for each $1\leq j\leq q_{\mathrm{sk}}$, we consider the expression (9.25) of
$\mathcal{K}_{\mathcal{G}}$ (with extra restrictions), and the part of it
_involving the realization_ $\mathcal{G}_{\mathrm{sk}}[\mathbb{U}_{j}]$, which
is precisely defined as $\mathcal{K}_{(\mathcal{G})}^{\mathbb{U}_{j}}$ in
Section 9.2.1. We make three observations, which are easily verified using
relevant definitions and structure of the molecule
$\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$:

  * •

The expression $\mathcal{K}_{(\mathcal{G})}^{\mathbb{U}_{j}}$ is not affected
by LF twists at vines in $\mathbb{U}_{i}\,(i\neq j)$.

  * •

For any $(i,j)$, the variables

$(t_{\mathfrak{u}_{1}^{i}},t_{\mathfrak{u}_{21}^{i}},t_{\mathfrak{u}_{22}^{i}},k_{\mathfrak{u}_{1}^{i}},k_{\mathfrak{u}_{11}^{i}},k_{\mathfrak{u}_{21}^{i}},k_{\mathfrak{u}_{22}^{i}})$

which the function $\mathcal{G}_{\mathrm{sk}}[\mathbb{U}_{i}]$ depends on, do
not overlap with the variables $k_{\mathfrak{n}}$ for
$\mathfrak{n}\in\mathcal{U}_{j}$, nor the variables $t_{\mathfrak{n}}$ for
branching nodes $\mathfrak{n}\in\mathcal{U}_{j}$.

  * •

Similarly, the extra SG, LG and ZG restrictions defined above can also be
expressed using the variables that do not overlap with any $k_{\mathfrak{n}}$
nor $t_{\mathfrak{n}}$ for $\mathfrak{n}\in\mathcal{U}_{j}$.

Let $\mathcal{G}_{\mathrm{sb}}$ be the result of splicing all vines in
$\mathscr{V}_{0}$ from $\mathcal{G}_{\mathrm{sk}}$ as in Proposition 7.7; note
that $\mathcal{G}_{\mathrm{sb}}$ is a garden and is not affected by any LF
twist for $\mathcal{G}$ (see Remark 7.16 (c)). It has a pre-layering inherited
from $\mathcal{G}_{\mathrm{sk}}$ (i.e. when splicing each vine or VC as in
Proposition 7.7, all the nodes in the post-splicing garden, including
$(\mathfrak{u}_{1},\mathfrak{u}_{11},\mathfrak{u}_{21},\mathfrak{u}_{22})$ in
Proposition 7.7, are in the same layer as in the pre-splicing garden). By the
above observations, and note that for each $j$ we are also summing
$\mathcal{K}_{(\mathcal{G})}^{\mathbb{U}_{j}}$ for $\mathcal{G}$ being an LF
twist at $\mathbb{U}_{j}$, we can rewrite

$\mathscr{K}_{p+1,n}^{\mathrm{eqc}}(k_{1},\cdots,k_{2R})=\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{\mathrm{sb}}}\zeta(\mathcal{G}_{\mathrm{sb}})\sum_{\mathscr{I}_{\mathrm{sb}}}\widetilde{\epsilon}_{\mathscr{I}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{I}_{\mathrm{sb}}}\cdot\int_{\mathcal{I}_{\mathrm{sb}}}\prod_{\mathfrak{n}\in\mathcal{N}_{\mathrm{sb}}}e^{\pi
i\zeta_{\mathfrak{n}}\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}}\\\
\times\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sb}}}^{(+)}\mathcal{K}_{\mathcal{Q}_{(\mathfrak{l},\mathfrak{l}^{\prime})}}^{*}(t_{\mathfrak{l}^{\mathrm{pr}}},t_{(\mathfrak{l}^{\prime})^{\mathrm{pr}}},k_{\mathfrak{l}})\prod_{\mathfrak{m}\in\mathcal{N}_{\mathrm{sb}}}\mathcal{K}_{\mathcal{T}_{(\mathfrak{m})}}^{*}(t_{\mathfrak{m}^{\mathrm{pr}}},t_{\mathfrak{m}},k_{\mathfrak{m}})\prod_{j=1}^{q_{\mathrm{sk}}}\widetilde{\mathcal{K}}^{\mathbb{U}_{j}}(t_{\mathfrak{u}_{1}^{j}},t_{\mathfrak{u}_{21}^{j}},t_{\mathfrak{u}_{22}^{j}},k_{\mathfrak{u}_{1}^{j}},k_{\mathfrak{u}_{11}^{j}},k_{\mathfrak{u}_{21}^{j}},k_{\mathfrak{u}_{22}^{j}}).$
(10.1)

Here in (10.1), all the notions $n_{\mathrm{sb}}$, $\mathscr{I}_{\mathrm{sb}}$
and $\mathcal{I}_{\mathrm{sb}}$ etc. are just like in (9.25) but for the
garden $\mathcal{G}_{\mathrm{sb}}$, except that (i) the factor
$\widetilde{\epsilon}_{\mathscr{I}_{\mathrm{sb}}}$ contains the product of
factors on the right hand side of (3.22) _only for_
$\mathfrak{n}\not\in\\{\mathfrak{u}_{1}^{j}:1\leq j\leq q_{\mathrm{sk}}\\}$,
and (ii) the extra factors $\widetilde{\mathcal{K}}^{\mathbb{U}_{j}}$ are the
ones defined in (9.26) with $\mathbb{U}$ replaced by $\mathbb{U}_{j}$, and
$\mathcal{Z}_{\mathscr{I}_{\mathrm{sb}}}$ is the indicator function of all the
extra SG, LG and ZG restrictions defined above (this $\mathcal{Z}$ depends
only on $k[\mathcal{G}_{\mathrm{sb}}]$). Note that the factor $e^{-\pi
i\cdot\delta L^{2\gamma}t_{\mathfrak{u}_{1}}\Gamma}$ in (9.26) exactly cancels
the factor $e^{\pi i\zeta_{\mathfrak{n}}\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}$ in (10.1) for
$\mathfrak{n}=\mathfrak{u}_{1}^{j}$.

We perform some further reductions to (10.1). First, by the decay properties
of $\mathcal{K}_{\mathcal{Q}_{(\mathfrak{l},\mathfrak{l}^{\prime})}}^{*}$ in
Propositions 5.1 and 6.1, we may decompose (10.1) by restricting each
$k_{\mathfrak{l}}$ to a unit ball $|k_{\mathfrak{l}}-k_{\mathfrak{l}}^{0}|\leq
1$; we only need to treat a single term of this form, as summability is
provided by the above decay properties. By Lemma A.4, with a loss of at most
$C_{0}^{n}$, we may also restrict $k_{\mathfrak{n}}$ to a unit ball
$|k_{\mathfrak{n}}-k_{\mathfrak{n}}^{0}|\leq 1$ for each _node_
$\mathfrak{n}$.

Second, by the weighted Fourier $L^{1}$ bounds for
$\mathcal{K}_{\mathcal{Q}_{(\mathfrak{l},\mathfrak{l}^{\prime})}}^{*}$ and
$\mathcal{K}_{\mathcal{T}_{(\mathfrak{m})}}^{*}$ (see Propositions 5.1 and
6.1), and the same bounds for $\widetilde{\mathcal{K}}^{\mathbb{U}_{j}}$ with
each vector variable $k_{\mathfrak{u}_{1}^{j}}$ etc. restricted to a unit ball
(see Proposition 9.5), we can decompose these functions using time Fourier
transform, and reduce (10.1) to the following expression (with some fixed
parameters $\xi$ and $\xi_{\mathfrak{n}}$)

$\displaystyle\widetilde{\mathscr{K}}_{p+1,n}^{\mathrm{eqc}}$
$\displaystyle:=\zeta(\mathcal{G}_{\mathrm{sb}})\cdot e^{\pi
i\xi(p+1)}\cdot\big{(}\max_{\mathfrak{n}\in\mathcal{N}_{\mathrm{sb}}}(\langle\xi\rangle+\langle\xi_{\mathfrak{n}}\rangle)\big{)}^{-2\eta^{8}}\cdot(C_{1}\sqrt{\delta})^{n-n_{\mathrm{sb}}}\cdot
L^{-\eta^{8}g_{\mathrm{sk}}-\eta^{2}b_{\mathrm{sk}}+\eta^{7}q_{\mathrm{sk}}}\cdot\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}},$
$\displaystyle\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}$
$\displaystyle:=\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{\mathrm{sb}}}\sum_{\mathscr{I}_{\mathrm{sb}}}\widetilde{\epsilon}_{\mathscr{I}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{I}_{\mathrm{sb}}}\cdot\prod_{\mathfrak{m}}\mathcal{K}_{(\mathfrak{m})}(k_{\mathfrak{m}})\cdot\mathcal{K}^{\dagger}(k[\mathcal{W}])\int_{\mathcal{I}_{\mathrm{sb}}}\prod_{\mathfrak{n}\in\mathcal{N}_{\mathrm{sb}}}e^{\pi
i\zeta_{\mathfrak{n}}\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}\cdot e^{\pi
i\xi_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}}.$ (10.2)

Here $\mathcal{K}_{(\mathfrak{m})}$ is a function of $k_{\mathfrak{m}}$ for
each node $\mathfrak{m}$ (branching node or leaf), which is supported in the
unit ball $|k_{\mathfrak{m}}-k_{\mathfrak{m}}^{0}|\leq 1$, and has all
derivatives up to order $30d$ bounded by $1$. The function
$\mathcal{Z}_{\mathscr{I}_{\mathrm{sb}}}$ is the same as above, and
$\mathcal{K}^{\dagger}$ is a function of $k[\mathcal{W}]$ where
$\mathcal{W}=\\{\mathfrak{u}_{1}^{j},\mathfrak{u}_{11}^{j},\mathfrak{u}_{21}^{j},\mathfrak{u}_{22}^{j}:1\leq
j\leq q_{\mathrm{sk}}\\}$. This function $\mathcal{K}^{\dagger}$ is supported
in the unit balls $|k_{\mathfrak{u}_{1}^{j}}-k_{\mathfrak{u}_{1}^{j}}^{0}|\leq
1$ etc., and is bounded by $1$ (without derivatives). The quantity
$g_{\mathrm{sk}}$ represents the total incoherency index for all the regular
couples $\mathcal{Q}_{(\mathfrak{l},\mathfrak{l}^{\prime})}$ and regular trees
$\mathcal{T}_{(\mathfrak{m})}$ for
$\mathfrak{l},\mathfrak{l}^{\prime}\in\mathcal{L}_{\mathrm{sk}}$ and
$\mathfrak{m}\in\mathcal{N}_{\mathrm{sk}}$, plus the total incoherency index
for all the vines in $\mathscr{V}_{0}$. The quantity $b_{\mathrm{sk}}$
represents the number of _bad_ vines in $\mathscr{V}_{0}$, and
$q_{\mathrm{sk}}$ is the number of VC’s in $\mathscr{V}_{0}$ as above.

###  10.3. Analysis of ladders: $L^{1}$ bound and extra decay

For simplicity, we denote
$\mathbb{M}(\mathcal{G}_{\mathrm{sb}}):=\mathbb{M}_{\mathrm{sb}}$ below. To
estimate $\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}$ in (10.2), we need to deal
with the ladders in the molecule $\mathbb{M}_{\mathrm{sb}}$, and for this we
need to interpolate between two arguments, which will be presented in Section
11 below. Both arguments involve the process of representing (10.2) from the
point of view of the molecule $\mathbb{M}_{\mathrm{sb}}$, which we now
describe.

Recall that we identify $(k_{1},\cdots,k_{2R})$-decorations
$\mathscr{I}_{\mathrm{sb}}$ of $\mathcal{G}_{\mathrm{sb}}$ with
$(k_{1},\cdots,k_{2R})$-decorations $\mathscr{O}_{\mathrm{sb}}$ of
$\mathbb{M}_{\mathrm{sb}}$ via Definition 7.4. For
$\mathfrak{n}=\mathfrak{n}(v)$ the corresponding values of
$\zeta_{\mathfrak{n}}\Omega_{\mathfrak{n}}$ and $-\Gamma_{v}$ differ by a
constant $\Gamma_{v}^{0}$ which depends only on $(k_{1},\cdots,k_{2R})$, see
(7.2) and (7.4). Also by definition, the time domain
$\mathcal{I}_{\mathrm{sb}}$ as in (3.25) can be written as the set
$\mathcal{O}_{\mathrm{sb}}$ which contains the tuples $(t_{v})$ for atoms
$v\in\mathbb{M}_{\mathrm{sb}}$ satisfying that (i)
$t_{v}\in[\mathfrak{L}_{v},\mathfrak{L}_{v}+1]$ where $\mathfrak{L}_{v}$ is
the layer of $v$ as in Definition 7.2, and (ii) $t_{v_{1}}>t_{v_{2}}$ if there
is a PC bond between $v_{1}$ and $v_{2}$ with $v_{2}$ labeled C (i.e.
$\mathfrak{n}(v_{1})$ is the parent of $\mathfrak{n}(v_{2})$, see Definition
7.2).

Next, The parameters $\xi_{\mathfrak{n}}$ are renamed as $\xi_{v}$, the
functions $\mathcal{K}_{(\mathfrak{m})}(k_{\mathfrak{m}})$ can be written as
$\mathcal{K}_{(\ell)}(k_{\ell})$ for all bonds $\ell$ which satisfy the same
assumptions (such as support in $|k_{\ell}-k_{\ell}^{0}|\leq 1$ with fixed
values of $k_{\ell}^{0}$ etc.), and the function $\mathcal{K}^{\dagger}$ can
be written as a uniformly bounded function of the variables $k[\mathbb{W}]$
(which we still denote by $\mathcal{K}^{\dagger}$), where $\mathbb{W}$ is the
set of bonds $\ell\leftrightarrow v$ for all atoms
$v\in\mathbb{M}_{\mathrm{sb}}$ such that
$\mathfrak{n}(v)\in\\{\mathfrak{u}_{1}^{j}:1\leq j\leq q_{\mathrm{sk}}\\}$. We
may also define $\widetilde{\epsilon}_{\mathscr{O}_{\mathrm{sb}}}$ to be equal
to $\widetilde{\epsilon}_{\mathscr{I}_{\mathrm{sb}}}$ in (10.2); in the
molecule setting it can be written as a product of $\epsilon_{v}$ for all
atoms $v\in\mathbb{M}_{\mathrm{sb}}$ such that
$\mathfrak{n}(v)\not\in\\{\mathfrak{u}_{1}^{j}:1\leq j\leq
q_{\mathrm{sk}}\\}$, where each $\epsilon_{v}$ is a suitable function of
$(k_{\ell}:\ell\leftrightarrow v)$ valued in $\\{-1,0,1\\}$. Similarly we
define $\mathcal{Z}_{\mathscr{O}_{\mathrm{sb}}}$ to be equal to
$\mathcal{Z}_{\mathscr{I}_{\mathrm{sb}}}$ in (10.2), which can be written as a
product of $\mathcal{Z}_{v}$ for some of the atoms
$v\in\mathbb{M}_{\mathrm{sb}}$. Here each $\mathcal{Z}_{v}$ is a suitable
function of $(k_{\ell_{1}},k_{\ell_{2}})$ for some opposite-direction bonds
$\ell_{1},\ell_{2}\leftrightarrow v$, which has value in $\\{0,1\\}$ depending
on the SG, LG and ZG properties of $v$.

With all the above interpretations, we can rewrite (10.2) as

$\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}:=\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{\mathrm{sb}}}\sum_{\mathscr{O}_{\mathrm{sb}}}\widetilde{\epsilon}_{\mathscr{O}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{O}_{\mathrm{sb}}}\cdot\prod_{\ell}\mathcal{K}_{(\ell)}(k_{\ell})\cdot\mathcal{K}^{\dagger}(k[\mathbb{W}])\int_{\mathcal{O}_{\mathrm{sb}}}\prod_{v\in\mathbb{M}_{\mathrm{sb}}}e^{\pi
i\delta L^{2\gamma}(-\Gamma_{v}+\Gamma_{v}^{0})t_{v}}e^{\pi
i\xi_{v}t_{v}}\,\mathrm{d}t_{v}.$ (10.3)

Our first argument depends on an estimate which is in the same spirit as
Proposition 9.5 (but technically much easier as it does not involve Fourier
$L^{1}$ norms).

######  Proposition 10.4.

Let the molecule $\mathbb{M}_{\mathrm{sb}}$ be as in Section 10.2. Let
$\mathbb{L}$ be a ladder in $\mathbb{M}_{\mathrm{sb}}$, which has length $m$
and incoherency index $g$, such that
$\mathfrak{n}(v)\not\in\\{\mathfrak{u}_{1}^{j}:1\leq j\leq q_{\mathrm{sk}}\\}$
for any atom $v\in\mathbb{L}$. Let $v_{j}$ be the (at most $4$) atoms that are
not in $\mathbb{L}$ but connected to atoms in $\mathbb{L}$ by a bond, and let
$\ell_{j}$ be the (at most $4$) bonds that connect atoms in $\mathbb{L}$ to
atoms not in $\mathbb{L}$. Define the expression

$\mathscr{K}^{\mathbb{L}}=\mathscr{K}^{\mathbb{L}}(t_{v_{1}},t_{v_{2}},t_{v_{3}},t_{v_{4}},k_{\ell_{1}},k_{\ell_{2}},k_{\ell_{3}},k_{\ell_{4}})$

similar to (10.3), but with the following changes:

  * •

We replace the power $(\delta/(2L^{d-\gamma}))^{n_{\mathrm{sb}}}$ by
$(\delta/(2L^{d-\gamma}))^{2m+1}$ and remove the $\mathcal{K}^{\dagger}$
factor.

  * •

In the summation $\sum_{\mathscr{O}_{\mathrm{sb}}}(\cdots)$, we only sum over
the variables $k_{\ell}$ for $\ell\in\mathbb{L}$ (i.e. $\ell$ connecting two
atoms in $\mathbb{L}$), and treat the other $k_{\ell}$ variables as fixed. We
also replace $\widetilde{\epsilon}_{\mathscr{O}_{\mathrm{sb}}}$ and
$\mathcal{Z}_{\mathscr{O}_{\mathrm{sb}}}$ by the product of $\epsilon_{v}$ and
$\mathcal{Z}_{v}$ (defined above) over all $v\in\mathbb{L}$.

  * •

In the integration $\int_{\mathcal{O}_{\mathrm{sb}}}(\cdots)$, we only
integrate over the variables $t_{v}$ for $v\in\mathbb{L}$, and treat the other
$t_{v}$ variables as fixed.

  * •

In the product $\prod_{v\in\mathbb{M}_{\mathrm{sb}}}(\cdots)$, we only include
those factors where $v\in\mathbb{L}$; in the product $\prod_{\ell}$, we only
include those factors where $\ell\in\mathbb{L}$.

