{e:K3-02}.
\end{proof}

\begin{lemma}
\label{lem:mom-222333}
For every $\tau$ in groups \hyperlink{(3-3)}{(3-3)}, \hyperlink{(2-3)}{(2-3)} and \hyperlink{(2-2)}{(2-2)},  the bounds \eqref{e:mom-bounds} hold.
\end{lemma}

\begin{proof}
In the proof
we will often use the positive powers of $\e$ to improve the singularities of the kernels as before without
explicitly mentioning.
The 3rd chaos is bounded as follows, using  \eqref{e:Diff-K} with
$\alpha=1$: \footnote{Here \eqref{e:Diff-K} with
$\alpha=0$ i.e. bounding the two terms in  $\tikz[baseline=-3]\draw[kernel1] (0,0) --(.8,0);$ separately
would lead to a bound $\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big)  \lesssim \e^{4-6\kappa}  \delta_\e(z_1-z_2)$, but this is not sufficient for the
desired (small scale) bound.}
\begin{equ}[e:I(PsiXi1)Xi1-3rd]
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (.6, 1.2)  [var] (a)  {};
\node at (1.2,0) [var1] (b) {};
\node at (0.6,-1.2) [var1] (x) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);

\node at (4-.6,1.2) [var] (d) {};
\node at (4-1.2,0)  [var1] (c)  {};
\node at (4-.6,-1.2) [var1] (y) {};
\draw (c) -- (d);
\draw[kernel1] (c) -- (y);

\draw[delta]  (a) to (d) ;
\draw[delta]  (b) to (c) ;
\draw[delta]  (x) to (y) ;
\end{tikzpicture}
\qquad
\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big)
\lesssim  %\|z_1- z_2\|_\e^{-2-\kappa}
(\|z_1\|_\e^{2} +\|z_2\|_\e^{2})
\delta_\e(z_1-z_2)  \e^{2-6\kappa}.
\end{equ}
%From this we can deduce a bound $\lambda^{-6\kappa}$ as in \eqref{e:mom-cheY3}.
Now \eqref{e:CovarianceBounds} applies with $(\zeta,\alpha,n_\delta,n_\e)=(2,0,1,2-6\kappa)$.
%(in particular for the small scale we bound $\|z_1- z_2\|_\e^{-2-\kappa}\le \e^{-2-\kappa}$).

For one term in the 1st chaos the second moment is bounded as follows
\begin{equ}[e:I(PsiXi1)Xi1-1st]
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (.6, 1.2)  [var] (a)  {};
\node at (1.2,0) [var1] (b) {};
\node at (0.6,-1.2) [var1] (x) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);

\node at (4-.6,1.2) [var] (d) {};
\node at (4-1.2,0)  [var1] (c)  {};
\node at (4-.6,-1.2) [var1] (y) {};
\draw (c) -- (d);
\draw[kernel1] (c) -- (y);

\draw[bend right =40, delta]  (a) to (x) ;
\draw[bend left =40, delta]  (b) to (c) ;
\draw[bend left =40, delta]  (d) to (y) ;
\end{tikzpicture}
\qquad
\big(\CK^{(\eps; 1)}\tau\big)\big(z_1, z_2\big)
\lesssim
% \|z_1- z_2\|_\e^{-2-4\kappa}
% (\|z_1\|_\e^{2} +\|z_2\|_\e^{2}+\|z_1-z_2\|_\e^{2})
\|z_1- z_2\|_\e^{-4\kappa} +  \|z_1\|_\e^{-4\kappa}+ \| z_2\|_\e^{-4\kappa}
\end{equ}
%from which we obtain the desired bounds by \eqref{e:CovarianceBounds} and \eqref{e:kappa3LSSS}.
essentially the same as \eqref{e:kappa3}.
For the other term %which is graphically denoted as
%\begin{tikzpicture}  [scale=0.33,baseline=0]
\tikz[scale=0.27,baseline=-7]
\node at (-.5, 0)  [var] (a)  {};
\node at (1,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw[bend left = 60, delta, thick]  (b) to (x) ;
} ,
recalling \eqref{e:M-XiIXi} and \eqref{e:C-XiIXi},,
upon cancellation by  $C^\e[\,\<I(PsiXi1)Xi1less>\,]$ we are left with
$\e^{2-2\kappa}\int K(-z)K(z-w)\xi(w)\mrd w$,
and the covariance is bounded by $\|z_1\|^{-3\kappa}\|z_2\|^{-3\kappa}$ which can be treated easily.
%bounded as in \eqref{e:kappa3LSSS}.
The bound for
$\<I(PsiXi1)Xi1less>$ is similar and even simpler.