It is easy to check that the function $\mathscr{K}^{\mathbb{L}}$ defined this
way indeed only depends on the time variables $t_{v_{1}},\cdots,t_{v_{4}}$ and
vector variables $k_{\ell_{1}},\cdots,k_{\ell_{4}}$. Then, uniformly in all
$t_{v_{j}}$ and $k_{\ell_{j}}$ variables, we have

$|\mathscr{K}^{\mathbb{L}}|\lesssim_{1}(C_{1}\delta)^{m}L^{-(\gamma_{1}-\sqrt{\eta})g+20d}.$
(10.4)

###### Proof.

We shall only sketch the proof here, as the argument is a combination of the
simpler parts of the proofs of Propositions 9.4 and 9.5. First, note that the
gap associated with the ladder is fixed by the difference
$r:=k_{\ell_{1}}-k_{\ell_{2}}$. Next, we argue as in Step 2 of the proof of
Proposition 9.5 to re-parametrize the sum over the decoration as we did for
the ladder component of Vine (II), see Figure 18 (A). This leads to the new
set of variables $(x_{j},y_{j})_{1\leq j\leq m}$, where each $(x_{j},y_{j})$
corresponds to a double bond in the ladder connecting two atoms $v$ and $w$,
such that $\Gamma_{v}=2x_{j}\cdot y_{j}$ and $\Gamma_{w}=-2x_{j}\cdot
y_{j}+2r\cdot\zeta_{j}$. Here
$\zeta_{j}=\alpha_{j}x_{j}+\beta_{j}y_{j}+\theta_{j}r$ for some
$\alpha_{j},\beta_{j},\theta_{j}\in\\{0,\pm 1\\}$ with
$\alpha_{j}^{2}+\beta_{j}^{2}\neq 0$.

Recall that in Step 2 of the proof of Proposition 9.4 we estimated each
quantity $\mathcal{M}_{j}^{i}$ pointwise, leading to the final $L^{\infty}$
bound of $\mathcal{I}$. Here we can do exactly the same thing, and estimate
the sum in $(x_{j},y_{j})$ by either $\delta^{-1}L^{2(d-\gamma)}$ (if the
corresponding atoms $v$ and $w$ are in the same layer) or
$\delta^{-1}L^{2(d-\gamma)}L^{-(\gamma_{1}-\sqrt{\eta})}$ (if $v$ and $w$ are
in different layers). In the end we are left with the last remaining sum,
which is taken over the decoration of the last double bond in the ladder, and
can be bounded trivially by $L^{2d}$. Putting together, this then implies
(10.4). ∎

Our second argument depends on the following $L^{1}$ estimate for time
integral, which is basically the same as in [15, 18] but with some differences
due to layerings.

######  Proposition 10.5.

Let the molecule $\mathbb{M}_{\mathrm{sb}}$ be as in Section 10.2. Consider
all the maximal ladders in $\mathbb{M}_{\mathrm{sb}}$, which we denote by
$\mathbb{L}_{j}\,(1\leq j\leq q_{\mathrm{sb}})$ with length $z_{j}$. Let
$\rho_{\mathrm{sb}}:=q_{\mathrm{sb}}+m_{\mathrm{sb}}$, where $m_{\mathrm{sb}}$
is the number of atoms not in any of these maximal ladders. Note the
decoration $\mathscr{O}_{\mathrm{sb}}$ in
$\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}$ in (10.3) satisfies
$|k_{\ell}-k_{\ell}^{0}|\leq 1$ and
$\widetilde{\epsilon}_{\mathscr{O}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{O}_{\mathrm{sb}}}\neq
0$; we fix a dyadic number $P_{j}\in[L^{-1},1]\cup\\{0\\}$ for each $1\leq
j\leq q_{\mathrm{sb}}$, and further restrict $\mathscr{O}_{\mathrm{sb}}$ such
that the gap $r_{j}$ of the ladder $\mathbb{L}_{j}$ satisfies
$|r_{j}|\sim_{0}P_{j}$ (or $|r_{j}|\gtrsim_{0}1$ if $P_{j}=1$).

Consider the integral $\int_{\mathcal{O}_{\mathrm{sb}}}(\cdots)$ in (10.3),
which is a function of $\alpha_{v}:=\delta
L^{2\gamma}(-\Gamma_{v}+\Gamma_{v}^{0})+\xi_{v}$ where
$v\in\mathbb{M}_{\mathrm{sb}}$, and denote it by
$\mathbb{B}(\alpha[\mathbb{M}_{\mathrm{sb}}])$. Let
$\lfloor\alpha_{v}\rfloor:=\sigma_{v}$, and we restrict $\Gamma_{v}$ (and
hence $\alpha_{v}$ and $\sigma_{v}$) to those that _may occur_ for _some_
decoration $\mathscr{O}_{\mathrm{sb}}$ satisfying all the above restrictions,
including the further restrictions on gaps for $\mathbb{L}_{j}$. Then, under
these assumptions, we have

$\sum_{\sigma[\mathbb{M}_{\mathrm{sb}}]}\sup_{\alpha[\mathbb{M}_{\mathrm{sb}}]:|\alpha_{v}-\sigma_{v}|\leq
1}|\mathcal{B}(\alpha[\mathbb{M}_{\mathrm{sb}}])|\lesssim_{1}(C_{1}\delta^{-1/2-5\nu})^{n_{\mathrm{sb}}}\cdot
L^{\delta^{5\nu}\rho_{\mathrm{sb}}}\prod_{j=1}^{q_{\mathrm{sb}}}\mathfrak{X}_{j}^{z_{j}},$
(10.5)

where $\mathfrak{X}_{j}:=\min((\log L)^{2},1+\delta L^{2\gamma}P_{j})$.

###### Proof.

The proof is divided into several steps.

Step 1: Reduction and localization. First recall the following elementary
inequality (see also (10.5) of [15]): for any unit box $[0,1]^{n}$, any
function $F:[0,1]^{\mathbb{M}_{\mathrm{sb}}}\to\mathbb{R}$ and any
$\mathfrak{L}_{v}\in\mathbb{R}$ we have

$\sup_{\alpha\in[0,1]^{\mathbb{M}_{\mathrm{sb}}}}|F(\alpha)|\leq\sum_{A\subset\mathbb{M}_{\mathrm{sb}}}\int_{[0,1]^{\mathbb{M}_{\mathrm{sb}}}}\bigg{|}\prod_{v\in
A}(\partial_{\alpha_{v}}-\pi
i\mathfrak{L}_{v})F(\alpha)\bigg{|}\,\mathrm{d}\alpha$ (10.6)

For the proof, simply get rid of $\mathfrak{L}_{v}$ by multiplying by the
$e^{-\pi i\mathfrak{L}_{v}\alpha_{v}}$ factor, and then proceed by induction
in $n$.

Using (10.6), with an acceptable loss of $C_{0}^{n_{\mathrm{sb}}}$, we can
reduce the left hand side of (10.5) to the integral of
$|\mathcal{B}(\alpha[\mathbb{M}_{\mathrm{sb}}])|$ on the union of the unit
boxes $\lfloor\alpha_{v}\rfloor=\sigma_{v}$ for all possible $(\sigma_{v})$.
The derivatives $\partial_{\alpha_{v}}-\pi i\mathfrak{L}_{v}$ turn into
factors $\pi i(t_{v}-\mathfrak{L}_{v})$ in the
$\int_{\mathcal{O}_{\mathrm{sb}}}(\cdots)$ integral in (10.3), which are
harmless due to the restriction
$t_{v}\in[\mathfrak{L}_{v},\mathfrak{L}_{v}+1]$, so we will ignore them below.
Denote by $\mathcal{W}$ the above union of unit boxes, next we will specify
some localization properties of $\mathcal{W}$.

Suppose that $\alpha=(\alpha_{v})$ is defined from some decoration
$\mathscr{O}_{\mathrm{sb}}$ satisfying all the restrictions stated in the
assumptions, and fix one ladder $\mathbb{L}_{j}$ satisfying
$\mathfrak{X}_{j}=1+\delta L^{2\gamma}P_{j}\leq(\log L)^{2}$, and any atoms
$v$ and $w$ connected by a double bond, we have
$\Gamma_{v}+\Gamma_{w}=2r_{j}\cdot y_{w}$ where $r_{j}$ is the gap of
$\mathbb{L}_{j}$ and $y_{w}$ is the difference of the two vectors $k_{\ell}$
for two different single bonds at $v$ and $w$. Using that
$|k_{\ell}-k_{\ell}^{0}|\leq 1$, we see that
$\Gamma_{v}+\Gamma_{w}=2r_{j}\cdot z_{w}^{0}+\mathcal{O}_{0}(P_{j})$ where
$z_{w}^{0}$ is a fixed vector (note however that $r_{j}$ is not a fixed
vector), and thus $\alpha_{v}+\alpha_{w}=r_{j}\cdot
y_{w}^{0}+c_{w}+\mathcal{O}_{0}(\mathfrak{X}_{j})$ for some fixed vector
$y_{w}^{0}$ and constant $c_{w}$ that are independent of the decoration. The
integer part of this $\mathcal{O}_{0}(\mathfrak{X}_{j})$ has at most
$\mathcal{O}_{0}(\mathfrak{X}_{j})$ choices, so we may fix one choice for each
$j$ and each $(v,w)$, at an additional cost of

$C_{0}^{n_{\mathrm{sb}}}\cdot\prod_{j:\,\mathfrak{X}_{j}=1+\delta
L^{2\gamma}P_{j}}\mathfrak{X}_{j}^{z_{j}+1};$ (10.7)

this can be viewed as dividing $\mathcal{W}$ into subsets, where the total
number of subsets is given by (10.7). Once each integer part is fixed, we can
then absorb it into $c_{w}$ and write $\alpha_{v}+\alpha_{w}=r_{j}\cdot
y_{w}^{0}+c_{w}+\mathcal{O}_{0}(1)$. Note also that the $z_{j}+1$ in (10.7)
may be replaced by $z_{j}$, as the loss caused by this is at most $(\log
L)^{2q_{\mathrm{sb}}}$ which is acceptable.

Now, by Lemma A.8, for each $j$ we can find $q=q_{j}\leq d$ and atoms
$w_{1}^{j},\cdots,w_{q}^{j}\in\mathbb{L}_{j}$, such that for any
$w\in\mathbb{L}_{j}$ we can write

$y_{w}^{0}=\sum_{i=1}^{q}\lambda_{i}y_{w_{i}^{j}}^{0},\quad|\lambda_{i}|\leq
C_{0}.$

This implies that for each pair of atoms $(v,w)$ in $\mathbb{L}_{j}$ as above,
we have

$\alpha_{v}+\alpha_{w}=\sum_{i=1}^{q}\lambda_{i}\big{(}\alpha_{v_{i}^{j}}+\alpha_{w_{i}^{j}}\big{)}+d_{w}+\mathcal{O}_{0}(1),$
(10.8)

where $d_{w}$ is another constant that is independent of the decoration.
Clearly (10.8) does not change if each $\alpha_{v}$ is shifted by
$\mathcal{O}_{0}(1)$, so it is satisfied by any point in $\mathcal{W}$. In
addition, the difference of any two vectors in the decoration
$\mathscr{O}_{\mathrm{sb}}$ is a vector that belongs to $\mathbb{Z}_{L}^{d}$
and a fixed unit ball, so for each $v$ the quantity $\Gamma_{v}$ may take at
most $L^{20d}$ values for all possible decorations. This means that if
$\alpha=(\alpha_{v})\in\mathcal{W}$, then for each $v$ we must have
$\alpha_{v}\in\mathcal{S}_{v}$ where $\mathcal{S}_{v}$ is the union of at most
$L^{20d}$ unit intervals that depends only on $v$.

Step 2: Monotonicity. Fix a ladder $\mathbb{L}_{j}$ with
$\mathfrak{X}_{j}=1+\delta L^{2\gamma}P_{j}$, and let $w_{i}^{j}\,(1\leq i\leq
q=q_{j})$ be as above. Consider the sequence $\\{\max(t_{v},t_{w})\\}$ where
$(v,w)$ runs over all pairs of atoms in $\mathbb{L}_{j}$ connected by a double
bond. We claim that this sequence can be divided into two subsequences, each
of which is _monotnonic_ when viewed from left to right as in Figure 15.

In fact, fix any pair $(v,w)$, and assume (say) $t_{v}>t_{w}$. Note that
$\mathbb{M}_{\mathrm{sb}}=\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$; define
$v^{+}$ to be the unique atom such that $\mathfrak{n}(v^{+})$ is the parent
node of $\mathfrak{n}(v)$ in $\mathcal{G}_{\mathrm{sb}}$. Then $v^{+}\neq w$
and is adjacent to $v$, so it must be the atom to the left or right of $v$ in
Figure 15. Let $w^{+}$ be the atom connected to $v^{+}$ by a double bond, then
$\max(t_{v^{+}},t_{w^{+}})\geq t_{v^{+}}>t_{v}=\max(t_{v},t_{w})$. This means
that the sequence $\\{\max(t_{v},t_{w})\\}$ has _no local maximum_ , thus it
is formed by at most two monotonic subsequences when viewed from let to right
as in Figure 15.

Now, for each pair $(v,w)$ in $\mathbb{L}_{j}$ such that
$w\not\in\\{w_{i}^{j}:1\leq i\leq q=q_{j}\\}$, we may specify whether
$t_{v}>t_{w}$ or $t_{v}<t_{w}$ in addition to the existing inequality
relations between $(t_{v})$ forced by parent-child relations. With an
acceptable loss of $C_{0}^{n_{\mathrm{sb}}}$, we may fix such a set of
inequalities; then for each such $(v,w)$, we choose the atom corresponding to
the bigger time variable. Define the collection of these atoms by
$\mathbb{X}$, and let
$\mathbb{Y}=\mathbb{M}_{\mathrm{sb}}\backslash\mathbb{X}$. Note that
$\mathbb{Y}$ consists of (i) the atoms in $(v,w)$ corresponding to the smaller
time variable, (ii) the atoms $w_{i}^{j}$ and corresponding $v_{i}^{j}$ for
$1\leq i\leq q$, (iii) all atoms in ladders $\mathbb{L}_{j}$ where
$\mathfrak{X}_{j}=(\log L)^{2}$, and (iv) all atoms that are not in any ladder
$\mathbb{L}_{j}$.

Step 3: Partial time integration. Now we analyze the target expression, which
is

$\int_{\mathcal{W}}\bigg{|}\int_{\widetilde{\mathcal{O}}_{\mathrm{sb}}}\prod_{v\in\mathbb{X}\cup\mathbb{Y}}e^{\pi
i\alpha_{v}t_{v}}\,\mathrm{d}t_{v}\bigg{|}\,\mathrm{d}\alpha[\mathbb{X}]\mathrm{d}\alpha[\mathbb{Y}]$
(10.9)

thanks to the reduction in Step 1. Here
$\widetilde{\mathcal{O}}_{\mathrm{sb}}$ is the set $\mathcal{O}_{\mathrm{sb}}$
but with the extra inequalities in Step 2 added. In this integral, we will
first integrate in the $t[\mathbb{Y}]$ variables; clearly we may only
integrate the product for $v\in\mathbb{Y}$, and the result may depend on
$\alpha[\mathbb{Y}]$ and $t[\mathbb{X}]$, so we denote it by

$\mathcal{H}=\mathcal{H}(t[\mathbb{X}],\alpha[\mathbb{Y}])=\int_{(t[\mathbb{X}],t[\mathbb{Y}])\in\widetilde{\mathcal{O}}_{\mathrm{sb}}}\prod_{v\in\mathbb{Y}}e^{\pi
i\alpha_{v}t_{v}}\,\mathrm{d}t_{v}.$ (10.10)

Note that $\alpha_{v}\in\mathcal{S}_{v}$ for any
$\alpha[\mathbb{M}_{\mathrm{sb}}]\in\mathcal{W}$ as shown above, moreover we
have

$\int_{\alpha_{v}\in\mathcal{S}_{v}\,\\!(\forall
v\in\mathbb{Y})}|\mathcal{H}(t[\mathbb{X}],\alpha[\mathbb{Y}])|\,\mathrm{d}\alpha[\mathbb{Y}]\leq(C_{0}\log
L)^{|\mathbb{Y}|}$ (10.11)

uniformly in $t[\mathbb{X}]$. This can be proved by integrating in the
$t[\mathbb{Y}]$ variables in a certain order in (10.10) and proceeding by
induction, as in Lemma 10.2 of [15] or Proposition 2.3 of [14]. The only
caution is that each time we should integrate in the $t_{v}$ variable where
$v$ is a _minimal element_ under the ancestor-descendant partial ordering
(i.e. $v_{1}\succeq v_{2}$ if and only if $\mathfrak{n}(v_{1})$ is an ancestor
of $\mathfrak{n}(v_{2})$ in $\mathcal{G}_{\mathrm{sb}}$). Note that the lower
bound of this integration in $t_{v}$ is always a constant, and the upper bound
is either a constant or another variable $t_{v^{+}}$ where
$\mathfrak{n}(v^{+})$ is the parent of $\mathfrak{n}(v)$. After integrating in
$t_{v}$, we obtain a factor that is bounded by
$\langle\alpha_{v}+\beta\rangle^{-1}$, where $\beta$ is either $0$ or some
quantity determined by the $\alpha_{v_{*}}$ variables where $t_{v_{*}}$ has
been integrated out in previous steps (this bound is true even when
$|\alpha_{v}+\beta|\leq 1$ because the interval of integration in $t_{v}$ has
length at most $1$). This factor can be bounded upon integrating in
$\alpha_{v}$, using the elementary inequality

$\sup_{\beta\in\mathbb{R}}\int_{\mathcal{S}_{v}}\frac{1}{\langle\alpha_{v}+\beta\rangle}\,\mathrm{d}\alpha_{v}\leq
C_{0}\log L.$

We are then left with an expression that is the sum of at most four integrals
with similar forms (possibly depending on the size of $t_{v^{+}}$) but one
fewer integrated variable; next we simply integrate in $t_{v^{\prime}}$ where
$v^{\prime}$ is a minimal element in the same partial ordering after removing
$v$, and so on.