Let $\tau=\<r_z_large>$.
%For the 4th chaos, $\big(\CK^{(\eps; 4)}\tau\big)\big(z_1, z_2\big)$ is equal to
% \e^{4-4\kappa} (K^\e)^{*2} (0)^2
%\int K^\e(z_1-w)K^\e(z_2-w) \delta_\e (z_1-z_2)\mrd w\mrd z_1\mrd z_2
%plus (up to possible minus sign) three other terms given by the same expression with $z_1$ or $z_2$ (or both) in the function $K^\e$
%replaced by $z_*$.
%Here $\delta_\e$ is as in \eqref{e:def-delta-eps}. Since $|K^\e * K^\e |\lesssim \e^{-\kappa}$,
The 4th chaos can be bounded in the same way as the 3rd chaos of
$\<I(PsiXi1)Xi1>$ above. The 2nd chaos has $5$ terms:
\[
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (0, 1.2)  [var] (a)  {};
\node at (1,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\node at (-1,0) [var] (y) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw (x) -- (y);
\draw[delta]  (a) to (y) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (0, 1.2)  [var] (a)  {};
\node at (1,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\node at (-1,0) [var] (y) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw (x) -- (y);
\draw[delta]  (a) to (x) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (0, 1.2)  [var] (a)  {};
\node at (1,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\node at (-1,0) [var] (y) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw (x) -- (y);
\draw[delta]  (b) to (y) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (0, 1.2)  [var] (a)  {};
\node at (1,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\node at (-1,0) [var] (y) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw (x) -- (y);
\draw[bend left = 60, delta]  (b) to (x) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (0, 1.2)  [var] (a)  {};
\node at (1,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\node at (-1,0) [var] (y) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw (x) -- (y);
\draw[bend right = 60, delta]  (y) to (x) ;
\end{tikzpicture}
\]
For the first term,  the bound follows in the same way as \eqref{e:I(PsiXi1)Xi1-3rd}.
For the second term the bound follows in the same way as \eqref{e:I(PsiXi1)Xi1-1st}.
The fourth term follows in the same way as %\eqref{e:I(PsiXi1)Xi1-1st1}.
\tikz[scale=0.27,baseline=-7]
\node at (-.5, 0)  [var] (a)  {};
\node at (1,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw[bend left = 60, delta, thick]  (b) to (x) ;
} above.
These are all because $|\e^\kappa K(-\cdot)*K|\lesssim 1$.
For the third term,
by  \eqref{e:Diff-K} with
$\alpha=1$, one can deduce a bound on
$\big(\CK^{(\eps; 2)}\tau\big)$ by the right-hand side of \eqref{e:I(PsiXi1)Xi1-3rd}.
The fifth term is cancelled by our choice of  $C^\e [ \,\<PsiXi1>\, ]$
in \eqref{e:M-XiIXi} and \eqref{e:C-XiIXi}.
Turning to the 0th chaos, there are two terms
\[
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (0, 1.2)  [var] (a)  {};
\node at (1,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\node at (-1,0) [var] (y) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw (x) -- (y);
\draw[delta]  (a) to (y) ;
\draw[bend left = 60, delta]  (b) to (x) ;
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (0, 1.2)  [var] (a)  {};
\node at (0,0) [var1] (b) {};
\node at (0,-1.2) [var1] (x) {};
\node at (-1.2,0) [var] (y) {};
\draw (b) -- (a);
\draw[kernel1] (b) -- (x);
\draw (x) -- (y);
\draw[delta]  (b) to (y) ;
\draw[delta,bend left = 60]  (a) to (x) ;
\end{tikzpicture}
\]
Our choice of renormalisation \eqref{e:M-four} and \eqref{e:def-C-four}
precisely cancel these terms, up to remainders from the positively renormalised kernels,
which clearly satisfy the desired bounds.
The bounds for $\<PsiPsibarXibar>$ are similar and simpler.

For $\tau=\<I(R2-1)Xi1new>$, the desired bounds follow similarly as \eqref{e:mom-Y333} and \eqref{e:mom-Y11}.

Consider $\<cherry231>$ which includes the following trees (which appear in \eqref{e:tildeR2-exp}):
\minilab{e:cherry231-list}
\begin{equs}
(\CD_j^{\pm} \mcb{I} (\Psi_i \bar\Xi_i) )(\bar\CD_j \bar\Psi_j),
(\CD_j^{\pm} \bar\Psi_i)( \CD_j^+ \CS_{\bs}  \mcb{I} (\Psi_j \bar\Xi_j)),		\label{e:cherry231-list1}
\\
(\CD_j^+ \CS_{\southeast} \mcb{I}  (\Psi_j \bar\Xi_j)) (\CD_j^+  \CS_{\southwest}\bar\Psi_j),
(\CD_j^+  \CS_{\southeast} \bar\Psi_j) (\CD_j^+ \CS_{\southwest} \mcb{I}  (\Psi_j \bar\Xi_j)),	\label{e:cherry231-list2}
\end{equs}
where $\bs\in \{\Southwest,\Southeast\}$.
Recall that they are not renormalised.
Their 3rd chaoses can be all bounded as the {\it 1st} graph in \eqref{e:grouped-graph1} by $\big(\CK^{(\eps; 3)}\tau\big)\big(z_1, z_2\big)
\lesssim \|z_1- z_2\|_\e^{-4\kappa}$. \vspace{-5pt}
\begin{equ}[e:grouped-graph1]
\begin{tikzpicture}  [scale=0.4,baseline=0]
\node at (.6, 1)  [var] (a)  {};
\node at (0,0) [var1] (b) {};
\node at (1.2,0) [var1] (x) {};
\draw (b) -- (a);
\draw[very thick] (b) -- (0.6,-1);
\draw[very thick] (x) -- (0.6,-1);