Step 4: Completing the proof. With (10.11) we can now complete the proof of
(10.5). First we perform the reduction in Step 1 and reduce to estimating
(10.9) with a loss of $\prod_{j=1}^{q_{\mathrm{sb}}}\mathfrak{X}_{j}^{z_{j}}$.
Then we apply (10.11) and also apply the fact from Step 2 that, all the
variables $t[\mathbb{X}\cap\mathbb{L}_{j}]$ belong to $[0,\mathfrak{D}]$ and
can be arranged into _at most two monotonic sequences_. Note also that, by
definition of $\mathbb{X}$ and $\mathbb{Y}$, once $\alpha[\mathbb{Y}]$ is
fixed, each $\alpha_{v}\,(v\in\mathbb{X})$ must belong to an interval of
length $\mathcal{O}_{0}(1)$ thanks to (10.8). Putting together, this leads to
the bound

$(\ref{targettimeint})\leq\int_{(\mathrm{monotonicity})}\,\mathrm{d}t[\mathbb{X}]\int_{\mathcal{W}}|\mathcal{H}(t[\mathbb{X}],\alpha[\mathbb{Y}])|\,\mathrm{d}\alpha[\mathbb{X}]\mathrm{d}\alpha[\mathbb{Y}]\leq
C_{0}^{n_{\mathrm{sb}}}(\log
L)^{|\mathbb{Y}|}\prod_{j:\,\mathfrak{X}_{j}=1+\delta
L^{2\gamma}P_{j}}\frac{\mathfrak{D}^{z_{j}}}{(z_{j}-d)!},$

where the integral in $t[\mathbb{X}]$ is taken under the above monotonicity
conditions. Note that

$(\log L)^{|\mathbb{Y}|}\leq\prod_{j:\,\mathfrak{X}_{j}=(\log
L)^{2}}\mathfrak{X}_{j}^{z_{j}}\cdot\prod_{j:\,\mathfrak{X}_{j}=1+\delta
L^{2\gamma}P_{j}}(\log L)^{z_{j}}\cdot(\log L)^{10d\rho_{\mathrm{sb}}}$

due to the definition of $\mathbb{Y}$; note also that in view of the
$(\delta^{-1/2-5\nu)^{n_{\mathrm{sb}}}}$ factor in (10.5), we have

$\delta^{(1/2+5\nu)n_{\mathrm{sb}}}\leq\prod_{j:\,\mathfrak{X}_{j}=1+\delta
L^{2\gamma}P_{j}}\delta^{(1+10\nu)z_{j}}.$

Therefore, the desired bound (10.5) follows from the elementary inequality

$\frac{(\log
L)^{z_{j}}\mathfrak{D}^{z_{j}}\delta^{(1+10\nu)z_{j}}}{(z_{j}-d)!}\leq(\log
L)^{2d}\exp\big{(}\mathfrak{D}\delta^{1+10\nu}\log L\big{)}\leq
L^{\delta^{8\nu}}$

for each $j$, where we recall that
$\mathfrak{D}\cdot\delta=\tau_{*}\ll\delta^{-\nu}$ from (3.18). ∎

###  10.4. Properties of $\mathbb{M}_{\mathrm{sb}}$

We state some properties of the molecule
$\mathbb{M}_{\mathrm{sb}}=\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$, which are
essentially the same as in [18].

######  Proposition 10.6.

Consider the garden $\mathcal{G}_{\mathrm{sb}}$ and the corresponding molecule
$\mathbb{M}_{\mathrm{sb}}=\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$. Consider
also a $(k_{1},\cdots,k_{2R})$-decoration $\mathscr{I}_{\mathrm{sb}}$ of
$\mathcal{G}_{\mathrm{sb}}$ such that
$\widetilde{\epsilon}_{\mathscr{I}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{I}_{\mathrm{sb}}}\neq
0$, which we identify with a $(k_{1},\cdots,k_{2R})$-decoration
$\mathscr{O}_{\mathrm{sb}}$ of $\mathbb{M}_{\mathrm{sb}}$ satisfying
$\widetilde{\epsilon}_{\mathscr{O}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{O}_{\mathrm{sb}}}\neq
0$ via Definition 7.4 (this can be viewed as inherited from certain decoration
of $\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$ or $\mathcal{G}_{\mathrm{sk}}$).
Then we have the followings:

  1. (1)

The molecule $\mathbb{M}_{\mathrm{sb}}$ is connected, has $n_{\mathrm{sb}}$
atoms and $2n_{\mathrm{sb}}-R$ bonds.

  2. (2)

$\mathbb{M}_{\mathrm{sb}}$ contains a collection $\mathscr{C}_{1}$ of disjoint
SG vine-like objects, such that each of them is either an HV, or a (CN) vine
or a root (CL) vine. We can also specify a total of $q_{\mathrm{sk}}$ atoms
which we call _hinge atoms_ ; any triple bond must have an endpoint that is a
hinge atom.

  3. (3)

We say an atom $v$ is _degenerate_ if there are two bonds
$(\ell_{1},\ell_{2})$ at $v$ of opposite directions and
$k_{\ell_{1}}=k_{\ell_{2}}$ (note this is equivalent to being ZG as in
Definition 10.3), and that it is _tame_ if the values of $k_{\ell}$ for all
$\ell\leftrightarrow v$ are equal, and equals a fixed value in
$(k_{1},\cdots,k_{2R})$ if $d(v)<4$. Then any degenerate/ZG atom that is not a
hinge atom must be tame.

  4. (4)

Any SG vine-like object which is not a subset of an SG vine-like object in
$\mathscr{C}_{1}$, and also _does not contain_ a hinge atom, must be a subset
of a DV or an LG or ZG vine-like object131313Note that a ZG vine-like object
means that the joints are degenerate, hence tame by (3). in
$\mathbb{M}_{\mathrm{sb}}$, which is disjoint with the vine-like objects in
$\mathscr{C}_{1}$ and does not contain any hinge atom. Note that here we must
be in one of the cases defined in Lemma A.5.

  5. (5)

For any hinge atom $v$, we can find two bonds $(\ell_{1},\ell_{2})$ at $v$,
such that the triple $(v,\ell_{1},\ell_{2})$ always has SG. If $v$ is also a
joint of an SG vine (or HV) in $\mathscr{C}_{1}$, then $(\ell_{1},\ell_{2})$
is chosen to belong to this vine (or the adjoint vine of this HV).

  6. (6)

Any SG vine-like object $\mathbb{U}$ which is not a subset of an SG vine-like
object in $\mathscr{C}_{1}$, and also _contains_ a hinge atom, _must_ contain
a hinge atom $v$, such that either $v$ is an interior atom of $\mathbb{U}$, or
$v$ is a joint of $\mathbb{U}$ and exactly one bond in $(\ell_{1},\ell_{2})$
belongs to $\mathbb{U}$, where $(\ell_{1},\ell_{2})$ is defined in (5) above.

###### Proof.

Recall the reduction process from $\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$ to
$\mathbb{M}_{\mathrm{sb}}$ by merging all VC’s in $\mathscr{V}_{0}$. Now each
VC in $\mathscr{V}_{0}$ is reduced to an atom, and we define them to be the
hinge atoms (in fact they are just those atoms $v$ where
$\mathfrak{n}(v)\in\\{\mathfrak{u}_{1}^{j}:1\leq j\leq q_{\mathrm{sk}}\\}$).
Moreover, by Corollary 7.10, we know that each SGVC, SGHV and SGHVC in
$\mathscr{C}$ is reduced to either a single atom, or an SGHV, or an SG vine
that is also a (CN) vine, or an SG vine that is also a root (CL) vine; we then
define $\mathscr{C}_{1}$ to be the collection of all such vine-like objects
(other than single atoms). For each hinge atom $v$, it is merged from an SGVC
in $\mathscr{C}$ (say $\mathbb{U}$), then we choose $(\ell_{1},\ell_{2})$ in
(5) such that in the molecule $\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$ they are
the two bonds at one joint of $\mathbb{U}$ that do not belong to $\mathbb{U}$
(this must exist because $\mathbb{U}$ is not a root VC). Note that if $d(v)=4$
then the other two bonds $(\ell_{3},\ell_{4})$ also have the same properties.
If $v$ is also a joint of an SG vine (or HV) in $\mathscr{C}_{1}$, then we
specify $(\ell_{1},\ell_{2})$ to belong to this vine (or the joint vine of
this HV); if not, we can choose freely between $(\ell_{1},\ell_{2})$ and
$(\ell_{3},\ell_{4})$.

The above definitions are the same as in Proposition 9.1 of [18] (with the
trivial difference that the DV’s are not included in the collection
$\mathscr{C}_{1}$ defined in (2), so the case of an SG vine-like object that
is a subset of a DV in $\mathbb{M}_{\mathrm{sb}}$ is now included in (4)).
This means that conclusions (2)–(6) also follow directly from the proof of
Proposition 9.1 of [18]. Note that here we are dealing with gardens, but the
proof in [18] does not distinguish between gardens and couples. As for
conclusion (1), it simply follows from Proposition 7.3 and the fact that
$\mathcal{G}_{\mathrm{sb}}$ (just as $\mathcal{G}_{\mathrm{sk}}$) is an
irreducible garden. ∎

##  11\. Stage 2 reduction and proof of Proposition 3.11

###  11.1. Reduction to counting estimates

To prove Proposition 3.11, we need to consider the expression
$\mathscr{K}_{p+1,n}^{\mathrm{tr}}$ defined in (3.46), and the difference term
$\mathscr{R}_{p+1,n}$ defined in (3.48). These involve four types of terms:

  1. (a)

The term

$\mathscr{K}_{p+1,n}^{\mathrm{tr}}(\zeta_{1},\cdots,\zeta_{2R},k_{1},\cdots,k_{2R})$
(11.1)

as in (3.46) for $R\geq 2$;

  2. (b)

The term

$\mathscr{K}_{p+1,n}^{\mathrm{tr,nr}}(+,-,k,k)$ (11.2)

as in (3.46) for $R=1$, where we only sum over _non-regular_ couples;

  3. (c)

The term

$\mathscr{K}_{p+1,n}^{\mathrm{tr,nc}}(+,-,k,k)$ (11.3)

as in (3.46) for $R=1$, where we only sum over _regular, non-coherent_ layered
couples;

  4. (d)

The term

$\mathscr{K}_{p+1,n}^{\mathrm{tr,co}}(+,-,k,k)-\mathcal{U}_{n/2}(p+1,k)$
(11.4)

as in (3.46) for $R=1$, where we only sum over the _regular, coherent_ layered
couples, and $n$ is even, $\mathcal{U}_{n/2}$ is as in (3.48).

The terms (11.3) and (11.4) will be estimated using Propositions 4.9, 5.1, 6.1
and 6.2. The terms (11.1) and (11.2) will be estimated in this section. In
both cases, note that the estimates (3.47) and (3.49) involves the weights
$(\langle k_{1}\rangle\cdots\langle k_{2R}\rangle)^{4\Lambda_{p+1}}$ for both
gardens and couples, and the derivatives $\partial_{k}^{\rho}\,(|\rho|\leq
4\Lambda_{p+1})$ for couples, which we now discuss.

First, for each $1\leq j\leq 2R$ we have

$\langle
k_{j}\rangle^{4\Lambda_{p+1}}\leq(3n_{j})^{4\Lambda_{p+1}}\cdot\langle
k_{\mathfrak{l}_{j}}\rangle^{4\Lambda_{p+1}}$

for some leaf $\mathfrak{l}_{j}\in\mathcal{T}_{j}$, where $n_{j}$ is the order
of the tree $\mathcal{T}_{j}$, since $k_{j}$ is an algebraic sum of all such
$k_{\mathfrak{l}_{j}}$. Since $4\Lambda_{p+1}\leq C_{2}$, we have
$(3n_{j})^{4\Lambda_{p+1}}\leq 2^{n_{j}}\cdot C_{2}$, and putting together all
$1\leq j\leq 2R$ yields the factors $2^{n}$ and $C_{2}^{R}$. Moreover, if we
include the extra factor $\langle
k_{\mathfrak{l}_{j}}\rangle^{4\Lambda_{p+1}}$ to the input function
$F_{\mathfrak{L}_{\mathfrak{l}_{j}}}(k_{\mathfrak{l}_{j}})$ in (3.24), then
this function is still bounded in $\mathfrak{S}^{40d,40d}$ because of
$F_{q}(k)=\varphi(\delta q,k)+\mathscr{R}_{q}(k)$, the bound (3.29) and
$\Lambda_{p}=40\Lambda_{p+1}$, but with norm being $C_{2}$ instead of $C_{1}$
for the $\varphi(\delta q,k)$ term. However this happens for at most $R$ input
factors, so again the resulting loss is at most $C_{2}^{R}$. These losses can
be covered by proving slightly stronger bounds with $L^{-(R-1)/4}$ replaced by
$L^{-(R-1)(1/4+1/100)}$ in (3.47), and $L^{-(4\theta_{p+1}+\theta_{p})/5}$
replaced by $L^{-(3\theta_{p+1}+2\theta_{p})/5}$ in (3.49), and $\delta^{2\nu
n}$ replaced by $\delta^{3\nu n}$ in both, but these stronger bounds will
follow from same arguments below.

Similarly, note that for couples ($R=1$), the function
$\mathscr{K}_{p+1,n}^{\mathrm{tr}}$ is defined for all $k\in\mathbb{R}^{d}$ if
we extend the input functions $F_{q}(k)=\varphi(\delta
q,k)+\mathscr{R}_{q}(k)$ for $q\leq p$ by using (3.29). If we translate $k$
and all $k_{\mathfrak{l}}$ by the same vector the whole expression is
invariant, so taking any $\partial_{k}$ derivative is equivalent to taking
exactly one $\partial_{k_{\mathfrak{l}}}$ derivative for some leaf
$\mathfrak{l}\in\mathcal{G}$. If we take at most $4\Lambda_{p+1}\leq C_{2}$
derivatives, then we get at most $n^{C_{2}}$ terms, where at most $C_{2}$
input factors are affected for each term, and each affected factor is still
bounded in $\mathfrak{S}^{40d,40d}$ but by $C_{2}$ instead of $C_{1}$, for the
same reason as above. This again leads to loss of at most $C_{2}n^{C_{2}}\leq
C_{2}2^{n}$, which is acceptable provided we prove the slightly stronger
bounds above.

Therefore, from now on we will focus on the proof of (3.47) and (3.49) with
$4\Lambda_{p+1}$ replaced by $0$, i.e. the supremum norm in $k$ (or $k_{j}$).
In the proof below we will fix the values of $k$ (or $k_{j}$), and all our
estimates will be uniform in them; the input factors $F_{q}(k)=\varphi(\delta
q,k)+\mathscr{R}_{q}(k)$ are then bounded in $\mathfrak{S}^{40d,40d}$ by
$C_{1}$.

With the above reductions, we now start with the term
$\mathscr{K}_{p+1,n}^{\mathrm{tr}}$ in (11.1) or
$\mathscr{K}_{p+1,n}^{\mathrm{tr,nr}}$ in (11.2), which is a sum of
$\mathcal{K}_{\mathcal{G}}$ as in (3.46). We then perform the stage 1
reduction in Section 10 and consider the sum over a single congruence class as
defined in Definition 10.2. Note that we still need to count the number of
congruence classes, which will be left to Section 11.3 below; for now we will
fix this single sum and reduce it to (10.3) as done in Section 10.

####  11.1.1. Ladders and $L^{1}$ bounds

Now start from the expression $\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}$ in
(10.3). Recall the various parameters defined in Proposition 10.5:
$n_{\mathrm{sb}}$ is the order of $\mathcal{G}_{\mathrm{sb}}$,
$q_{\mathrm{sb}}$ is the number of maximal ladders in
$\mathbb{M}_{\mathrm{sb}}=\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$,
$z_{j}\,(1\leq j\leq q_{\mathrm{sb}})$ are the lengths of these ladders,
$m_{\mathrm{sb}}$ is the number of atoms not in these ladders, and
$\rho_{\mathrm{sb}}=q_{\mathrm{sb}}+m_{\mathrm{sb}}$. Also recall the
parameters coming from the stage 1 reduction process in Sections 10.1–10.2:
$g_{\mathrm{sk}}$ is the total incoherency index for all the regular couples
$\mathcal{Q}_{(\mathfrak{l},\mathfrak{l}^{\prime})}$ and regular trees
$\mathcal{T}_{(\mathfrak{m})}$ for
$\mathfrak{l},\mathfrak{l}^{\prime}\in\mathcal{L}_{\mathrm{sk}}$ and
$\mathfrak{m}\in\mathcal{N}_{\mathrm{sk}}$, plus the total incoherency index
for all the vines in $\mathscr{V}_{0}$, $b_{\mathrm{sk}}$ represents the
number of bad vines in $\mathscr{V}_{0}$, and $q_{\mathrm{sk}}$ is the number
of VC’s in $\mathscr{V}_{0}$. Note that $q_{\mathrm{sk}}$ also equals the
number of hinge atoms in $\mathbb{M}_{\mathrm{sb}}$ (Proposition 10.6 (2)). In
addition, we also define $g_{\mathrm{sb}}$ to be the total incoherency index
for all the maximal ladders in Proposition 10.5.

The goal until the end of Section 11.3 is to prove the following two
estimates:

$|\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}|\lesssim_{2}(C_{1}\sqrt{\delta})^{n_{\mathrm{sb}}}L^{-\eta\cdot
g_{\mathrm{sb}}+80d\cdot(\rho_{\mathrm{sb}}+q_{\mathrm{sk}})},$ (11.5)
$|\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}|\lesssim_{1}(C_{1}\delta^{-10\nu})^{n_{\mathrm{sb}}}L^{-\eta^{6}\cdot(\rho_{\mathrm{sb}}+q_{\mathrm{sk}})}\cdot
L^{-(R-1)(1/4+1/50)},$ (11.6)

which we then interpolate to prove Proposition 3.11. The proof of (11.5) is
very easy with Proposition 10.4, which we describe below.

Starting with (10.3), we choose all the maximal ladders in Proposition 10.5;
if any of them contains hinge atoms, then we replace it by the shorter ladders
separated by these hinge atoms. This leads to a new set of $q^{\prime}\leq
q_{\mathrm{sb}}+q_{\mathrm{sk}}$ maximal ladders not containing hinge atom,
such that the total incoherency index equals $g^{\prime}\geq
g_{\mathrm{sb}}-2q_{\mathrm{sk}}$. For each new ladder $\mathbb{L}$, we apply
Proposition 10.4, noting that the function
$\mathcal{K}^{\dagger}(k[\mathbb{W}])$ does not depend on any of the
$k_{\ell}$ variables for $\ell\in\mathbb{L}$; the resulting functions
$\mathscr{K}^{\mathbb{L}}$ is then estimated using (10.4). The rest of (10.3),
namely summation and integration over the $k_{\ell}$ and $t_{v}$ variables for
$\ell$ and $v$ not in any ladder, is estimated trivially, using the fact that
each $k_{\ell}$ belongs to a fixed unit ball and that each $t_{v}$ belongs to
a fixed unit interval. Note that the number of atoms not in these new ladders
is $m^{\prime}\leq m_{\mathrm{sb}}+2q_{\mathrm{sk}}$, so we obtain that

$|\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}|\leq(C_{1}\sqrt{\delta})^{n_{\mathrm{sb}}-m^{\prime}}\delta^{m^{\prime}}L^{-(\gamma_{1}-\sqrt{\delta})g^{\prime}+20dq^{\prime}}\cdot
L^{20dm^{\prime}},$ (11.7)

which easily implies (11.5) due to the upper bounds for
$(q^{\prime},m^{\prime})$ and lower bound for $g^{\prime}$.

From now on we will focus on the proof of (11.6). Starting with (10.3), which
we may rewrite as

$\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}=\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{\mathrm{sb}}}\sum_{\mathscr{O}_{\mathrm{sb}}}\widetilde{\epsilon}_{\mathscr{O}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{O}_{\mathrm{sb}}}\cdot\prod_{\ell}\mathcal{K}_{(\ell)}(k_{\ell})\cdot\mathcal{K}^{\dagger}(k[\mathbb{W}])\cdot\mathcal{B}(\alpha[\mathbb{M}_{\mathrm{sb}}])$
(11.8)

using the definition in Proposition 10.5, with the notions $\alpha_{v}:=\delta
L^{2\gamma}(-\Gamma_{v}+\Gamma_{v}^{0})+\xi_{v}$ etc. Then we may fix the
values of $P_{j}\,(1\leq j\leq q_{\mathrm{sb}})$ as in Proposition 10.5; this
loses a factor $(\log L)^{q_{\mathrm{sb}}}$, which can be covered in view of
the gain in (11.6) (say if we prove the stronger bound with $\eta^{6}$
replaced by $2\eta^{6}$, which follows from same arguments).

Next, we shall fix the values of $\sigma_{v}=\lfloor\alpha_{v}\rfloor$, and
bound (11.8) by

$|\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}|\leq\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{\mathrm{sb}}}\cdot\sum_{\sigma[\mathbb{M}_{\mathrm{sb}}]}\sup_{\alpha[\mathbb{M}_{\mathrm{sb}}]:|\alpha_{v}-\sigma_{v}|\leq
1}|\mathcal{B}(\alpha[\mathbb{M}_{\mathrm{sb}}])|\\\
\times\sup_{\sigma[\mathbb{M}_{\mathrm{sb}}]}\sum_{\mathscr{O}_{\mathrm{sb}}:|\alpha_{v}-\sigma_{v}|\leq
1}\big{|}\widetilde{\epsilon}_{\mathscr{O}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{O}_{\mathrm{sb}}}\cdot\prod_{\ell}\mathcal{K}_{(\ell)}(k_{\ell})\cdot\mathcal{K}^{\dagger}(k[\mathbb{W}])\big{|}.$
(11.9)

Note that the $\sum_{\sigma[\mathbb{M}_{\mathrm{sb}}]}(\cdots)$ factor in
(11.9) is bounded by (10.5), so to prove (11.6) we only need to bound the
$\sup_{\sigma[\mathbb{M}_{\mathrm{sb}}]}(\cdots)$ factor uniformly in
$(\sigma_{v})$. This corresponds to solving a counting problem for all the
$(k_{1},\cdots,k_{2R})$-decorations $\mathscr{O}_{\mathrm{sb}}$ of the
molecule $\mathbb{M}_{\mathrm{sb}}$. Moreover, the decoration
$\mathscr{O}_{\mathrm{sb}}$ must satisfy the following requirements:

  1. (a)

It is restricted by some fixed $(k_{\ell}^{0})$ and $(\beta_{v})$ in the sense
of Definition 7.4, where $(k_{\ell}^{0})$ and $(\beta_{v})$ are fixed
parameters (the $(\beta_{v})$ is determined by
$(\sigma_{v},\Gamma_{v}^{0},\xi_{v})$ etc.).

  2. (b)

It is in the support of
$\widetilde{\epsilon}_{\mathscr{O}_{\mathrm{sb}}}\cdot\mathcal{Z}_{\mathscr{O}_{\mathrm{sb}}}$,
in particular it must satisfy all the requirements in Proposition 10.6.

  3. (c)

The gap $r_{j}$ or the ladder $\mathbb{L}_{j}$ is such that
$|r_{j}|\sim_{0}P_{j}$.

For later use, we shall impose some further requirements on this decoration
$\mathscr{O}_{\mathrm{sb}}$, which we describe below. Consider all the SG
vine-like objects in $\mathbb{M}_{\mathrm{sb}}$ that are subsets of DV or LG
or ZG vine-like objects (Proposition 10.6 (4), note that here we must be in
one of the cases defined in Lemma A.5), and select the ones that are maximal
(i.e. those that are not subsets of larger SG vine-like objects with same
properties). Here for any DV, if both vines (V) forming this DV are SG, then
we select only one of them, plus all the maximal SG vine-like proper subsets
of the other. Define by $\mathscr{C}_{2}$ the collection of SG vine-like
objects formed in this way. Then:

  1. (4)

We fix each atom in $\mathbb{M}_{\mathrm{sb}}$ as either SG, LG or ZG; here we
say an atom $v$ is SG if there is at least one choice of $(\ell_{1},\ell_{2})$
such that $(v,\ell_{1},\ell_{2})$ is SG. Note that an atom may be SG in more
than one way, in which case we will further specify it below.

  2. (5)

For each SG atom $v$, we specify the two bonds $(\ell_{1},\ell_{2})$ at $v$
such that $(v,\ell_{1},\ell_{2})$ is SG. Here we require that (i) for any
hinge atom $v$ we must specify $(\ell_{1},\ell_{2})$ as in Proposition 10.6
(5), and (ii) for any joint $v$ of any SG vine-like object in
$\mathscr{C}_{1}$ or $\mathscr{C}_{2}$, we must specify $(\ell_{1},\ell_{2})$
to belong to this vine-like object141414This is consistent with (i) due to
Proposition 10.6 (5). Also, if this vine-like object is an HV or HVC, then
$(\ell_{1},\ell_{2})$ is chosen to belong to the corresponding adjoint VC.;
for any other atom $v$, we can specify $(\ell_{1},\ell_{2})$ arbitrarily.

  3. (6)

Finally, for each SG atom $v$ which is not an interior atom of an SG vine-like
object in $\mathscr{C}_{1}$ or $\mathscr{C}_{2}$, we fix its gap as
$|r|\sim_{0}Q_{v}$, where $Q_{v}\in[L^{-1},L^{-\gamma+\eta}]$ is a dyadic
number.

Note that putting all the extra requirements in (d)–(f) leads to a loss of at
most $C_{0}^{n_{\mathrm{sb}}}(\log L)^{p_{\mathrm{sb}}}$, where
$p_{\mathrm{sb}}$ is the total number of SG atoms satisfying the condition in
(f).

Now, define $\mathfrak{E}_{\mathrm{sb}}$ to be the number of
$(k_{1},\cdots,k_{2R})$-decorations $\mathscr{O}_{\mathrm{sb}}$ of
$\mathbb{M}_{\mathrm{sb}}$ that satisfy all the requirements (a)–(f) defined
above, and define

$\mathfrak{A}_{\mathrm{sb}}:=\mathfrak{E}_{\mathrm{sb}}\cdot
L^{-(d-\gamma)\chi(\mathbb{M}_{\mathrm{sb}})}\cdot(C_{1}^{-1}\delta^{1/2})^{\chi(\mathbb{M}_{\mathrm{sb}})}\prod_{j=1}^{q_{\mathrm{sb}}}\mathfrak{X}_{j}^{z_{j}},$
(11.10)

where we recall the circular rank $\chi$ defined in Proposition 7.3. Then,
using (11.9) and (10.5), and that
$\chi(\mathbb{M}_{\mathrm{sb}})=n_{\mathrm{sb}}-R+1$, we know that (11.6)
would follow if we can prove that

$\mathfrak{A}_{\mathrm{sb}}\leq
L^{-4\eta^{6}(\rho_{\mathrm{sb}}+p_{\mathrm{sb}})}\cdot
L^{(R-1)(d-\gamma-1/4-1/40)}.$ (11.11)

Here note that each hinge atom is SG by Proposition 10.6 (5), and cannot be an
interior atom of any SG vine-like object in $\mathscr{C}_{1}$ or
$\mathscr{C}_{2}$, so we always have $q_{\mathrm{sk}}\leq p_{\mathrm{sb}}$.

###  11.2. Reduction to large gap molecules

Our goal now is to prove (11.11). To bound the number of decorations
$\mathfrak{E}_{\mathrm{sb}}$ in the counting problem, we will reduce
$\mathbb{M}_{\mathrm{sb}}$ by performing suitable _cutting_ operations
introduced in [18], at all the SG vine-like objects and SG atoms. The proof is
essentially the same as in Sections 9–10 of [18], with only minor differences
due to the fact that here we are dealing with general gardens instead of
couples. We start by recalling the definition of cuts.

######  Definition 11.1 (The cutting operation [18]).

Given a molecule $\mathbb{M}$ and an atom $v$. Suppose $v$ has two bonds
$\ell_{1}$ and $\ell_{2}$ of opposite directions, then we may _cut_ the atom
$v$ along the bonds $\ell_{1}$ and $\ell_{2}$, by replacing $v$ with two atoms
$v_{1}$ and $v_{2}$, such that the endpoint $v$ for the bonds $\ell_{1}$ and
$\ell_{2}$ is moved to $v_{1}$, and the endpoint $v$ for the other bond(s) is
moved to $v_{2}$, see Figure 19. We say this cut is an $\alpha$\- (resp.
$\beta$-) cut, if it does not (resp. does) generate a new connected component,
and accordingly we say the resulting atoms $v_{1}$ and $v_{2}$ are $\alpha$\-
or $\beta$\- atoms. If we are also given a decoration $(k_{\ell})$ of
$\mathbb{M}$, then we may define the gap of this cut to be
$r:=k_{\ell_{1}}-k_{\ell_{2}}$.

Figure 19. A cutting operation executed at a degree $4$ atom $v$, see
Definition 11.1.

######  Remark 11.2.

Compared to [18], here we also allow to perform cutting at a degree $2$ atom
$v$. This case is in fact trivial: after the cut $v_{2}$ becomes an isolated
atom (without any bond) which forms a single connected component by itself.

With Definition 11.1, we shall now reduce $\mathbb{M}_{\mathrm{sb}}$ by
performing various cutting operations, and possibly removing some connected
components created in this process, until reaching a final molecule
$\mathbb{M}_{\mathrm{fin}}$. At each step, let the molecule before and after
the operation be $\mathbb{M}_{\mathrm{pre}}$ and $\mathbb{M}_{\mathrm{pos}}$,
then any decoration $(k_{\ell})$ of $\mathbb{M}_{\mathrm{pre}}$ can be
naturally inherited to obtain a decoration of $\mathbb{M}_{\mathrm{pos}}$.

For any molecule $\mathbb{M}$ in this process (which could be
$\mathbb{M}_{\mathrm{pre}}$ or $\mathbb{M}_{\mathrm{pos}}$), consider the
possible $(c_{v})$-decorations of $\mathbb{M}$, also restricted by some
$(\beta_{v})$ and $(k_{\ell}^{0})$; we also assume that this decoration is
inherited from a $(k_{1},\cdots,k_{2R})$-decoration of
$\mathbb{M}_{\mathrm{sb}}$ that satisfies all assumptions (a)–(f) in Section
11.1.1. Then we consider the number of such restricted decorations, take
supremum over the parameters $(c_{v},\beta_{v},k_{\ell}^{0})$, and define it
to be $\mathfrak{E}$. Similar to (11.10) we also define

$\mathfrak{A}:=\mathfrak{E}\cdot
L^{-(d-\gamma)\chi(\mathbb{M})}(C_{1}^{-1}\delta^{1/2})^{\chi(\mathbb{M})}\prod_{j=1}^{q}\mathfrak{X}_{j}^{z_{j}},$
(11.12)

where $\chi(\mathbb{M})$ is the circular rank of $\mathbb{M}$, and
$z_{j}\,(1\leq j\leq q)$ are the lengths of the maximal ladders
$\mathbb{L}_{j}$ in $\mathbb{M}$, and $\mathfrak{X}_{j}$ are defined from
$P_{j}$ as in Proposition 10.5, while $P_{j}$ are fixed such that the gap
$r_{j}$ of $\mathbb{L}_{j}$ is such that $|r_{j}|\sim_{0}P_{j}$. Define also
$m$ to be the number of atoms not in the maximal ladders, and $\rho=q+m$. The
notations $(\mathfrak{A},\mathfrak{E})$ and $(\rho,q,m)$ will apply to the
molecules appearing below, with possible subscripts matching those of
$\mathbb{M}$ (so $\mathfrak{A}_{\mathrm{pre}}$ is defined for the molecule
$\mathbb{M}_{\mathrm{pre}}$ etc.)

We will prove, for each operation, an inequality of form

$\mathfrak{A}_{\mathrm{pre}}\leq\mathfrak{I}\cdot\mathfrak{A}_{\mathrm{pos}}$
(11.13)

for some quantity $\mathfrak{I}$, which we define to be the _deviation_ of
this operation. Clearly, if we can bound the deviation in each operation step,
as well as $\mathfrak{A}_{\mathrm{fin}}$ for the final molecule
$\mathbb{M}_{\mathrm{fin}}$, then we can deduce from these an upper bound for
$\mathfrak{A}_{\mathrm{sb}}$ using (11.13).

We will define the needed cutting operations, and prove (11.13) for each
operation, in Section 11.2.1; then in Section 11.3 we obtain an upper bound
for $\mathfrak{A}_{\mathrm{fin}}$ and complete the proof of (11.11).

####  11.2.1. Cutting operations and estimates

Recall the notion of $V,E,F$ and $\chi$ as in Definition 7.1; define also
$V_{\alpha}$ and $V_{\beta}$ to be the number of $\alpha$\- and $\beta$-atoms
(see Definition 11.1, and also the description of operations below), and use
$\Delta$ to denote increments (such as $\Delta
F=F_{\mathrm{pos}}-F_{\mathrm{pre}}$).

Operation 1: Removing SG vine-like objects. In Operation 1, we collect all the
SG vine-like objects in $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$, as defined in
Section 11.1.1; note that these also include any possible triple bonds in
$\mathbb{M}_{\mathrm{sb}}$ (see Proposition 10.6 (2)). For each of these
objects $\mathbb{U}$, we cut the molecule at each of its joints $v$, along the
two bonds $(\ell_{1},\ell_{2})$ fixed as in requirement (e) defined in Section
11.1.1 (i.e. the two bonds that belong to $\mathbb{U}$). This disconnects a
VC, denoted by $\mathbb{U}^{\prime}$, from the rest of the molecule, and then
we remove $\mathbb{U}^{\prime}$. The two joints then become two atoms of
degree at most $2$ (it is possible that they have degree $0$, i.e. isolated
atoms); we define them as $\alpha$-atoms if no new connected component is
created, and $\beta$-atoms otherwise. If they are $\alpha$-atoms, we also
label each of them by the dyadic number $Q\in[L^{-1},L^{-\gamma+\eta}]$ such
that $|r|\sim_{0}Q$ for the gap $r$ of $\mathbb{U}^{\prime}$ (note $r\neq 0$).

Operation 2: Removing triangles. In Operation 2, we consider the possible
triangles $v_{1}v_{2}v_{3}$ in the molecule, such that there are bonds
$\ell_{j}$ connecting $v_{j+1}$ and $v_{j+2}$ (where $v_{4}=v_{1}$ and
$\ell_{4}=\ell_{1}$ etc.), and $(v_{j},\ell_{j+1},\ell_{j+2})$ is an SG triple
for $j\in\\{1,2\\}$. Let $|k_{\ell_{j+1}}-k_{\ell_{j+2}}|\sim_{0}Q_{j}$ for
$j\in\\{1,2\\}$ with $Q_{j}\in[L^{-1},L^{-\gamma+\eta}]$. We then cut the
molecule at each $v_{j}$ along the bonds $(\ell_{j+1},\ell_{j+2})$, which
disconnects the triangle formed by $v_{j}$ and $\ell_{j}$ from the rest of the
molecule, and remove the triangle. This leaves $3$ atoms $v_{j}$ of degree at
most $2$. We define $v_{3}$ as a $\beta$-atom, and define $v_{j}\,(1\leq j\leq
2)$ as an $\alpha$\- (resp. $\beta$-) atom if it belongs to the same (resp.
different) component with $v_{3}$, except when $v_{1}$ and $v_{2}$ are in the
same component different from $v_{3}$, in which case we define $v_{1}$ as an
$\alpha$-atom and define $v_{2}$ as a $\beta$-atom. For any $\alpha$-atom
$v_{j}$ we label it by the corresponding $Q_{j}$.

Operation 3: Remaining SG cuts. In Operation 3, we select each of the
remaining SG atoms $v$, and cut them along the designated bonds
$(\ell_{1},\ell_{2})$ as in requirement (e) in Section 11.1.1. Note that each
cut may be $\alpha$\- or $\beta$-cut; we define the resulting atoms as
$\alpha$\- or $\beta$-atoms, and label any $\alpha$-atom by the dyadic number
$Q\in[L^{-1},L^{-\gamma+\eta}]$, as in Definition 11.1.

Operation 4: Remaining $\beta$-cuts. In Operation 4, we look for all the
possible atoms where a $\beta$-cut is possible, and perform the corresponding
$\beta$-cut (defining the resulting atoms as $\beta$-atoms), until this can no
longer be done.

Order of these operations. We first perform Operation 1 for all SG vine-like
objects in $\mathscr{C}_{1}$ (first for the (CN) vines and root (CL) vines,
and then for the HV’s), then perform Operation 1 for all SG vine-like objects
in $\mathscr{C}_{2}$. Next we perform Operation 2 for all eligible triangles,
and then perform Operation 3 for each of the remaining SG atoms. Finally we
perform Operation 4 for all remaining atoms where a $\beta$-cut is possible.

######  Remark 11.3.

After all the cutting operations, the resulting graph will contain some
$\alpha$-atoms. For each $\alpha$-atom $v$ and a given decoration we define an
auxiliary number $\iota_{v}\in\\{0,1\\}$, such that $\iota_{v}=1$ if a cutting
operation happens before the cutting at the atom $v$, such that the gaps
$r,r^{\prime}$ of these cuttings satisfy $0<|r\pm r^{\prime}|\leq
L^{-50\eta}Q_{v}$; if no such cutting operation exists then define
$\iota_{v}=0$. Note that the number of choices for all $(\iota_{v})$ is at
most $2^{n_{\mathrm{sb}}}$ which can be safely ignored; therefore we may
assume a choice of $(\iota_{v})$ is fixed in the proof below.

######  Remark 11.4.

Note that the joints of the SG vine-like object in Operation 1, the atoms
$v_{1}$ and $v_{2}$ in Operation 2 and the atom $v$ in Operation 3 are all SG
with fixed $(\ell_{1},\ell_{2})$. We will also assume that no atom $v_{3}$ in
Operation 2 or atom $v$ in Operation 4 is ZG; in fact, if any such atom $v$ is
ZG, then it must be tame by Proposition 10.6 (3), and thus can be treated by
the arguments in Section 10.5 of [18]. Namely, we consider the connected
components after removing the atom $v$. If (i) $d(v)<4$, or if some component
contains exactly one atom adjacent to $v$, then the values of $k_{\ell}$ for
$\ell\leftrightarrow v$ must all be fixed in the given decoration, so it will
not matter whether the resulting $\alpha$\- or $\beta$-atoms have zero gap. We
then cut at $v$ and proceed normally. If (ii) $d(v)=4$ and no component
contains exactly one atom adjacent to $v$, then removing $v$ will yield
$\Delta\chi\leq-2$ by direct calculation, which leads to a power gain
$\mathfrak{I}\leq L^{-(d-2\gamma-\eta)}$, as $(k_{\ell})$ for
$\ell\leftrightarrow v$ have at most $L^{d}$ choices. We then remove $v$
without assigning any new $\alpha$\- or $\beta$-atoms and proceed with the
rest of the molecule. The big gain $L^{-(d-2\gamma-\eta)}$ is enough to cover
any loss that may occur, see Remark 11.8.

######  Proposition 11.5.

The followings hold regarding Operations 1–4, and the process of applying them
as described above:

  1. (1)

After each Operation 1–4, _except for an Operation 1 that creates a new
connected component_ , we have that

$\mathfrak{I}\leq
L^{-\eta^{3}}\cdot\prod_{v}^{(\alpha)}L^{(\gamma+3\eta)/2-\kappa_{v}\eta}Q_{v}\cdot
L^{(2\gamma_{0}+5\eta^{2})\Delta F-(\gamma_{0}+2\eta^{2})\Delta V_{\beta}}$
(11.14)

for any $Q_{v}\in[L^{-1},L^{-\gamma+\eta}]$, where the product is taken over
all the newly created $\alpha$-atoms $v$, and $Q_{v}$ denotes the label of
$v$. The parameter $\kappa_{v}$ equals $0$ if $\iota_{v}=0$, and equals $50$
if $\iota_{v}\neq 0$.