\node at (4,0)  [var1] (c)  {};
\node at (4-.6,1) [var] (d) {};
\node at (4-1.2,0) [var1] (y) {};
\draw (c) -- (d);
\draw[very thick] (c) -- (4-.6,-1);
\draw[very thick] (y) -- (4-0.6,-1);

\draw[bend left =40, delta]  (b) to (c) ;
\draw[bend left =40, delta]  (x) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (.6, 1)  [var2] (a)  {};
\node at (-.6,1) [var2] (b) {};
\node at (1.2,0) [var1] (x) {};
\draw[very thick] (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[very thick] (0,0) -- (0.6,-1);
\draw[very thick] (x) -- (0.6,-1);

\node at (4.6,1)  [var2] (c)  {};
\node at (4-.6,1) [var2] (d) {};
\node at (4-1.2,0) [var1] (y) {};
\draw[very thick] (4,0) -- (d);
\draw[very thick] (4,0) -- (c);
\draw[very thick] (4,0) -- (4-.6,-1);
\draw[very thick] (y) -- (4-0.6,-1);

\draw[bend left =40, delta]  (b) to (c) ;
\draw[bend left =40, delta]  (x) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}  [scale=0.33,baseline=0]
\node at (.6, 1)  [var2] (a)  {};
\node at (-.6,1) [var2] (b) {};
\node at (-1.2,0) [var1] (x) {};
\draw[very thick] (0,0) -- (a);
\draw[very thick] (0,0) -- (b);
\draw[very thick] (0,0) -- (-0.6,-1);
\draw[very thick] (x) -- (-0.6,-1);

\node at (3.6,1)  [var2] (c)  {};
\node at (3-.6,1) [var2] (d) {};
\node at (3+1.2,0) [var1] (y) {};
\draw[very thick] (3,0) -- (d);
\draw[very thick] (3,0) -- (c);
\draw[very thick] (3,0) -- (3.6,-1);
\draw[very thick] (y) -- (3+0.6,-1);

\draw[bend right =50, delta]  (b) to (x) ;
\draw[bend left =50, delta]  (c) to (y) ;
\draw[bend left =40, delta]  (a) to (d) ;
\end{tikzpicture}
\end{equ}
The 1st chaos of each of these trees
has two terms,
one of which,
given by contracting $\Psi$ and $\bar\Psi$,
can be essentially bounded as in \eqref{e:mom-supercheche}.
We now turn to the other term in the 1st chaos,
obtained by contracting $\bar\Psi$  and $\bar\Xi$.
This term vanishes for \eqref{e:cherry231-list1}
by independence of $\xi_i$ and $\xi_j$.
For \eqref{e:cherry231-list2},
the same argument as for
$\<R2-1new>$ in the proof of Lemma~\ref{lem:mom-xi},
which uses
and the fact that $x^{\southeast}\neq x^{\southwest}$,
applies here,
so  we again have a bound of the form \eqref{e:mom-2424}.
The case $\tau=\<cherry232>$ is simpler.

Finally, for $\<I'Xi1I'(R2-1)new>$
we also have
the desired bounds
by observing the {\it 2nd and 3rd} graphs in \eqref{e:grouped-graph1}
and absorbing powers of $\e$ into kernels as before.
\end{proof}

\section{Stochastic heat equation}
\label{sec:SHE}

In this section we establish uniform in $$ estimates on the discrete stochastic heat equation (SHE) as a function of time with values in $Ω_N;α$ (Proposition \ref{prop:SHE}).
Our results can be seen as the discrete versions of~\cite[Sec.~4]{CCHS_2D},
but several statements and proofs below differ
%to those of~\cite{CCHS_2D}
and give sharper results (e.g. Lemmas~\ref{lem:SHE_gr}-\ref{lem:SHE_rho}).