  2. (2)

If an Operation 1 creates a new connected component, then we have that

$\mathfrak{I}\leq
L^{\gamma_{0}-\eta^{3}}\cdot\prod_{v}^{(\alpha)}L^{(\gamma+3\eta)/2-\kappa_{v}\eta}Q_{v}\cdot
L^{(2\gamma_{0}+5\eta^{2})\Delta F-(\gamma_{0}+2\eta^{2})\Delta V_{\beta}},$
(11.15)

where the notations are the same as in (11.14).

  3. (3)

In the whole process, the number of Operations 1 that indeed create new
connected components, is at most $2(R-1)$.

###### Proof.

(1) This is exactly Proposition 9.4 of [18], and follows from exactly the same
proof. Note that in the couple setting in Section 9 of [18] (i.e. $R=1$), no
Operation 1 can create a new component, and the proof of Proposition 9.4 of
[18] relies on this fact.

(2) Note that all the SG vine-like objects in $\mathscr{C}_{1}$ and
$\mathscr{C}_{2}$ are disjoint (and the DV’s and LG and ZG vine-like objects
containing those objects in $\mathscr{C}_{2}$ are also disjoint with the
objects in $\mathscr{C}_{1}$), so removing any of them will not affect the
others. By Lemma A.5, any Operations 1 concerning $\mathscr{C}_{2}$ will _not_
create a new connected component, because the corresponding DV or LG or ZG
vine-like object will still remain connected.

Therefore, we only need to consider Operations 1 concerning $\mathscr{C}_{1}$,
which is always removing a single vine $\mathbb{U}$. By definition we have
$\Delta F=1$ and $\Delta V_{\beta}=2$, and no new $\alpha$-atom is created. If
$\mathbb{U}$ contains $m$ atoms (including joints). Consider the collection
$(k_{\ell})$ for all bonds $\ell\in\mathbb{U}$ in the decoration of
$\mathbb{M}_{\mathrm{pre}}$, and let the number of choices for them be
$\mathfrak{F}$, then we have

$\mathfrak{I}\lesssim_{1}\mathfrak{F}\cdot
L^{-(m-1)(d-\gamma)}\cdot(C_{1}^{-1}\delta^{1/2})^{m-1}\prod_{j:\mathbb{L}_{j}\subset\mathbb{U}}\mathfrak{X}_{j}^{z_{j}}$
(11.16)

where the product is taken over all $j$ such that the maximal ladder
$\mathbb{L}_{j}$ (Proposition 10.5) is a subset of $\mathbb{U}$ (so there are
at most three such $j$, see Figure 15), and $z_{j}$ is the length of
$\mathbb{L}_{j}$.

Now we calculate $\mathfrak{F}$. Consider the two bonds $\ell_{1}$ and
$\ell_{2}$ at one joint of $\mathbb{V}\mathbb{C}$ that belong to $\mathbb{U}$.
Note that removing $\mathbb{U}$ will create a new connected component; thus in
the $(c_{v})$-decoration of $\mathbb{M}_{\mathrm{pre}}$, we must have that
$k_{\ell_{1}}\pm k_{\ell_{2}}$ equals a fixed constant and that
$|k_{\ell_{1}}|^{2}\pm|k_{\ell_{2}}|^{2}$ belongs to a fixed interval of
length at most $n_{\mathrm{sb}}\cdot\delta^{-1}L^{-2\gamma}$, both depending
only on the parameters $(c_{v})$ and $(\beta_{v})$. By losing another factor
$n_{\mathrm{sb}}\leq(\log L)^{C_{2}}$ which can be ignored in view of the
$L^{-\eta^{3}}$ factor in (11.15), we may assume that
$|k_{\ell_{1}}|^{2}\pm|k_{\ell_{2}}|^{2}$ belongs to a fixed interval of
length $\delta^{-1}L^{-2\gamma}$. By (A.24) in Lemma A.10, the number of
choices for $(k_{\ell_{1}},k_{\ell_{2}})$ is at most
$L^{d-\gamma+\gamma_{0}+\eta^{3}}$.

Now, once $(k_{\ell_{1}},k_{\ell_{2}})$ is fixed, we can invoke the re-
parametrization introduced in the proof of Proposition 9.5 (note that we are
dealing with a single vine now), and define the new variables
$(x_{j},y_{j})\,(1\leq j\leq\widetilde{m})$ if $\mathbb{U}$ is bad vine, or
$(x_{j},y_{j})\,(1\leq j\leq\widetilde{m})$ and $(u_{1},u_{2},u_{3})$ if
$\mathbb{U}$ is normal vine, where $m=2\widetilde{m}+2$ for bad vine and
$m=2\widetilde{m}+5$ for normal vine. In either case, since each $\Gamma_{v}$
belongs to a fixed interval of length $\delta^{-1}L^{-2\gamma}$, we know that
each $(x_{j},y_{j})$ satisfies a system of form (A.21), and
$(u_{1},u_{2},u_{3})$ satisfies a system of form (A.22). Moreover for each
$j$, if the atoms associated with $(x_{j},y_{j})$ (see the proof of
Proposition 9.5) belongs to a certain ladder $\mathbb{L}$, then the value of
$r$ in (A.21) must equal the gap of $\mathbb{L}$. Therefore, by Lemma A.9, we
get in either case that

$\mathfrak{F}\lesssim_{1}L^{d-\gamma+\gamma_{0}+\eta^{3}}L^{(m-2)(d-\gamma)}\cdot
L^{\eta^{3}}(C_{1}\delta^{-1/2})^{m-1}\prod_{j:\mathbb{L}_{j}\subset\mathbb{U}}\mathfrak{X}_{j}^{-z_{j}},$

hence $\mathfrak{I}\lesssim_{1}L^{\gamma_{0}+2\eta^{3}}$, so (11.15) is true.

(3) By definition, we are doing Operations 1 concerning $\mathscr{C}_{1}$
before those concerning $\mathscr{C}_{2}$. Moreover, those Operations 1
concerning $\mathscr{C}_{2}$ and HV’s in $\mathscr{C}_{1}$ will not create any
new connected component, so we only need to consider the Operations 1
concerning (CN) and root (CL) vines in
$\mathbb{M}_{\mathrm{sb}}=\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$, which are
done before all the others. By Proposition 7.8, after removing all these (CN)
and root (CL) vines, the number of connected components increases to at most
$2R-1$, so in this process the number of Operations 1 that create new
connected components is at most $(2R-1)-1=2(R-1)$. ∎

###  11.3. The rigidity theorem

Starting from $\mathbb{M}_{\mathrm{sb}}$, we perform all the Operations 1–4 as
described in Section 11.2.1, and obtain a final molecule
$\mathbb{M}_{\mathrm{fin}}$. This molecule may not be connected, and let
$\mathbb{M}$ be any of its connected components. Consider also a decoration of
$\mathbb{M}_{\mathrm{fin}}$ (and $\mathbb{M}$) that is inherited from a
decoration of $\mathbb{M}_{\mathrm{sb}}$ as in Section 11.2. Then, they
satisfy certain crucial properties, which we summarize in the following
proposition.

######  Proposition 11.6.

In any (connected) component $\mathbb{M}$ of $\mathbb{M}_{\mathrm{fin}}$, each
$\alpha$-atom $v$ is labeled by a dyadic number
$Q_{v}\in[L^{-1},L^{-\gamma+\eta}]$, such that if $v$ has two bonds
$(\ell_{1},\ell_{2})$ then $|k_{\ell_{1}}-k_{\ell_{2}}|\sim_{0}Q_{v}$ in the
decoration; recall also $\iota_{v}$ and $\kappa_{v}$ introduced in Remark 11.3
and Proposition 11.5. Define any atom that is not an $\alpha$\- nor
$\beta$-atom to be an _$\varepsilon$ -atom_. Then, any $\varepsilon$-atom in
$\mathbb{M}_{\mathrm{fin}}$ _must_ be LG or ZG in the decoration, and must be
tame if it is ZG. Moreover $\mathbb{M}_{\mathrm{fin}}$ contains no triple
bond.

We say a component is _perfect_ if all $\alpha$\- and $\beta$\- atoms have
degree $2$, and all $\varepsilon$-atoms have degree $4$. Otherwise we say
$\mathbb{M}$ is _imperfect_. Then, there are at most $R$ imperfect components.
Moreover, any perfect component has at least one $\beta$-atom. If a perfect
component $\mathbb{M}$ is a cycle (i.e. all atoms have degree $2$), then it is
either a double bond (which is also Vine (I)), or a cycle of length at least
$4$, or a triangle with at most one $\alpha$-atom. If $\mathbb{M}$ is not a
cycle, then all its $\alpha$\- and $\beta$-atoms form several disjoint chains,
such that each chain has two distinct $\varepsilon$-atom at both ends.
Finally, no $\alpha$\- or $\beta$-atom is ZG with the assumptions in Remark
11.4, and if any (perfect or imperfect) component $\mathbb{M}$ is a _vine_
with two joints having degree $2$, then it _must_ be LG in the decoration.

###### Proof.

We only need to prove that there are at most $R$ imperfect components. All the
other statements follow from exactly the same proof as in Proposition 9.5 of
[18] (note also that any ZG $\varepsilon$-atom cannot be a hinge atom, thus
must be tame by Proposition 10.6 (3)).

The molecule $\mathbb{M}_{\mathrm{sb}}$ has $n=n_{\mathrm{sb}}$ atoms and
$2n-R$ bonds, as in Proposition 10.6 (1). It is easy to check that in each
Operation 1–4, the value of $4V-2E-2(V_{\alpha}+V_{\beta})$ is preserved. This
value is $2R$ for $\mathbb{M}_{\mathrm{sb}}$, and has to be the same for
$\mathbb{M}_{\mathrm{fin}}$. But this value also equals the sum of $4-d(v)$
over all $\varepsilon$-atoms $v$, plus the sum of $2-d(v)$ over all $\alpha$\-
and $\beta$\- atoms $v$. If we add up these quantities over each component,
then by definition, we get $0$ for a perfect component, and at least $2$ for
an imperfect component. This implies that the number of imperfect components
is at most $R$ (in fact, it is exactly $R$ if $R=1$). ∎

With Proposition 11.6, we can reduce the proof of (11.11) to the estimates for
quantities $\mathfrak{A}$ and $\mathfrak{E}$ associated with the counting
problems for each component $\mathbb{M}$ of $\mathbb{M}_{\mathrm{fin}}$. These
estimates are stated as a _rigidity theorem_ in Proposition 9.6 of [18]; its
proof is a major technical component of [18]. Fortunately, here we can use it
(almost) as a black box.

######  Proposition 11.7.

For each component $\mathbb{M}$ of $\mathbb{M}_{\mathrm{fin}}$, define
$(\mathfrak{A},\mathfrak{E})$ and $(\rho,q,m)$ as in Section 11.2 and (11.12),
but associated with $\mathbb{M}$. Let also each $\alpha$-atom $v$ be labeled
by the dyadic number $Q_{v}$, then we have

$\mathfrak{A}\leq\prod_{v}^{(\alpha)}L^{-(\gamma+3\eta)/2+\kappa_{v}\eta}Q_{v}^{-1}\cdot
L^{(\gamma_{0}+2\eta^{2})V_{\beta}-(2\gamma_{0}+5\eta^{2})G}\cdot
L^{-\eta^{5}\rho+\eta Z},$ (11.17)

where the product is taken over all $\alpha$-atoms $v$, and $V_{\beta}$ is the
number of $\beta$-atoms in $\mathbb{M}$, and $G$ is $0$ or $1$ depending on
whether $\mathbb{M}$ is imperfect or perfect component. Finally $Z$ equals $0$
if $R=1$; if $R\geq 2$ then $Z$ equals the number of $\varepsilon$-atoms _not_
of degree $4$ in $\mathbb{M}$.

###### Proof.

For perfect components, or imperfect components when $R=1$ (which are called
_odd_ components in [18]), this is exactly the same as Proposition 9.6 of
[18]; see also Section 10.5 of [18] that addresses tame atoms. Now consider an
imperfect component when $R\geq 2$. By repeating the same proof in Proposition
9.6 of [18] we can obtain that $\mathfrak{A}\lesssim_{1}1$ (the proof in [18]
trivially extends to general imperfect components). On the other hand, we may
also apply Lemma 10.2 of [18] to get

$\mathfrak{A}\lesssim_{1}L^{-\eta^{2}\rho+C_{0}(V_{\alpha}+V_{\beta}+Z)},$

noticing that $V_{\alpha}+V_{\beta}+Z\geq 1$. Since for any $Q_{v}\leq
L^{-\gamma+\eta}$ we have

$L^{-(\gamma+3\eta)/2+\kappa_{v}\eta}Q_{v}^{-1}\geq L^{(\gamma-5\eta)/2}\geq
L^{\gamma/4},$

and also $G=0$, by interpolation we obtain that

$\mathfrak{A}\lesssim_{1}L^{-\eta^{4}\rho+C_{0}\eta^{2}(V_{\alpha}+V_{\beta}+Z)}\ll_{1}\mathrm{RHS\
of\ }(\ref{finalcount}).\qed$

With Proposition 11.7, we are finally ready to prove (11.11).

###### Proof of the bound (11.11).

By iterating (11.13) for each of the Operations 1–4 in reducing
$\mathbb{M}_{\mathrm{sb}}$ to $\mathbb{M}_{\mathrm{fin}}$, and using the fact
that $\mathbb{M}_{\mathrm{fin}}$ is formed by all the components $\mathbb{M}$,
we have

$\mathfrak{A}_{\mathrm{sb}}\leq\mathfrak{A}_{\mathrm{fin}}\cdot\prod_{\mathrm{(all\
operations)}}\mathfrak{I},\qquad\mathfrak{A}_{\mathrm{fin}}\leq\prod_{\mathrm{(all\
components)}}\mathfrak{A},$ (11.18)

where the products are taken over all Operations 1–4 and all components
$\mathbb{M}$ of $\mathbb{M}_{\mathrm{fin}}$. Putting together (11.14), (11.15)
and (11.17), we get that

$\mathfrak{A}_{\mathrm{sb}}\leq
L^{(2\gamma_{0}+5\eta^{2})(F_{\mathrm{im}}-1)}\cdot L^{\gamma_{0}\cdot
p_{\mathrm{op}}^{+}}\cdot L^{\eta Z_{\mathrm{tot}}}\cdot
L^{-\eta^{5}\rho_{\mathrm{fin}}-\eta^{3}p_{\mathrm{op}}}.$ (11.19)

Here in (11.19):

  * •

$\rho_{\mathrm{fin}}$ is the $\rho$ value for $\mathbb{M}_{\mathrm{fin}}$
defined as in Section 11.2, and $F_{\mathrm{im}}$ is the number of imperfect
components in $\mathbb{M}_{\mathrm{fin}}$, so $F_{\mathrm{im}}\leq R$ by
Proposition 11.6.

  * •

$p_{\mathrm{op}}$ is the total number of Operations 1–4, and
$p_{\mathrm{op}}^{+}$ is the total number of Operations 1 that create new
components, so $p_{\mathrm{op}}^{+}\leq 2(R-1)$ by Proposition 11.5 (3).

  * •

$Z_{\mathrm{tot}}$ is the sum of $Z$, defined in Proposition 11.7, over all
components $\mathbb{M}$, so $Z_{\mathrm{tot}}=0$ if $R=1$; if $R\geq 2$ then
$Z_{\mathrm{tot}}$ is the total number of $\varepsilon$-atoms not of degree
$4$, so by Proposition 10.6 (1) we have $Z_{\mathrm{tot}}\leq 2R$. In any case
we have $Z_{\mathrm{tot}}\leq 4(R-1)$.

Note also that if an SG atom $v$ satisfies requirement (f) in Section 11.1.1
(i.e. is not an interior atom of an SG vine-like object in $\mathscr{C}_{1}$
or $\mathscr{C}_{2}$), then we must be cutting at $v$ in one of the Operations
1–4, so we have $p_{\mathrm{sb}}\leq p_{\mathrm{op}}$. Moreover each of
Operations 1–4 changes the value of $\rho$ by at most $C_{0}$, since each vine
contains at most $3$ ladders and no more than $20$ atoms apart from these
ladders, thus $\rho_{\mathrm{fin}}\geq\rho_{\mathrm{sb}}-C_{0}\cdot
p_{\mathrm{op}}$. Putting together we get that

$\mathfrak{A}_{\mathrm{sb}}\leq L^{(R-1)(4\gamma_{0}+4\eta+5\eta^{2})}\cdot
L^{-\eta^{5}(\rho_{\mathrm{sb}}-C_{0}\cdot
p_{\mathrm{op}})-\eta^{3}p_{\mathrm{op}}}\leq
L^{-\eta^{5}(\rho_{\mathrm{sb}}+p_{\mathrm{sb}})}\cdot
L^{(R-1)(4\gamma_{0}+1/10)},$ (11.20)

which implies (11.11) because $d-\gamma-4\gamma_{0}\geq 1/2$. This completes
the proof. ∎

######  Remark 11.8.

Recall the discussion of ZG/tame atoms in Remark 11.4. In case (i) the proof
is modified trivially. In case (ii), by the arguments in Remark 11.4, we know
that this operation satisfies (11.14) with an extra power gain
$L^{-(d-2\gamma-2\eta)-(2\gamma_{0}+5\eta^{2})}$ (assuming $\Delta F=1$,
otherwise $\Delta\chi=-3$ and the estimate is much better). On the other hand,
this increases the number of imperfect components by $2$, leading to a loss
$L^{2(2\gamma_{0}+5\eta^{2})}$ in (11.19), but this is acceptable as
$d-2\gamma-2\gamma_{0}\geq 1$.

####  11.3.1. Proof of Proposition 3.11

By interpolating (11.5) with (11.6), which follows from (11.11), we get that

$|\mathscr{K}_{\mathcal{G}_{\mathrm{sb}}}|\lesssim_{2}(C_{1}\delta^{10\nu})^{n_{\mathrm{sb}}}\cdot
L^{-40\nu\eta\cdot g_{\mathrm{sb}}}\cdot
L^{-\eta^{6}(\rho_{\mathrm{sb}}+q_{\mathrm{sk}})/2}\cdot
L^{-(R-1)(1/4+1/60)}.$ (11.21)

Combining with (10.2) and arguments in Section 10.2, we get that

$\displaystyle|\mathscr{K}_{p+1,n}^{\mathrm{eqc}}|$
$\displaystyle\lesssim_{2}(C_{1}\sqrt{\delta})^{n-n_{\mathrm{sb}}}(C_{1}\delta^{10\nu})^{n_{\mathrm{sb}}}\cdot
L^{-\eta^{8}g_{\mathrm{sk}}-\eta^{2}b_{\mathrm{sk}}+\eta^{7}q_{\mathrm{sk}}}$
$\displaystyle\qquad\qquad\qquad\quad\times L^{-40\nu\eta\cdot
g_{\mathrm{sb}}}\cdot L^{-\eta^{6}(\rho_{\mathrm{sb}}+q_{\mathrm{sk}})/2}\cdot
L^{-(R-1)(1/4+1/60)}$ $\displaystyle\lesssim_{2}(C_{1}\delta^{10\nu})^{n}\cdot
L^{-\nu\eta(g_{\mathrm{sk}}+g_{\mathrm{sb}})}\cdot
L^{-\eta^{6}(\rho_{\mathrm{sb}}+q_{\mathrm{sk}})/4}\cdot
L^{-(R-1)(1/4+1/60)},$ (11.22)

where $g_{\mathrm{sk}}$ and $b_{\mathrm{sk}}$ are as in Section 10.2.