%We fix $\e=2^{-N}$ throughout this section
%and let $\T^2_\e$ be a generic square lattice in $\T^2$ with spacing $\eps$.
Recall $=2^-N$ and  $^2_$ as in  Section~\ref{sec:funcSpaces}.
Equip $^2_$ with the natural measure $x = ^2 m(x)$ approximating the Lebesgue measure on $^2$, where $m$ is the counting measure on $^2_$.
Recall the discrete Laplace operator $Δ≡Δ_$ from~\eqref{e:laplacian} and let ${e^tΔ}_t≥0$ denote the associated heat semigroup.
By translation invariance, we can view $e^tΔ$ as a function
$e^tΔ^2_→[0,∞)$ which acts on $L^2(^2_)$ by convolution $e^tΔf(x) = ∫_^2_ e^tΔ(x-y)f(y) y$.

Let $ξ$ be the white noise over the Hilbert space $L^2(×^2_)$.
Consider the solution to the SHE
$Ψ≡Ψ_N _+×^2_→$
\begin{equ}\label{eq:disc_SHE}
\d_t \Psi = \Delta \Psi + \xi\;,
\end{equ}
with some initial condition $Ψ(0,·)^2_→$.

If $Ψ(0)=0$, then $Ψ(t,x) = ∫_0^ts∫_^2_ y e^(t-s)Δ(x-y)ξ(s,y)$, and therefore, by It\^o isometry and the semi-group property of $e^tΔ$,
for all $x∈^2_, t≥0$
\begin{equs}[eq:C_t_x]
C_t(x) &\eqdef
\E[\Psi(t,x)\Psi(t,0)] = \int_0^t \mrd s\int_{\T^2_\e} \mrd ye^{(t-s)\Delta}(x-y)e^{(t-s)\Delta}(y)
\\
&= \int_0^t\mrd s e^{2(t-s)\Delta}(x)
= \frac12\int_0^{2t} e^{u\Delta}(x)\mrd u\;.
\end{equs}

\begin{lemma}
Uniformly over $x\in\T^2_\e$ and $t\in [0,1]$
\begin{equ}\label{eq:hk_upper}
e^{t\Delta}(x) \lesssim \eps^{-2} \wedge t^{-1} \wedge |x|^{-2}\;,
\end{equ}
and uniformly over $y\in\T^2_\e$ with $|y|\geq |x|$
\begin{equ}\label{eq:hk_diff}
|e^{t\Delta}(x) - e^{t\Delta}(y)| \lesssim
(\e^{-3}\wedge t^{-3/2} \wedge |x|^{-3})|x-y|\;.
\end{equ}
\end{lemma}

\begin{proof}
This follow from~\cite[Lem.~B.1]{SSSX21}.
\end{proof}
Note that $^-2$ in~\eqref{eq:hk_upper} is only useful when $x=0$ and $t<^2$.
%Furthermore, by the trivial bound $|e^{t\Delta}(x) - e^{t\Delta}(y)| \leq |e^{t\Delta}(x)| + |e^{t\Delta}(y)|$,
%if $t^{1/2}\leq |x-y|$ we can replace $t^{-3/2}|x-y|$ by $t^{-1}$.
The identity~\eqref{eq:C_t_x} and bound~\eqref{eq:hk_upper} imply that, for all $x∈^2_,t≥0$,
\begin{equ}
%|C_t(0)|& \lesssim \int_0^{t} \e^{-2}\wedge s^{-1}\mrd s =
%t\e^{-2} & \text{ if } t < \e^2\\
%1+\log(t\e^{-2}) & \text{ if } t \geq \e^2\;,
\label{eq:C_x_bound}
|C_t(x)| \lesssim \int_0^{t} s^{-1}\wedge |x|^{-2} \mrd s =
\begin{cases}
t|x|^{-2} & \text{ if } t < |x|^2\;,\\
1+\log(t|x|^{-2}) & \text{ if } t \geq |x|^2\;,
\end{cases}
\end{equ}
and if $x=0$, one can replace $|x|$ on the right-hand side by $$.
Consider further now $|y|≥|x|$ with $|x-y|^2 ≤t$
for which \eqref{eq:hk_upper}-\eqref{eq:hk_upper} imply
\begin{equs}
&\lesssim \int_0^{|x-y|^2} s^{-1} \wedge |x|^{-2} \mrd s
+ \int_{|x-y|^2}^{t} (s^{-3/2}\wedge |x|^{-3})|x-y| \mrd s\;.
\end{equs}
For the case $|x| < |x-y|$, we obtain
\begin{equs}[eq:C_x_C_y_1]
1+\log(|x-y||x|^{-1}) + |x-y|(|x-y|^{-1}-t^{-1/2})
\\
&\lesssim 1+\log(|x-y||x|^{-1})\;.
\end{equs}
For the case $ |x-y|≤|x| < t^1/2$, $|C_t(x)-C_t(y)|$ is bounded by a multiple of
\begin{equ}{}
|x-y|^2|x|^{-2} + |x-y||x|^{-3}(|x|^2-|x-y|^2) + |x-y|(|x|^{-1}-t^{-1/2})\;.
\end{equ}
Finally, for the case $ |x-y|≤t^1/2≤|x|$, we obtain
\begin{equ}
|C_t(x)-C_t(y)| \lesssim |x-y|^2|x|^{-2} + |x-y||x|^{-3}(t-|x-y|^2)\;.
\end{equ}
Note that in the final two cases we obtain the bound
\begin{equ}\label{eq:crude_diff}
|C_t(x)-C_t(y)|\lesssim |x-y||x|^{-1}
\end{equ}
(which is sharp for the range $|x-y|≪|x|≪t^1/2$).
Finally, we remark that
\begin{equs}[eq:C_0_C_y]
\int_0^{\e^2\wedge t}\e^{-2}\mrd s
+ \int_{\e^2\wedge t}^{|y|^2\wedge t} s^{-1}\mrd s
+ \int_{|y|^2\wedge t}^t s^{-3/2}|y|\mrd s
\\
&= \begin{cases}
t\e^{-2} & \text{ if } t<\e^2\;,\\
1+\log ((t\wedge |y|^2)\e^{-2}) & \text{ if } \e^2 \leq t \leq |y|^2\;,\\
1+ \log (|y|^2\e^{-2}) + (1-|y|t^{-1/2}) & \text{ if } |y|^2 < t\;.
\end{cases}
\end{equs}