With (11.22) we can now finish the proof of Proposition 3.11.

###### Proof of Proposition 3.11.

We only need to consider the terms (a)–(d) in (11.1)–(11.4). For each term, we
can treat the $(\langle k_{1}\rangle\cdots\langle
k_{2R}\rangle)^{4\Lambda_{p+1}}$ weights and $\partial_{k}^{\rho}$ derivative
for couples as described in Section 11, so we just need to bound it pointwise
for each fixed $(k_{1},\cdots,k_{2R})$. By translation invariance, for
(11.2)–(11.4) we will assume $k\in\mathbb{Z}_{L}^{d}$ (the general
$k\in\mathbb{R}^{d}$ case is the same).

First consider (11.3) and (11.4). If in (11.3) we restrict to couples of
incoherency index $g\geq 1$, then the contribution of each individual couple
is bounded by $(C_{1}\delta)^{n}\cdot L^{-\gamma_{1}g/2}$ by (5.1). For fixed
$g$, the number of choices for the regular couple (without layering) is at
most $C_{0}^{n}$ by Proposition 4.5, and the number of possible layerings is
at most $C_{0}^{n}C_{2}^{g}$ by Proposition 4.10. This implies that

$|(\ref{finaltermc})|\lesssim_{2}\sum_{g=1}^{n}(C_{1}\delta)^{n}\cdot
L^{-\gamma_{1}g/2}\cdot C_{0}^{n}C_{2}^{g}\leq
L^{-\gamma_{1}/4}\cdot\delta^{n/2}.$ (11.23)

As for (11.4), let the (canonical and) regular, coherent layered couple be
$\mathcal{Q}$. By Proposition 4.9, since both $q$ and $q^{\prime}$ are now
replaced by $p$, we know that all nodes in $\mathcal{Q}$ must be in layer $p$,
in particular the number of choices for $\mathcal{Q}$ is at most $C_{0}^{n}$.
Then $\mathcal{K}_{\mathcal{Q}}$ is simply a restriction of
$\mathcal{K}_{\mathcal{Q}}^{*}$ defined in Definition 3.8, so by (6.1), (6.2),
(6.6) and (6.7) we get

$|(\ref{finaltermd})|\leq
L^{-(2\theta_{p+1}+3\theta_{p})/5}\cdot\delta^{n/2}.$ (11.24)

Now we consider (11.1) and (11.2). By reductions in Sections 9 and 10, each of
them can be reduced to a summation of terms of form
$\mathscr{K}_{p+1,n}^{\mathrm{eqc}}$ which satisfy (11.22). Let
$\bar{g}:=g_{\mathrm{sk}}+g_{\mathrm{sb}}$ and
$\bar{\rho}:=\rho_{\mathrm{sb}}+q_{\mathrm{sk}}$ with the relevant parameters
defined as before; clearly these values are not changed under congruence
relation defined in Definition 10.2. Also when $R=1$ and for any non regular
couple $\mathcal{Q}$, the skeleton $\mathcal{Q}_{\mathrm{sk}}$ is nontrivial
and hence $\bar{\rho}\geq 1$.

With $\bar{g}$ and $\bar{\rho}$ fixed, we consider the number of choices for
the canonical layered garden $\mathcal{G}$. First consider the molecule after
removing all maximal ladders from
$\mathbb{M}_{\mathrm{sb}}=\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$, which is a
molecule of at most $C_{0}\bar{\rho}$ atoms and thus has at most
$(C_{0}\bar{\rho})!$ choices; then, by adding back at most $C_{0}\bar{\rho}$
disjoint maximal ladders of total length at most $n$, we get at most
$C_{0}^{n}(C_{0}\bar{\rho})!$ choices for
$\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$. Similarly by adding back at most
$C_{0}\bar{\rho}$ disjoint VC’s in $\mathscr{V}_{0}$ in Section 10.1, we get
at most $C_{0}^{n}(C_{0}\bar{\rho})!$ choices for
$\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$. Then by Proposition 7.3 we get at
most $C_{0}^{n}(C_{0}\bar{\rho})!(C_{0}R)!$ choices for the garden
$\mathcal{G}_{\mathrm{sk}}$, and by Proposition 4.5 we get at most
$C_{0}^{n}(C_{0}\bar{\rho})!(C_{0}R)!$ choices for the garden $\mathcal{G}$ if
we do not consider layering.

Now we consider the possible canonical layerings of $\mathcal{G}$. By losing
another factor $2^{n}$, we may fix the exact positions of all the $\bar{g}$
incoherencies. If we fix the pre-layering of $\mathcal{G}_{\mathrm{sk}}$
(which is equivalent to the layering of atoms of the molecule
$\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$), then by Propositions 4.9 and 4.10,
we know that the number of layerings for all the regular couples
$\mathcal{Q}_{(\mathfrak{l},\mathfrak{l}^{\prime})}$ and
$\mathcal{T}_{(\mathfrak{m})}$ attached to $\mathcal{G}_{\mathrm{sk}}$ is
bounded by

$C_{0}^{n}C_{2}^{\bar{g}}\cdot\prod_{\ell}C_{0}^{|\mathfrak{L}_{v}-\mathfrak{L}_{v^{\prime}}|},$
(11.25)

where the product is taken over all bonds $\ell$ of
$\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$ connecting atoms
$v,v^{\prime}\in\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$ similar to (8.1).
Therefore, we only need to bound the summation

$\sum_{(\mathfrak{L}_{v})}\prod_{\ell}C_{0}^{|\mathfrak{L}_{v}-\mathfrak{L}_{v^{\prime}}|},$
(11.26)

where the summation is taken over all possible layerings of atoms of
$\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$.

Next, consider all the VC’s in $\mathscr{V}_{0}$ defined in Definition 10.2
which are merged in reducing $\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$ to
$\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$ in Sections 10.1–10.2. Since they are
formed by (CL) vines only, by Proposition 7.7 we know that for each of the VC
in $\mathscr{V}_{0}$, the layers of the joint atoms of its vine ingredients
form a monotonic sequence. Thus, by Proposition 8.3, we know that the number
of possible layerings of atoms in these VC’s, multiplied by the product in
(11.26) involving bonds in these VC’s, is bounded by
$C_{0}^{n}C_{2}^{\bar{g}}\cdot C_{2}^{q_{\mathrm{sk}}}$ (where
$q_{\mathrm{sk}}$ is the number of all these VC’s).

Finally, consider all the maximal ladders in
$\mathbb{M}_{\mathrm{sb}}=\mathbb{M}(\mathcal{G}_{\mathrm{sb}})$. By repeating
the above arguments but now using Proposition 8.2, we can bound the the number
of possible layerings of atoms in these ladders, multiplied by the product in
(11.26) involving bonds in these ladders, by $C_{0}^{n}C_{2}^{\bar{g}}\cdot
C_{2}^{q_{\mathrm{sb}}}$ (where $q_{\mathrm{sb}}$ is the number of all these
maximal ladders). The number of atoms not in these maximal ladders is
$m_{\mathrm{sb}}$, and the number of layerings for them, multiplied by the
product in (11.26) involving bonds at these atoms, is trivially bounded by
$C_{2}^{m_{\mathrm{sb}}}$.

Putting altogether, we know that with fixed values of $n$, $R$, $\bar{g}$ and
$\bar{\rho}$, the number of possible canonical layered gardens $\mathcal{G}$
is bounded by

$C_{0}^{n}(C_{0}\bar{\rho})!(C_{0}R)!C_{2}^{\bar{g}+\bar{\rho}}.$

The number of congruence classes, which is the number of terms of form
$\mathscr{K}_{p+1,n}^{\mathrm{eqc}}$ contained in (11.1) and (11.2), is also
bounded by this number. Combining with (11.22), we get that

$|(\ref{finalterma})|\mathrm{\ or\
}|(\ref{finaltermb})|\lesssim_{2}(C_{1}\delta^{10\nu})^{n}L^{-\eta^{7}}\cdot
L^{-\nu\eta(\bar{\rho}+\bar{g})}C_{2}^{\bar{g}+\bar{\rho}}\cdot
L^{-(R-1)(1/4+1/60)}(C_{0}R)!\cdot(C_{0}\bar{\rho})!.$ (11.27)

Since $\max(R,\bar{\rho},\bar{g})\leq n\leq N_{p+1}^{4}\leq(\log L)^{C_{2}}$
by the assumption in Proposition 3.11, from (11.27) we easily deduce that

$|(\ref{finalterma})|\mathrm{\ or\
}|(\ref{finaltermb})|\lesssim_{2}\delta^{8\nu n}L^{-\eta^{7}}\cdot
L^{-(R-1)(1/4+1/70)}.$ (11.28)

This completes the proof of Proposition 3.11. ∎

##  12\. Linearization and proof of Proposition 3.12

###  12.1. Flower trees and construction of parametrix

We turn to the proof of Proposition 3.12. Like in [15, 18], the key point is
to construct a suitable parametrix of $1-\mathscr{L}_{1}$ for the
$\mathbb{R}$-linear operator $\mathscr{L}_{1}$ defined in (3.39), which
involves the notions of of flower trees and flower gardens.

######  Definition 12.1 (Flower trees and flower gardens [15, 18]).

We define a _flower tree_ to be a ternary tree $\mathcal{T}$ with one
particular leaf $\mathfrak{f}$ fixed. This leaf $\mathfrak{f}$ is called the
_flower_ of $\mathcal{T}$. There is a unique path from the root $\mathfrak{r}$
to the flower $\mathfrak{f}$, which we call the _stem_ of $\mathcal{T}$.
Similarly we can define a _flower garden_ $\mathcal{G}$ of width $2R$ to be a
collection of $2R$ flower trees whose roots have signs $(+,-,\cdots,+,-)$,
such that all the _non-flower_ leaves are completely paired151515We may also
pair the flowers (for example pair the flower of the first tree with the
second, the third with the fourth, etc.) to make $\mathcal{G}$ a genuine
garden, as was done in [18], but this is not necessary in the proof below. as
in Definition 3.4. For any $k,k^{\prime}\in\mathbb{Z}_{L}^{d}$ we define a
$(k,k^{\prime})$-decoration of $\mathcal{T}$ or $\mathcal{G}$ to be a
decoration in the sense of Definition 3.5 such that $k_{\mathfrak{r}}=k$ and
$k_{\mathfrak{f}}=k^{\prime}$ for the root $\mathfrak{r}$ and flower
$\mathfrak{f}$ of each tree.

We define the layering of a flower garden as in Definition 3.6, but with the
additional requirement that all the nodes on the stem of each tree (from the
root to the flower) are in layer $p$. We also define the collection
$\mathscr{H}_{p+1}$ of _canonical layered flower gardens_ $\mathcal{G}$ in the
same way as in Definition 3.7 (but without any assumption about irreducibility
of $\mathcal{G}$), where we start from $2R$ flower trees with all branching
nodes, flowers and paired leaves in layer $p$, then divide the set of
remaining (non-flower) leaves into subsets of size $\geq 4$ and replace each
subset by an irreducible proper layered garden in $\mathscr{G}_{p}$. Note that
Proposition 4.8 also holds for $\mathscr{H}_{p+1}$ with essentially the same
proof.

For any flower tree $\mathcal{T}$ and layered flower garden $\mathcal{G}$ of
order $n$, and $k,k^{\prime}\in\mathbb{Z}_{L}^{d}$ and $p+1>t>s>p$, define the
quantities

$\widetilde{\mathcal{J}}_{\mathcal{T}}(t,s,k,k^{\prime})=\bigg{(}\frac{\delta}{2L^{d-1}}\bigg{)}^{n}\zeta(\mathcal{T})\sum_{\mathscr{D}}\epsilon_{\mathscr{D}}\int_{\mathcal{D}}\prod_{\mathfrak{n}\in\mathcal{N}}e^{\zeta_{\mathfrak{n}}\pi
i\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}\mathrm{d}t_{\mathfrak{n}}\cdot\bm{\delta}(t_{\mathfrak{f}^{\mathrm{pr}}}-s)\prod_{\mathfrak{f}\neq\mathfrak{l}\in\mathcal{L}}a_{k_{\mathfrak{l}}}(p),$
(12.1)
$\widetilde{\mathcal{K}}_{\mathcal{G}}(t,s,k,k^{\prime})=\bigg{(}\frac{\delta}{2L^{d-1}}\bigg{)}^{n}\zeta(\mathcal{G})\sum_{\mathscr{I}}\epsilon_{\mathscr{I}}\int_{\mathcal{I}}\prod_{\mathfrak{n}\in\mathcal{N}}e^{\zeta_{\mathfrak{n}}\pi
i\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}\mathrm{d}t_{\mathfrak{n}}\prod_{\mathfrak{f}}\bm{\delta}(t_{\mathfrak{f}^{\mathrm{pr}}}-s){\prod_{\mathfrak{f}\neq\mathfrak{l}\in\mathcal{L}}^{(+)}F_{\mathfrak{L}_{\mathfrak{l}}}(k_{\mathfrak{l}})},$
(12.2)

which are slight modifications of (3.33) and (3.24). Here in (12.1),
$\mathscr{D}$ is a $(k,k^{\prime})$-decoration of $\mathcal{T}$, $\mathcal{D}$
is defined as in (3.34), and the other objects are associated with the tree
$\mathcal{T}$. In (12.2), $\mathscr{I}$ is a $(k,k^{\prime})$-decoration of
$\mathcal{G}$, the other objects are associated with the garden $\mathcal{G}$,
and the set $\mathcal{I}$ is defined as in (3.25); the second product is taken
over all the flowers $\mathfrak{f}$ in $\mathcal{G}$, where
$\mathfrak{f}^{\mathrm{pr}}$ is the parent node of $\mathfrak{f}$, and in the
last product we assume $\mathfrak{l}$ has sign $+$ and is not any flower
$\mathfrak{f}$ in $\mathcal{G}$.

Now we define the parametrix of the operator $1-\mathscr{L}_{1}$. Define the
operator

$(\mathscr{X}\textit{{b}})_{k}(t)=\sum_{\zeta\in\\{\pm\\}}\sum_{k^{\prime}}\int_{p}^{t}\mathscr{X}_{kk^{\prime}}^{\zeta}(t,s)b_{k^{\prime}}(s)\,\mathrm{d}s;\quad\textit{{b}}=b_{k^{\prime}}(s),$
(12.3)

with the kernel given by

$\mathscr{X}_{kk^{\prime}}^{\zeta}(t,s)=\sum_{\mathcal{T}}\widetilde{\mathcal{J}}_{\mathcal{T}}(t,s,k,k^{\prime}),$
(12.4)

where the summation is taken over all flower trees $\mathcal{T}$ of order
$\leq N_{p+1}$ such that $\zeta_{\mathfrak{r}}=+$ and
$\zeta_{\mathfrak{f}}=\zeta$ for the root $\mathfrak{r}$ and flower
$\mathfrak{f}$.

Let $\mathscr{Y}=(1-\mathscr{L}_{1})\mathscr{X}-1$ and
$\mathscr{W}=\mathscr{X}(1-\mathscr{L}_{1})-1$. Note that the operator
$\mathscr{L}_{1}$ defined in (3.39) formally corresponds to attaching two
trees to a single node as siblings, as in the proof of Proposition 11.2 of
[15], we then see that $\mathscr{Y}$ and $\mathscr{W}$ both have the same
expression as in (12.3) and (12.4), but with the sum in (12.4) taken over
different sets of $\mathcal{T}$ (cf. Section 11.1.1 of [18]). Namely, for
$\mathscr{Y}$ we require that (Y-1) the order $n$ of each tree $\mathcal{T}$
in the sum satisfy $n>N_{p+1}$, and (Y-2) the subtree rooted at each child
node of $\mathfrak{r}$ has order $\leq N_{p+1}$; for $\mathscr{W}$ we require
that (W-1) the order $n>N_{p+1}$, and (W-2) the subtree rooted at each sibling
node of $\mathfrak{f}$ has order $\leq N_{p+1}$, and (W-3) the flower tree
obtained by replacing the parent $\mathfrak{f}^{\mathrm{pr}}$ of
$\mathfrak{f}$ with a new flower has also order $\leq N_{p+1}$. See Figure 20
for illustration. Note that the above requirement imposes that the order of
$\mathcal{T}$ satisfies $n\in[N_{p+1}+1,3N_{p+1}+1]$ for both $\mathscr{Y}$
and $\mathscr{W}$.

Figure 20. A flower tree $\mathcal{T}$ with root $\mathfrak{r}$ and flower
$\mathfrak{f}$. Let the order of subtrees $\mathcal{T}_{j}$ be $n_{j}$ etc.,
and the order of $\mathcal{T}$ be $m$. Then the requirements for
$\mathscr{Y}_{m}$ is that $N_{p+1}<m\leq
N_{p+1}+n_{1}^{\prime}+n_{2}^{\prime}+1$ and $n_{j}^{\prime}\leq N_{p+1}$, the
requirements for $\mathscr{W}_{m}$ is that $N_{p+1}<m\leq
N_{p+1}+n_{1}+n_{2}+1$ and $n_{j}\leq N_{p+1}$.

######  Proposition 12.2.

Let $P$ be an integer that is large enough depending on $C_{1}$, assume also
that $\delta$ is small enough depending on $P$. Let $\mathbb{E}_{p}$ be as in
Proposition 3.9. Then for any $(t,s,k,k^{\prime})$ we have

$\mathbb{E}_{p}|\mathscr{X}_{kk^{\prime}}^{\zeta}(t,s)|^{2P}\leq\langle
k-\zeta k^{\prime}\rangle^{-4\Lambda_{p+1}P}L^{(100d)^{3}P}.$ (12.5)

The same estimate holds for $\mathscr{Y}$ and $\mathscr{W}$, but with an extra
factor $e^{-100N_{p+1}}$ on the right hand side.

Next, let $M_{1},M_{2}\geq L$ be two fixed dyadic numbers. Consider the
definition (12.1) of $\widetilde{\mathcal{J}}_{\mathcal{T}}$ that occurs in
(12.4). Suppose we restrict to

$\max\\{|k_{\mathfrak{l}}|:\mathfrak{f}\neq\mathfrak{l}\in\mathcal{L}\\}\sim_{0}M_{1}$

(or with $\sim_{0}$ replaced by $\lesssim_{0}$ if $M_{1}=L$) by a smooth
cutoff function for the tree $\mathcal{T}$, where $\mathfrak{f}$ is the flower
of $\mathcal{T}$ and $\mathcal{L}$ the set of leaves, then (12.5) holds with
an extra factor $M_{1}^{-80dP}$ on the right hand side; the same is true for
$\mathscr{Y}$ and $\mathscr{W}$.