\begin{lemma}\label{lem:SHE_gr}
Suppose $\ell\subset \T^2_\e$ is a line containing $k$ points in $\T^2_\e$, i.e. $h\eqdef |\ell|=k \eps$.
Let $\delta_\ell\colon L^2(\T^2_\e)\to\R$ be the linear map $\delta_\ell(f) = \e\sum_{x\in\ell} f(x)$, which we also identify with a function in $L^2(\T^2_\e)$ by duality $\scal{\delta_\ell,f}=\delta_\ell(f)$.
Suppose $\Psi(0)=0$.
\begin{equs}
\E \scal{\delta_\ell,\Psi(t)}^2
\lesssim h\big(\big[h\log(th^{-2} + 1)\big] \wedge t^{1/2}\big)
\begin{cases}
h^2\log(th^{-2}+1) & \text{ if } h<\sqrt t\\
& \text{ if } h \geq \sqrt t\;.
\end{cases}
\end{equs}
\end{lemma}

\begin{proof}
Using~\eqref{eq:C_x_bound} and its sharpening for $x=0$,
\begin{equs}
\E  & \scal{\delta_\ell,\Psi(t)}^2
= \scal{\delta_\ell, C_t * \delta_\ell}
\e^{2}\sum_{x,y\in\ell} C_t(x-y)
\lesssim
h\e\sum_{j=0}^{k-1}(C_t(j\e e_1))\label{eq:d_ell_Psi_bound}
\\
&\lesssim h\e\big[t\e^{-2}\wedge (1+|\log(t\e^{-2})|)\big]
+h\int_0^{h}(tx^{-2} )\wedge (1+|\log(tx^{-2})|) \mrd x
\\
&\lesssim h\big[t\e^{-1}\wedge (\e+\e|\log(t\e^{-2})|)\big]
\begin{cases}
h^2\log(th^{-2}+1) & \text{ if } h<\sqrt t\\
& \text{ if } h \geq \sqrt t\;.
\end{cases}
\end{equs}
%The conclusion follows by remarking that,
We then note that
in the final line, the first term is controlled by the second.
\end{proof}

\begin{lemma}\label{lem:SHE_rho}
Suppose $\ell,\bar\ell\subset \T^2_\e$ are parallel lines each containing $k$ points in $\T^2_\e$ and distance $r$ apart, i.e. $h\eqdef |\ell|=k \eps$ and $\inf_{x\in\ell,y\in\bar\ell}|x-y|=r$.
Suppose $\Psi(0)=0$ and $r<h$.
\E \scal{\delta_\ell-\delta_{\bar\ell},\Psi(t)}^2
\lesssim
(ht^{1/2}) \wedge (hr(1+\log(h/r)))
%ht^{1/2} & \text{ if } t<r^2\\
%hr(1+\log(h/r)) & \text{ if } r^2 \leq t\;.
%h(t^{1/2} \wedge r)\;.
\end{lemma}

\begin{remark}\label{rem:correct_Thm_413}
We point out a typo in~\cite[Lem.~4.8,~4.12]{CCHS_2D} related to the above estimate.
First, $\kappa\in(0,1)$ therein should be $\kappa\in(0,\frac12)$,
and second, the second bound in~\cite[Lem.~4.8]{CCHS_2D} is incorrect.
Thus, in the proof of~\cite[Lem.~4.12]{CCHS_2D},
one can at best apply~\cite[Lem.~4.11]{CCHS_2D} with $M \sim C_\xi^{p/2} t^{p\kappa/2}$ and $\alpha=1-2\kappa$ with $\kappa\in (0,\frac12)$
(instead of $\alpha=1-\kappa$ as therein implicitly).
Since $\kappa = (1-\alpha)/2$,
this means that $t^{p(1-\alpha)/2}$ in the right-hand side of the bound in~\cite[Lem.~4.12]{CCHS_2D} should read $t^{p(1-\alpha)/4}$ for $\alpha\in (0,1)$.
We note, however, that this has no effect on the proof of~\cite[Thm.~4.13]{CCHS_2D} -- in fact,
the final term $|t-s|^{p(1-\beta)/2}$ now becomes $|t-s|^{p(1-\beta)/4}$ which is more natural
since it has the same homogeneity as the terms $|t-s|^{p(\alpha-\bar\alpha)/4}$ coming from the heat flow.
\end{remark}

%We expect the factor $(1+\log(h/r))$ can be dropped in Lemma~\ref{lem:SHE_rho} -- it arises from the term~\eqref{eq:log_factor} below that in turn follows from~\eqref{eq:crude_diff}. This final bound can likely be improved by sharpening~\eqref{eq:hk_diff}.

\begin{proof}
As in~\eqref{eq:d_ell_Psi_bound},
\E \scal{\delta_\ell-\delta_{\bar\ell},\Psi(t)}^2
%&= 2 \scal{\delta_\ell-\delta_{\bar\ell}, C_t * \delta_\ell}
%2\e^{2}\sum_{x,y\in\ell}\sum_{\bar y\in \bar \ell} C_t(x-y)-C_t(x-\bar y)
\lesssim
h\e\sum_{j=0}^{k-1}(C_t(j\e e_1) - C_t(j\e e_1 + re_2))$.
In all cases, in particular when $r^2>t$, by Lemma~\ref{lem:SHE_gr}
\begin{equ}
\E \scal{\delta_\ell-\delta_{\bar\ell},\Psi(t)}^2 \lesssim \E \scal{\delta_\ell,\Psi(t)}^2
\lesssim h\big(\big[h\log(th^{-2} + 1)\big] \wedge t^{1/2}\big) \leq ht^{1/2}\;.
\end{equ}
If $t\geq r^2$, then we gain something by using sharper bounds for $|C_t(x)-C_t(y)|$ above.
Considering the sum over $j\geq 1$, for $j^2\e^2 < r^2$ we have by~\eqref{eq:C_x_C_y_1}
\begin{equ}
|C_t(j\e e_1) - C_t(j\e e_1 + re_2)| \lesssim 1+\log(r|j\e|^{-1})\;,
\end{equ}
which contributes a multiple of
\int_0^{r} \{1+\log(rx^{-1})\} \mrd x = 2r
For $j^2\e^2 \geq r^2$, by~\eqref{eq:crude_diff}, we have
|C_t(j\e e_1) - C_t(j\e e_1 + re_2)| \lesssim
which contributes a multiple of
\int_r^{h} rx^{-1} \mrd x \asymp r \log (h/r)
In conclusion, the sum over $j\geq 1$ is bounded by a multiple of $hr(1+\log (h/r))$.
It remains to add the contribution from $j=0$ for $t\geq r^2$
for which, by~\eqref{eq:C_0_C_y},
\begin{equ}
h\e |C_t(0)-C_t(re_2)| \lesssim h\e [1+ \log (r^2\e^{-2}) + (1-rt^{-1/2})]
\lesssim hr\;.
\end{equ}
The conclusion follows since $hr(1+\log(h/r)) < ht^{1/2}$ only for $r^2<t$.
\end{proof}
%\begin{lemma}[Kolmogorov criterion]\label{lem:Kolmogorov_crit}
%Suppose $A$ is an $\Omega_N$-valued stochastic process
%such that there exists $\alpha\in(\frac12,1]$ and $M>0$ so that for all parallel $\ell,\bar\ell\in\olines_N$,
%\E |A(\ell)|^p \leq M |\ell|^{p\alpha}\;,\qquad \E |A(\ell)-A(\bar\ell)|^p \leq M \rho(\ell,\bar\ell)^{p\alpha}\;.
%Then for any $\bar\alpha\in (0,\alpha-\frac{16}{p})$ there exists $\lambda>0$, depending only on $p,\alpha,\bar\alpha$,
%such that
%\E |A|_{N;\bar\alpha}^p \leq \lambda M\;.
%Recall the function $\tilde \rho\colon \lines_N^2\to[0,\infty)$ from~\cite[Def.~3.3]{CCHS_2D} (denoted `$\rho$' therein).
%The given conditions readily imply that, for all $\ell,\bar\ell \in \olines_N$,
%\E |A(\ell) - A(\bar\ell)|^p \lesssim M\tilde\rho(\ell,\bar\ell)^{p\alpha}
%where the proportionality constant depends only on $p$.
%The proof now follows from~\cite[Lem.~4.11]{CCHS_2D}.
%Finally, consider a linear operator $K\colon \mfg^{\T^2_\e} \to \mfg^{\T^2_\e}$ that is translation invariant, i.e. $(Kf)(x+v) = [K(f\cdot+v)](x)$ for all $x,v\in\T^2_\e$.
%For $\alpha\in[0,1]$, define
%|K|_{\CC^\alpha\to L^\infty} = \sup\{|Kf|_{L^\infty_\e}\,:\, |f|_{\CC^\alpha_\e}=1\}
%where $|f|_{\CC^\alpha_\e}$ is the usual H\"older norm.
%We extend $K$ to $\hat K\colon \mfq_N\to\mfq_N$ by viewing $\mfq_N$ as two copies of $\mfg^{\T^2_\e}$.
%Then a proof identical to that of~\cite[Prop.~4.1]{CCHS_2D} implies that, for $0<\bar\alpha\leq\alpha\leq 1$,
%|\hat K|_{\Omega_{N;\alpha}\to \Omega_{N;\bar\alpha}} \lesssim |K|_{\CC^{(\alpha-\bar\alpha)/2}\to L^\infty}\;.