Moreover, suppose $|k|\geq M_{1}^{10}$, and in (12.1) we further restrict to

$\max_{\mathfrak{n}\in\mathcal{S}}|\Omega_{\mathfrak{n}}|\sim_{0}M_{2}$

(or with $\sim_{0}$ replaced by $\lesssim_{0}$ if $M_{2}=L$) by a smooth
cutoff function for the tree $\mathcal{T}$, where $\mathcal{S}$ is the set of
all branching nodes $\mathfrak{n}$ in the stem of $\mathcal{T}$ _other than_
the parent node $\mathfrak{f}^{\mathrm{pr}}$ of $\mathfrak{f}$, then (12.5)
holds with an extra factor $M_{2}^{-P/9}$ on the right hand side.

###### Proof.

We follow the arguments in Section 3.6 used to expand (3.41) based on (3.33),
and now apply similar arguments to expand the left hand side of (12.5) based
on (12.1); note that now we are dealing with moments instead of cumulants of
$(a_{k_{\mathfrak{l}}}(p))$, so we will make no irreducibility assumption
about flower gardens. This gives, upon application of (3.10), that

$\mathbb{E}_{p}|\mathscr{X}_{kk^{\prime}}^{\zeta}(t,s)|^{2P}=\sum_{\mathcal{G}}\widetilde{\mathcal{K}}_{\mathcal{G}}(t,s,k,k^{\prime})+\mathscr{E}^{(1)},$
(12.6)

where the sum is taken over all flower gardens
$\mathcal{G}=(\mathcal{T}_{1},\cdots,\mathcal{T}_{2R})\in\mathscr{H}_{p+1}^{\mathrm{tr}}$
of width $2P$, and $\mathscr{E}^{(1)}$ is a remainder term containing one of
$\mathscr{E}_{p}$ or $\widetilde{\mathscr{E}}_{p}$ in (3.28) and (3.30). Here
$\mathscr{H}_{p+1}^{\mathrm{tr}}$ is a subset of $\mathscr{H}_{p+1}$ defined
similar to $\mathscr{G}_{p+1}^{\mathrm{tr}}$ in Definition 3.10, where we
require in Definition 12.1 that each of the initial flower trees has order
$\leq N_{p+1}$, and that each of the layered gardens replacing leaf subsets
has order $\leq N_{p}$. For $\mathscr{Y}$ and $\mathscr{W}$ we have the same
formulas, but we also require the initial trees to satisfy (Y-1)–(Y-2) or
(W-1)–(W-3) above.

We may treat $\mathscr{E}^{(1)}$ just as in the proof of Proposition 3.9 in
Section 3.7; moreover, if the order of $\mathcal{G}$ is
$n(\mathcal{G})>N_{p+1}^{4}$, then the resulting term $\mathscr{E}^{(2)}$ can
also be treated just as in the proof of Proposition 3.9. Thus we may restrict
to $n(\mathcal{G})\leq N_{p+1}^{4}$ in (12.6), so the definition of
$\mathscr{H}_{p+1}^{\mathrm{tr}}$ is equivalent to each tree of $\mathcal{G}$
having $\leq N_{p}$ layer $p$ branching nodes. Similarly, the extra
requirements for $\mathscr{Y}$ and $\mathscr{W}$ are also just (Y-1)–(Y-2) or
(W-1)–(W-3), but with the notion of order in these requirements being replaced
by the number of layer $p$ branching nodes.

Now, to estimate (12.6), we simply repeat the arguments in Sections 4–11
devoted to the proof of Proposition 3.11. Below we only discuss the necessary
(minor) adjustments needed in the flower garden case here, as most of the
proof is completely analogous.

  1. (1)

Regarding the decay factor $\langle k-\zeta
k^{\prime}\rangle^{-4\Lambda_{p+1}P}$: this is obtained by the same arguments
as in Section 11.1, using the decay of
$F_{\mathfrak{L}_{\mathfrak{l}}}(k_{\mathfrak{l}})$ in (12.2); note that an
algebraic sum of $k_{\mathfrak{l}}$ for all non-flower leaves in each tree
equals $k-\zeta k^{\prime}$. The extra factor $M_{1}^{-80dP}$ when assuming
$\max\\{|k_{\mathfrak{l}}|:\mathfrak{f}_{j}\neq\mathfrak{l}\in\mathcal{L}_{j}\\}\sim_{0}M_{1}$
comes in the same way (note that if we insert some cutoff to the tree
$\mathcal{T}$ associated to $\mathscr{X}$, then the same cutoff will appear in
each tree $\mathcal{T}_{j}$ occurring in the garden $\mathcal{G}$ associated
with the expansion (12.6)).

  2. (2)

Regarding the extra power $L^{(100d)^{3}P}$: compared to (3.24), the
modification in (12.2) in the flower garden case only involves two nodes (the
flower $\mathfrak{f}$ and its parent $\mathfrak{f}^{\mathrm{pr}}$) for each
tree, and each such modification may cause a loss of $\leq L^{80d}$. For
example, this loss may come from taking the $\mathfrak{S}^{40d,40d}$ norm of
the input function $\mathbf{1}_{k_{\mathfrak{f}}=k^{\prime}}$ which occurs due
to the definition of $(k,k^{\prime})$-decorations, or from the atoms
corresponding to these one or two nodes in the molecule. They may also come
from the $\bm{\delta}(t_{\mathfrak{f}^{\mathrm{pr}}}-s)$ factor which removes
the integration in the $t_{\mathfrak{f}^{\mathrm{pr}}}$ variable, but
$\Omega_{\mathfrak{f}^{\mathrm{pr}}}$ has at most $L^{10d}$ possible values
once we fix each $k_{\mathfrak{l}}$ to a unit ball, so we may fix the value of
$\Omega_{\mathfrak{f}^{\mathrm{pr}}}$ and safely get rid of the factor
$e^{\zeta_{\mathfrak{n}}\pi i\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}$ for
$\mathfrak{n}=\mathfrak{f}^{\mathrm{pr}}$ at a loss of at most $L^{10d}$. In
any case the total loss is $\leq L^{200dP}$ for the total of $2P$ trees.

  3. (3)

Regarding the arbitrariness of $k$ and $k^{\prime}$: all the proofs in Section
4 through Section 11 are translation invariant161616Except those leading to
the $\langle k_{j}\rangle^{-4\Lambda_{p+1}}$ factors in Proposition 3.11; but
these factors are replaced by the $\langle k-\zeta
k^{\prime}\rangle^{-4\Lambda_{p+1}P}$ factor, which is treated in (1)., so
this is not a problem once $k$ and $k^{\prime}$ are fixed.

  4. (4)

Regarding LF twisting and cancellation: we only need to consider those LF
twists for $\mathcal{G}$ that may affect the property of belonging to
$\mathscr{H}_{p+1}^{\mathrm{tr}}$. Note that the definition of
$\mathscr{H}_{p+1}^{\mathrm{tr}}$, including the modified versions for
$\mathscr{Y}$ and $\mathscr{W}$, only involves the number of layer $p$
branching nodes in (i) each tree $\mathcal{T}_{j}$ of $\mathcal{G}$, (ii) each
of the trees rooted at the children nodes of each root $\mathfrak{r}_{j}$, and
(iii) each of the trees rooted at the sibling nodes of each flower
$\mathfrak{f}_{j}$, see Figure 20. Therefore, if the realization of a vine
$\mathbb{V}$ in $\mathcal{G}_{\mathrm{sk}}$ does not contain any of the
“special nodes”, i.e. the root $\mathfrak{r}_{j}$, any of the children nodes
of $\mathfrak{r}_{j}$, the flower $\mathfrak{f}_{j}$, the parent and any
sibling node of $\mathfrak{f}_{j}$, then any LF twist at this vine will not
affect the property of belonging to $\mathscr{H}_{p+1}^{\mathrm{tr}}$. Clearly
there are less than $20P$ vines containing one of these special nodes; for
each of them we can use Proposition 10.4 to treat its ladder parts, and sum
and integrate trivially over the $k_{\ell}$ and $t_{v}$ variables for $\ell$
and $v$ outside of the ladders (there are at most $20$ such variables per
vine). This leads to a loss of at most $L^{20d}$ per vine, and at most
$L^{400dP}$ in total.

  5. (5)

Similarly, the property that all the nodes on the stem of each tree must
belong to layer $p$, as in the definition of flower gardens in Definition
12.1, is also preserved under LF twisting. For example, consider $\mathcal{G}$
whose skeleton contains a Vine (II-a) in Figure 16, while the skeleton of its
LF twist $\mathcal{G}^{\prime}$ contains a Vine (II-b). We may assume the
flower is in the tree $\mathcal{T}_{1}$ or $\mathcal{T}_{2}$ in the notation
of Figure 16 (otherwise it is not affected by LF twisting); now if all stem
nodes in $\mathcal{G}$ are in layer $p$, then in particular
$\mathfrak{L}_{\mathfrak{u}_{2}}=p$, which implies that
$\mathfrak{L}_{\mathfrak{u}_{3}}=\mathfrak{L}_{\mathfrak{u}_{4}}=p$ as
$\mathfrak{L}_{\mathfrak{u}_{2}}\leq\min(\mathfrak{L}_{\mathfrak{u}_{3}},\mathfrak{L}_{\mathfrak{u}_{4}})$.
Thus all stem nodes in $\mathcal{G}^{\prime}$ must also be in layer $p$,
including those in the regular tree $\mathcal{T}_{(\mathfrak{u}_{2})}$,
because the layer of $\mathfrak{u}_{2}$ in the skeleton is given by the layer
of the lone leaf of the regular tree at $\mathfrak{u}_{2}$ in $\mathcal{G}$ or
$\mathcal{G}^{\prime}$.

  6. (6)

Regarding the extra factor $M_{2}^{-P/9}$: if we insert the $M_{2}$ cutoff,
then for each $j$ we fix a branching node $\mathfrak{n}_{j}\in\mathcal{S}_{j}$
and insert the cutoff such that $|\Omega_{\mathfrak{n}_{j}}|\sim_{0}M_{2}$
(otherwise these cutoffs play no role in the proof). Note for each $j$, the
expression (12.2) involves an integration of the exponential factor
$e^{\zeta_{\mathfrak{n}_{j}}\pi i\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}_{j}}t_{\mathfrak{n}_{j}}}$ in the time
variable $t_{j}$. By integrating in $t_{j}$ before manipulating everything
else, and arguing similarly as in the proof of Corollary 11.3 in [15], we gain
a factor $M_{2}^{-P/9}$ from this integration. This also leads to a loss of at
most $L^{20dP}$ by fixing the values of $\Omega_{\mathfrak{n}_{j}}$ as in (2)
, which is acceptable. We then proceed with the same arguments in Section 4
through Section 11 to treat the rest of the expression (12.2) involving the
rest of the garden. All the possible loss (such as those described in (2))
that may come along the way again add up to at most $L^{100dP}$ which is
acceptable.

  7. (7)

Regarding LF twisting in the context of (6): here we are assuming $|k|\geq
M_{1}^{10}$. For any (CL) vine $\mathbb{V}$ in
$\mathbb{M}(\mathcal{G}_{\mathrm{sk}})$ that is SG, let the nodes
$(\mathfrak{u}_{1},\mathfrak{u}_{11},\mathfrak{u}_{21},\mathfrak{u}_{22})$ be
as in Proposition 7.7, then we have the following: if the flower is a
descendant of $\mathfrak{u}_{1}$, then it must be a descendant of
$\mathfrak{u}_{11}$. In fact, if this is not true, then the flower must be a
descendant of $\mathfrak{u}_{21}$ or $\mathfrak{u}_{22}$, which means that
exactly one of the nodes $\mathfrak{u}_{21}$ and $\mathfrak{u}_{22}$ is on the
stem. But this is not possible, as
$|k_{\mathfrak{u}_{21}}-k_{\mathfrak{u}_{22}}|\leq 1$ by the SG assumption,
while the difference $|k_{\mathfrak{n}}-k_{\mathfrak{n}^{\prime}}|$ for any
node $\mathfrak{n}$ on stem and $\mathfrak{n}^{\prime}$ off stem is at least
$|k|/2$ since $|k|\geq M_{1}^{10}$. Therefore, see Figure 16, we know that any
LF twisting for $\mathcal{G}$ will not affect the structure of the stem. In
particular it will not affect the value of any $\Omega_{\mathfrak{n}}$ for any
$\mathfrak{n}$ on the stem of any tree, nor the smooth cutoff functions of
these $\Omega_{\mathfrak{n}}$ involving $M_{2}$. The rest of the discussion
for LF twisting is the same as in (4) and (5).

With the above discussions, we can then obtain the desired bound for (12.6):
if we restrict to $\mathcal{G}$ of order $n$, then this contribution is
bounded by

$\langle k-\zeta
k^{\prime}\rangle^{-4\Lambda_{p+1}P}L^{(100d)^{3}P}\cdot\delta^{2\nu n},$

with the corresponding extra factors if the cutoff functions involving $M_{1}$
or $M_{2}$ are present. By adding up all $n\geq 0$ for $\mathscr{X}$, and
adding up all $N_{p+1}+1\leq n\leq N_{p+1}^{4}$ for $\mathscr{Y}$ and
$\mathscr{W}$, this completes the proof of (12.5), as well as the variants
with the cutoff functions. ∎

###  12.2. Proof of Proposition 3.12

In this section we prove Proposition 3.12.

Part 1: Proof of (3.50) and (3.51). Recall that

$a_{k}^{\mathrm{tr}}(t)=\sum_{\mathcal{T}}(\mathcal{J}_{\mathcal{T}})_{k}(t)$
(12.7)

from (3.35) and (3.36), where the sum is taken over all trees of sign $+$ and
order $\leq N_{p+1}$. By repeating the same arguments in the proof of
Proposition 3.9 in Section 3.7, for each fixed $k\in\mathbb{Z}_{L}^{d}$ and
$t\in[p,p+1]$ we can bound

$\mathbb{E}_{p}|(a_{k}^{\mathrm{tr}})(t)|^{2}\leq L^{2}\langle
k\rangle^{-4\Lambda_{p+1}}.$ (12.8)

In the same way, by (3.39), we can write $(\mathscr{L}_{0})_{k}(t)$ in a form
similar to (12.7), but with the restriction that the order of $\mathcal{T}$ is
$>N_{p+1}$, while the order of the subtree rooted at each child node of the
root $\mathfrak{r}$ is $\leq N_{p+1}$. The same proof can be carried out, and
with the lower bound on the order of $\mathcal{T}$, using (3.45) and (3.49),
we get the same bound (12.8) for $\mathscr{L}_{0}$ but with extra factor
$e^{-100N_{p+1}}$ on the right hand side. In addition, we can treat the time
derivative $\partial_{t}a_{k}^{\mathrm{tr}}$ (which simply contains an extra
$\bm{\delta}(t_{\mathfrak{r}}-t)$ factor in $\mathcal{J}_{\mathcal{T}}$
similar to the one in (12.1)) using the same arguments in the proof of
Proposition 12.2, to get the same bound (12.8) with an additional $L^{100d}$
loss. The same holds also for $\partial_{t}(\mathscr{L}_{0})_{k}(t)$.

Now note that, (3.50) and (3.51), compared to (12.8), requires large deviation
estimates for the _supremum_ of $\langle
k\rangle^{(3/2)\Lambda_{p+1}}|a_{k}^{\mathrm{tr}}(t)|$ and $\langle
k\rangle^{(3/2)\Lambda_{p+1}}|(\mathscr{L}_{0})_{k}(t)|$. This can be solved
using the following trick in [15]: by Sobolev in $t$ we have

$\sup_{t,k}\big{(}\langle
k\rangle^{(3/2)\Lambda_{p+1}}|a_{k}^{\mathrm{tr}}(t)|\big{)}\lesssim_{0}\big{\|}\langle
k\rangle^{(3/2)\Lambda_{p+1}}a_{k}^{\mathrm{tr}}(t)\big{\|}_{L_{k,t}^{2}}+\big{\|}\langle
k\rangle^{(3/2)\Lambda_{p+1}}\partial_{t}a_{k}^{\mathrm{tr}}(t)\big{\|}_{L_{k,t}^{2}},$
(12.9)

where the norm is taken for $k\in\mathbb{Z}_{L}^{d}$ and $t\in[p,p+1]$ (here
$L_{k}^{2}$ is with respect to the counting measure). Then, using summability
in $k$, we can bound the second moment of the above supremum by the right hand
side of (12.8) with an additional $L^{200d}$ loss. The bounds (3.50) and
(3.51) then follow from the standard Chebyshev inequality.

Part 2: Proof of (3.52). Let $Z:=Z^{(5/4)\Lambda_{p+1}}([p,p+1])$, and recall
the definition of $\mathscr{X},\mathscr{Y},\mathscr{W}$ in Section 12.1. For
any operator $\mathscr{K}$ with kernel
$\mathscr{K}_{kk^{\prime}}^{\zeta}(t,s)$ as in (12.3), it is clear that

$\|\mathscr{K}\|_{Z\to Z}\leq L^{d}\sup_{\zeta,k,k^{\prime},t,s}\langle
k-\zeta
k^{\prime}\rangle^{(3/2)\Lambda_{p+1}}|\mathscr{K}_{kk^{\prime}}^{\zeta}(t,s)|:=L^{d}\|\mathscr{K}\|_{\mathfrak{N}}.$

Our goal is to prove large deviation estimates for
$\|\mathscr{X}\|_{\mathfrak{N}}$; the cases of $\mathscr{Y}$ and $\mathscr{W}$
are analogous.

Let $M_{1},M_{2}\geq L$. We insert smooth cutoff functions, which are just the
ones described in Proposition 12.2, to the
$\widetilde{\mathcal{J}}_{\mathcal{T}}$ terms defined in (12.1) that occur in
(12.4). Denote the resulting contribution by $\mathscr{X}_{M_{1}M_{2}}$ (for
$|k|\leq M_{1}^{10}$ we do not insert the $M_{2}$ cutoff, in which case we
write $\mathscr{X}_{M_{1}}$ instead of $\mathscr{X}_{M_{1}M_{2}}$). The large
deviation estimate we will prove is of form

$\mathbb{E}_{p}\|\mathscr{X}_{M_{1}M_{2}}\|_{\mathfrak{N}}^{2P}\leq
L^{C_{2}}M_{1}^{-P}M_{2}^{-P/18}$ (12.10)

(for $\mathscr{X}_{M_{1}}$ there is no $M_{2}^{-P/18}$ factor), and similar
bounds for $\mathscr{Y}$ and $\mathscr{W}$ but with the extra factor
$e^{-100N_{p+1}}$ on the right hand side. The proof of (12.10) is very similar
to the proof of Proposition 12.2 in [15], where we apply Lemma A.6 to
represent $k$ by a finite system and thus reduce to finitely many
possibilities of $k$ (under suitable cutoffs), and estimate the moment for
each fixed $k$ by Proposition 12.2. Below we will only sketch the main points.

  1. (1)

Like the cases of (3.50) and (3.51) above, if we fix $(k,k^{\prime},t,s)$,
then the desired bound follows from (12.5); the main difficulty comes from the
fact that we are taking $L^{\infty}$ norms in $(k,k^{\prime})$ and $(t,s)$.