%Suppose $K\colon \mfg^{\T^2_\e} \to \mfg^{\T^2_\e}$ is translation invariant
%and let $0<\bar\alpha\leq\alpha\leq 1$.
%|\hat K|_{\Omega_{N;\alpha}\to \Omega_{N;\bar\alpha}} \lesssim |K|_{\CC^{(\alpha-\bar\alpha)/2}\to L^\infty}
%and, if $\bar\alpha\in[\frac\alpha2,\alpha]$, then for all $A\in\mfq_N$
%|\hat K A|_{N;\gr{\bar\alpha}} \lesssim |K|_{\CC^{\alpha-\bar\alpha}\to L^\infty}|A|_{N;\alpha;\rho}^{2(\alpha-\bar\alpha)/\alpha}|A|_{N;\gr\alpha}^{(2\bar\alpha-\alpha)/\alpha}\;,
%where the proportionality constants are universal.
%The proof is identical to that of~\cite[Prop.~4.1]{CCHS_2D}.

\begin{lemma}\label{lem:hk_flow}
Uniformly in $t\in (0,1)$, $0<\bar\alpha\leq \alpha\in[0,1]$, and $\e>0$, one has $
|A-e^{t\Delta}A|_{N;\bar\alpha}\lesssim t^{(\alpha-\bar\alpha)/4}|A|_{N;\alpha}$,
where we treat $e^{t\Delta}$ a linear operator $e^{t\Delta}\colon \mfq_N\to\mfq_N$ by viewing $\mfq_N$ as two copies of $\mfg^{\T^2_\e}$.
\end{lemma}

\begin{proof}
We first claim that $|f-e^{t\Delta}f|_{L^\infty} \lesssim t^{\alpha/2}|f|_{\CC^\alpha_\e}$.
\begin{equ}
|(e^{t\Delta}f)(x) - f(x)| = |\E[f(x+\e X_{N(t\e^{-2})}) - f(x)]| \leq |f|_{\CC^\alpha_\e}\E[|\e X_{N(t\e^{-2})}|^\alpha]\;,
\end{equ}
where $X_0,X_1\ldots,$ is a simple symmetric random walk on $\Z^2$ and $\{N(t)\}_{t\geq 0}$ is an independent Poisson process.
The final term is bounded by a multiple of $|f|_{\CC^\alpha_\e} t^{\alpha/2}$ since
$\E[|X_n|^\alpha]\lesssim n^{\alpha/2}$ and $\E N(t)^{\alpha/2} \lesssim t^{\alpha/2}$ uniformly in $n,t\geq 0,\alpha\in[0,1]$, which proves the claim.
The conclusion follows by an analogue of \cite[Prop.~4.1]{CCHS_2D} with $\Omega^1_\alpha$ replaced by $\Omega_{N,\alpha}$.
\end{proof}
\begin{proposition}\label{prop:SHE}
For $i\in\{1,2\}$,
consider a white noise $\xi_i$
on $\R\times\obonds_i$
and let $\Psi^{(i)}$ be the solutions to the SHE~\eqref{eq:disc_SHE}
with $\xi=\xi_i$ and $\T^2_\e=\obonds_i$ (as usual, we identify $e\in\obonds_i$ with its midpoint).
For $t\geq 0$, denote $\Psi(t)=(\Psi^{(1)}(t),\Psi^{(2)}(t))\in \Omega_N$.

Then for $0<\bar\alpha<\alpha\leq 1$, $\kappa\in (0,\frac{\alpha-\bar\alpha}{4})$, $p \geq 1$, and $T>0$, we have
\begin{equ}
\E\Big[\sup_{0\leq s<t\leq T} |t-s|^{-p\kappa}|\e\Psi(t)-\e\Psi(s)|^p_{N;\bar\alpha}\Big] \lesssim |\e\Psi(0)|_{\alpha} + 1\;,
\end{equ}
where the proportionality constant depends only on $\bar\alpha,\alpha,\kappa,p,T$.
\end{proposition}

\begin{proof}
Suppose first that $\Psi(0)=0$ and $\alpha\in (\frac12,1)$. Then, by equivalence of Gaussian moments, Lemmas~\ref{lem:SHE_gr} and~\ref{lem:SHE_rho} imply that, for all parallel $\ell,\bar\ell\in\olines_N$ and $p\geq 1$, one has
\E|\e\Psi(t)(\ell)|^p \lesssim |\ell|^{p\alpha}t^{p(1-\alpha)/2}
and for any $\beta>0$
\begin{equ}
\E|\e\Psi(t)(\ell)-\e\Psi(t)(\bar\ell)|^p \lesssim \rho(\ell,\bar\ell)^{p(\alpha-\beta)}|\ell|^{p\beta + p(1-\alpha)/2} t^{p(1-\alpha)/4}\;.
\end{equ}
%where the proportionality constants depend only on $\alpha,p,\beta$.
It thus follows from a discrete version of the Kolmogorov criterion of \cite[Lem.~4.11]{CCHS_2D} (with the same proof) that
\begin{equ}\label{eq:SHE_t_control}
\E|\e\Psi(t)|^p_{N;\bar\alpha} \lesssim t^{p(1-\alpha)/4}\;.
%\;, \qquad \forall \bar\alpha \in (0,\alpha-\tfrac{16}{p})\;.
\end{equ}
The proof now follows in the same way as the proof of~\cite[Thm.~4.13]{CCHS_2D} with Lemma \ref{lem:hk_flow} and~\eqref{eq:SHE_t_control} substituting the role of~\cite[Cor.~4.2 \& Lem.~4.12]{CCHS_2D} respectively
(see also Remark~\ref{rem:correct_Thm_413}).
\end{proof}

\section{Convergence of discrete dynamic to a continuum limit}
\label{sec:diagonal_argument}

We fix throughout this section a mollifier $$ (not necessarily non-anticipative).
We prove in this section that the discrete dynamic $A^$ converges along a subsequence of $↓0$ to $(C)$ for \textit{some} operator $C∈L(^2,^2)$.
We make the following assumption throughout this section.
\begin{assumption}\label{as:subsequence}
We fix a sequence $\e=2^{-N}\downarrow0$ such that $c^\e_\star$ from Lemma~\ref{lem:renormalised-equ} converges along this sequence.
We continue to write this sequence as $\e$.
\end{assumption}
\begin{theorem}\label{thm:some_C}
There exists a subsequence of $\e\downarrow0$ and $C\in L(\mfg^2,\mfg^2)$
such that the statement of Theorem~\ref{thm:discrete_dynamics} holds along this subsequence with $\bar C$ replaced by $C$.
Furthermore, $\sup_{N\geq 1} \E |T_N|^{-p} < \infty$ for all $p >0$.
\end{theorem}
In Section~\ref{sec:gauge-covar} we will show that $C=C̅$ regardless of the subsequence $$ from Assumption~\ref{as:subsequence}, thus proving Theorem~\ref{thm:discrete_dynamics}.
We prove Theorem~\ref{thm:some_C} by a diagonal argument.
This argument has been used in several recent works,
e.g. \cite[Sec.~6]{MourratWeber}, \cite[Sec.~6]{HS15},  \cite[Sec.~7]{HM18}, \cite[Sec.~6]{CM18}, \cite[Sec.~4]{EH21}, \cite[Sec.~4]{ChoukGairingPerkowski} (see also \cite[Sec.~5.1]{MR4029148}
for an alternative approach based on
convergence of discrete multiple stochastic integrals \cite{MR3584558}).
%\footnote{Another interesting approach is
%\cite{MR3584558} which provides a general criterion for the {\it convergence} of discrete multiple stochastic integrals to multiple Wiener-It\^o integrals, which was adapted later to the setting of Wick products in \cite[Sec.~5.1]{MR4029148}.}
In our case we would like to preserve the exact gauge symmetry along the discrete approximations,
and thus are {\it not} allowed to renormalise the discrete dynamics;
this is why we first show
Theorem~\ref{thm:some_C} and then identify $C$ in Section~\ref{sec:gauge-covar}.

\subsection{Regularised discrete process \texorpdfstring{$A^{\e,\bar\e}$}{A eps, bar eps}}