  2. (2)

For the $L^{\infty}$ norm in $(k,k^{\prime})$, we only need to consider the
case when $|k|\geq M_{1}^{10}$. In fact if $|k|\leq M_{1}^{10}$ then
$(k,k^{\prime})$ has at most $M_{1}^{20d}$ choices, so we may replace the
$L_{k,k^{\prime}}^{\infty}$ norm by the $L_{k,k^{\prime}}^{2P}$ norm which
allows to fix $(k,k^{\prime})$ and apply Proposition 12.2. The subsequent
summation in $(k,k^{\prime})$ loses at most $M_{1}^{20d}$ which is acceptable
in (12.10).

  3. (3)

In the same way, we may also assume $|k|\geq M_{2}^{10}$ (otherwise we lost at
most $M_{2}^{20d}$ from summation in $(k,k^{\prime})$, which can be covered by
the $M_{2}^{-P/9}$ gain in Proposition 12.2 now that we know $|k|\geq
M_{1}^{10}$). Using the definition of $M_{2}$ and the formula for
$\Omega_{\mathfrak{n}}$, this in particular implies that all branching nodes
in the stem of $\mathcal{T}$ must have the same sign (as, if two consecutive
branching nodes in the stem have opposite signs, then the corresponding
resonance factor will satisfy $|\Omega_{\mathfrak{n}}|\geq|k|^{2}$ which is
not possible).

  4. (4)

Note that the expression (12.1), hence $\mathscr{X}_{M_{1}M_{2}}$, depends on
$k$ and $k^{\prime}$ _only via_ the $\Omega_{\mathfrak{n}}$ for branching
nodes $\mathfrak{n}$ in the stem; moreover each such $\Omega_{\mathfrak{n}}$
has the form $\langle k,z\rangle+\widetilde{\Omega}$, where $z$ is a vector
depending only on the $k_{\mathfrak{l}}$ variables for non-flower leaves
$\mathfrak{l}$ (in particular $|z|\leq M_{1}^{2}$), and $\widetilde{\Omega}$
also depends only on these $k_{\mathfrak{l}}$ variables (in particular
$|\widetilde{\Omega}|\leq M_{1}^{4}$).

  5. (5)

In addition, note from (12.1) and the
$\bm{\delta}(t_{\mathfrak{f}^{\mathrm{pr}}}-s)$ factor that
$\widetilde{\mathcal{J}}_{\mathcal{T}}=e^{\zeta_{\mathfrak{n}}\pi i\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}s}\cdot{\mathcal{J}}_{\mathcal{T}}^{\dagger}$,
where $\mathfrak{n}=\mathfrak{f}^{\mathrm{pr}}$ and
${\mathcal{J}}_{\mathcal{T}}^{\dagger}$ does not depend on
$\Omega_{\mathfrak{f}^{\mathrm{pr}}}$. Since
$\Omega_{\mathfrak{f}^{\mathrm{pr}}}$ has at most $M_{1}^{10d}$ choices (these
choices may depend on $k$ and $k^{\prime}$), we may fix its value at a loss of
at most $M_{1}^{10d}$ (which is amplified to $M_{1}^{20dP}$ with the $2P$
power, but still can be absorbed by the $M_{1}^{-80dP}$ factor in Proposition
12.2). Then we may get rid of the exponential factor and replace
$\widetilde{\mathcal{J}}_{\mathcal{T}}$ by
$\mathcal{J}_{\mathcal{T}}^{\dagger}$ in the definition of
$\mathscr{X}_{M_{1}M_{2}}$.

  6. (6)

Since $|\Omega_{\mathfrak{n}}|\leq M_{2}$ for all branching nodes
$\mathfrak{n}\neq\mathfrak{f}^{\mathrm{pr}}$ on the stem, we know that also
$|\langle k,z\rangle|\leq 2M_{2}$ (and $|z|\leq M_{1}^{2}$) in the context of
(4) above. By Lemma A.6, we can find $k_{*}\in\mathfrak{Rep}$, where
$\mathfrak{Rep}$ is a fixed subset of $\mathbb{Z}_{L}^{d}$ of at most
$M_{2}^{C_{0}}$ elements171717This part of argument is easier than the
irrational torus case [15], as in [15] we also have $y\in\mathbb{R}^{q}$ and
thus need an extra Sobolev argument in $y$., such that (A.20) holds. In this
case, using the above bounds for $\langle k,z\rangle$, we conclude that
$(\mathscr{X}_{M_{1}M_{2}})_{kk^{\prime}}^{\zeta}(t,s)=(\mathscr{X}_{M_{1}M_{2}})_{k_{*}k_{*}^{\prime}}^{\zeta}(t,s)$
with the $M_{2}$ cutoffs, where $k_{*}^{\prime}$ is such that
$k_{*}-k_{*}^{\prime}=k-k^{\prime}$ (we may assume
$k_{*}^{\prime}\in\mathfrak{Rep}$ also, as $|k-k^{\prime}|\leq M_{1}^{2}$). We
can then replace the supremum in $(k,k^{\prime})$ by the supremum in
$(k_{*},k_{*}^{\prime})$ and control it by the
$L_{(k_{*},k_{*}^{\prime})\in\mathfrak{Rep}^{2}}^{2P}$ norm, with a loss of at
most $M_{2}^{C_{0}}$ due to summation in $(k_{*},k_{*}^{\prime})$, which is
acceptable in view of the gain $M_{2}^{-P/9}$ in Proposition 12.2.

  7. (7)

Finally, for the $L^{\infty}$ norm in $(t,s)$, we shall apply Gagliardo-
Nirenberg inequality in slightly more precise way than (12.9) (this applies to
any of the cases considered above). Namely, concerning only the $(t,s)$ part,
we have

$\sup_{t,s}|\mathscr{K}(t,s)|^{2P}\leq
M_{2}^{P/100}\cdot\|\mathscr{K}\|_{L_{t,s}^{2P}}^{2P}+M_{2}^{-100P}\cdot\|\partial_{t,s}\mathscr{K}\|_{L_{t,s}^{2P}}^{2P}$
(12.11)

for any function $\mathscr{K}$. Note that the bound in Proposition 12.2
without $\partial_{t,s}$ derivative gains a factor $M_{2}^{-P/9}$ which
cancels the $M_{2}^{P/100}$ factor in (12.11).

  8. (8)

On the other hand, the term in (12.11) with $\partial_{t,s}$ derivative can be
estimated in the same way as in Proposition 12.2 _after replacing_
$\widetilde{\mathcal{J}}_{\mathcal{T}}$ by
$\mathcal{J}_{\mathcal{T}}^{\dagger}$ as in (5) above (here the $\partial_{t}$
derivative introduces an extra $\bm{\delta}(t_{\mathfrak{r}}-t)$ factor, while
the $\partial_{s}$ derivative introduces an extra
$\bm{\delta}(t_{\mathfrak{n}}-s)$ factor with $\mathfrak{n}$ being the parent
or a non-flower child of $\mathfrak{f}^{\mathrm{pr}}$, and all these factors
can be treated just as the one in (12.1); the case when
$\mathfrak{f}^{\mathrm{pr}}=\mathfrak{r}$ only needs some trivial
adjustments). This leads to a bound similar to (12.5) but with extra loss of
at most $L^{(100d)^{3}P}$ and possibly without the $M_{2}^{-P/9}$ decay, but
we still have the $M_{2}^{-100P}$ weight in (12.11) which provides the needed
decay.

By putting (1) through (8) together, we arrive at the proof of (12.10). Now,
with (12.10) and a simple Chebyshev argument, it is easy to see that by
excluding a subset $\mathfrak{G}\subset\mathfrak{F}_{p}$, which has gauge and
space translation symmetry and has measure $\leq e^{-4N_{p+1}}$ (so
$\mathfrak{F}_{p+1}=\mathfrak{F}_{p}\backslash\mathfrak{G}$), we have that

$\|\mathscr{Y}\|_{Z\to Z}+\|\mathscr{W}\|_{Z\to Z}\leq
1/2,\quad\|\mathscr{X}\|_{Z\to Z}\leq e^{2N_{p+1}}.$

By Neumann series, this means that the operators
$(1-\mathscr{L}_{1})\mathscr{X}$ and $\mathscr{X}(1-\mathscr{L}_{1})$ are both
invertible. Therefore $1-\mathscr{L}_{1}$ has both left and right inverses,
hence $1-\mathscr{L}_{1}$ is invertible and

$\|(1-\mathscr{L}_{1})^{-1}\|_{Z\to Z}\leq 2\|\mathscr{X}\|_{Z\to Z}\leq
e^{3N_{p+1}}.$

This proves (3.52).

Part 3: Proof of (3.53). This is essentially trivial once we have
(3.50)–(3.52), if we just use the easily established facts that

$\|\mathscr{L}_{3}(\textit{{b}},\textit{{b}},\textit{{b}})\|_{Z}\leq
L^{30d}\|\textit{{b}}\|_{Z}^{3},\quad\|\mathscr{L}_{2}(\textit{{b}},\textit{{b}})\|_{Z}\leq
L^{30d}\|\textit{{b}}\|_{Z}^{2}\cdot\|a_{k}^{\mathrm{tr}}\|_{Z}\leq
e^{4N_{p+1}}\|\textit{{b}}\|_{Z}^{2}.$

This completes the proof of Proposition 3.12.

##  Appendix A Miscellaneous results

In this appendix we collect some useful lemmas needed in the main proof.

###  A.1. Cumulants

We start with the properties of cumulants.

######  Lemma A.1 (Property of cumulants).

Let $S_{i}\,(1\leq i\leq r)$ be pairwise disjoint index sets, and
$X_{j}\,(j\in S_{i})$ be complex random variables. Let $Y_{i}:=\prod_{j\in
S_{i}}X_{j}$, then we have

$\mathbb{K}(Y_{1},\cdots,Y_{r})=\sum_{\mathcal{P}}\prod_{B\in\mathcal{P}}\mathbb{K}\big{(}(X_{j})_{j\in
B}\big{)},$ (A.1)

where the sum is taken over all partitions $\mathcal{P}$ of
$S_{1}\cup\cdots\cup S_{r}$, under the assumption that there is no proper
subset $A\subset\\{1,\cdots,r\\}$ such that $\cup_{i\in A}S_{i}$ equals the
union of _some of the sets_ $B\in\mathcal{P}$. The same results hold if the
cumulants are taken assuming a particular event $\mathfrak{F}$.

###### Proof.

Note that by (3.10) we have

$\mathbb{E}(Y_{1}\cdots Y_{r})=\mathbb{E}\bigg{(}\prod_{j\in
X_{1}\cup\cdots\cup
X_{r}}X_{j}\bigg{)}=\sum_{\mathcal{P}}^{(*)}\prod_{B\in\mathcal{P}}\mathbb{K}\big{(}(X_{j})_{j\in
B}\big{)},$ (A.2)

where the sum is taken over all partitions of $S_{1}\cup\cdots\cup S_{r}$,
without the restriction made in (A.1). If we define those partitions
satisfying the restriction in (A.1) as irreducible partitions, then any
partition $\mathcal{P}$ can be uniquely decomposed into irreducible
partitions; i.e. there is a unique partition $\mathcal{P}_{0}$ of
$\\{1,\cdots,r\\}$ such that
$\mathcal{P}=\cup_{A\in\mathcal{P}_{0}}\mathcal{P}_{A}$, where
$\mathcal{P}_{A}$ is an irreducible partition of $\cup_{i\in A}S_{i}$. This
implies that

$\mathbb{E}(Y_{1}\cdots
Y_{r})=\sum_{\mathcal{P}_{0}}\prod_{A\in\mathcal{P}_{0}}\sum_{\mathcal{P}_{A}}\prod_{B\in\mathcal{P}_{A}}\mathbb{K}\big{(}(X_{j})_{j\in
B}\big{)},$ (A.3)

where $\mathcal{P}_{0}$ and $\mathcal{P}_{A}$ is as above. Also by (3.10) we
have that

$\mathbb{E}(Y_{1}\cdots
Y_{r})=\sum_{\mathcal{P}_{0}}\prod_{A\in\mathcal{P}_{0}}\mathbb{K}\big{(}(Y_{i})_{i\in
A}\big{)}.$ (A.4)

Note that both (A.3) and (A.4) remain true (with obvious changes) if
$\\{1,\cdots,r\\}$ is replaced by any subset $A\subset\\{1,\cdots,r\\}$. It
then follows, by induction on $|A|$, that (A.1) also holds (with obvious
changes) if $\\{1,\cdots,r\\}$ is replaced by any subset
$A\subset\\{1,\cdots,r\\}$, and lastly setting $A=\\{1,\cdots,r\\}$ yields
(A.1). The results assuming $\mathfrak{F}$ follows in the same way. ∎

###  A.2. Analytical lemmas

Next we prove some analytical lemmas concerning time integrations.

######  Lemma A.2.

Consider the operators

$\displaystyle\mathcal{I}_{\chi}F(t,s)$
$\displaystyle=\int_{0}^{t}\int_{0}^{s}\chi(t)\chi(t^{\prime})\chi(s)\chi(s^{\prime})F(t^{\prime},s^{\prime})\,\mathrm{d}t^{\prime}\mathrm{d}s^{\prime},$
(A.5) $\displaystyle\mathcal{J}_{\chi}F(t,s,k)$
$\displaystyle=\int_{s}^{t}\int_{s}^{t^{\prime}}\chi(t)\chi(t^{\prime})\chi(s)\chi(s^{\prime})F(t^{\prime},s^{\prime},k)\,\mathrm{d}s^{\prime}\mathrm{d}t^{\prime},$

where $\chi$ is a smooth cutoff function. Then, for _complex valued_ $F$, we
have

$\|\mathcal{I}_{\chi}F\|_{\mathfrak{X}^{\eta}}\lesssim_{1}\|F\|_{L^{1+3\eta}(\mathbb{R}^{2})},\qquad\|\mathcal{J}_{\chi}F\|_{\mathfrak{X}^{-\eta,0,0}}\lesssim_{1}\|F\|_{L_{k}^{\infty}L^{1}(\mathbb{R}^{2})}.$
(A.6)

###### Proof.

For the first inequality, from Lemma 3.1 of [20] we have

$\widehat{\mathcal{I}_{\chi}F}(\xi,\sigma)=\int_{\mathbb{R}^{2}}K(\xi,\sigma,\xi^{\prime},\sigma^{\prime})\widehat{F}(\xi^{\prime},\sigma^{\prime})\,\mathrm{d}\xi^{\prime}\mathrm{d}\sigma^{\prime},$
(A.7)

where the kernel $K$ satisfies

$|K|\lesssim_{0}\frac{1}{\langle\xi\rangle\langle\xi-\xi^{\prime}\rangle\langle\sigma\rangle\langle\sigma-\sigma^{\prime}\rangle}.$
(A.8)

By the definition of $\mathfrak{X}^{\eta}$ norm (and assuming $F$ has compact
support), we then have

$\displaystyle\|\mathcal{I}_{\chi}F\|_{\mathfrak{X}^{\eta}}$
$\displaystyle\lesssim_{1}\int_{\mathbb{R}^{4}}\langle\xi\rangle^{\eta-1}\langle\sigma\rangle^{\eta-1}\langle\xi-\xi^{\prime}\rangle^{-1}\langle\sigma-\sigma^{\prime}\rangle^{-1}|\widehat{F}(\xi^{\prime},\sigma^{\prime})|\,\mathrm{d}\xi\mathrm{d}\sigma\mathrm{d}\xi^{\prime}\mathrm{d}\sigma^{\prime}$
$\displaystyle\lesssim_{1}\int_{\mathbb{R}^{2}}\langle\xi^{\prime}\rangle^{\eta-1}\langle\sigma^{\prime}\rangle^{\eta-1}|\widehat{F}(\xi^{\prime},\sigma^{\prime})|\,\mathrm{d}\xi^{\prime}\mathrm{d}\sigma^{\prime}$
$\displaystyle\lesssim_{1}\|\widehat{F}\|_{L^{1/(2\eta)}(\mathbb{R}^{2})}\lesssim_{1}\|F\|_{L^{1/(1-2\eta)}(\mathbb{R}^{2})}\lesssim_{1}\|F\|_{L^{1+3\eta}(\mathbb{R}^{2})}.$
(A.9)

As for the second inequality, we can calculate

$\displaystyle\widehat{\mathcal{J}_{\chi}F}(\xi,\sigma,k)$
$\displaystyle=\int_{\mathbb{R}^{2}}e^{-i(t\xi+s\sigma)}\int_{s}^{t}\int_{s}^{t^{\prime}}\chi(t)\chi(t^{\prime})\chi(s)\chi(s^{\prime})F(t^{\prime},s^{\prime},k)\,\mathrm{d}s^{\prime}\mathrm{d}t^{\prime}$
$\displaystyle=\int_{t^{\prime}>s^{\prime}}\chi(t^{\prime})\chi(s^{\prime})F(t^{\prime},s^{\prime},k)\,\mathrm{d}t^{\prime}\mathrm{d}s^{\prime}\int_{t^{\prime}}^{\infty}\int_{-\infty}^{s^{\prime}}\chi(t)\chi(s)e^{-i(t\xi+s\sigma)}\,\mathrm{d}s\mathrm{d}t$
$\displaystyle+\int_{t^{\prime}<s^{\prime}}\chi(t^{\prime})\chi(s^{\prime})F(t^{\prime},s^{\prime},k)\,\mathrm{d}t^{\prime}\mathrm{d}s^{\prime}\int_{-\infty}^{t^{\prime}}\int_{s^{\prime}}^{\infty}\chi(t)\chi(s)e^{-i(t\xi+s\sigma)}\,\mathrm{d}s\mathrm{d}t.$
(A.10)

From this we easily see that
$|\widehat{\mathcal{J}_{\chi}F}(\xi,\sigma,k)|\lesssim_{1}\langle\xi\rangle^{-1}\langle\sigma\rangle^{-1}\|F(t^{\prime},s^{\prime},k)\|_{L^{1}(\mathbb{R}^{2})}$
for each $k$, which finishes the proof in view of the definition (3.11). ∎

######  Lemma A.3.

Consider the following operators

$\displaystyle\mathcal{U}_{\chi}F(t,s,k)$
$\displaystyle=\int_{s}^{t}\int_{s}^{t_{1}}\chi(t)\chi(t_{1})\chi(s)\chi(s_{1})e^{i\Gamma_{1}(k)(t_{1}-s_{1})}F(t_{1},s_{1},k)\,\mathrm{d}s_{1}\mathrm{d}t_{1},$
(A.11) $\displaystyle\mathcal{V}_{\chi}F(t,s,k)$
$\displaystyle=\int_{s}^{t}\int_{s}^{t_{1}}\int_{s}^{t_{2}}\int_{s}^{s_{2}}\chi(t)\chi(t_{1})\chi(t_{2})\chi(s)\chi(s_{1})\chi(s_{2})$
$\displaystyle\qquad\qquad\qquad\qquad\times
e^{i\Gamma_{1}(k)(t_{1}-t_{2})}e^{i\Gamma_{2}(k)(s_{2}-s_{1})}F(t_{2},s_{2},k)\,\mathrm{d}s_{1}\mathrm{d}s_{2}\mathrm{d}t_{2}\mathrm{d}t_{1},$

then for complex valued $F$ we have

$\|\mathcal{U}_{\chi}F\|_{\mathfrak{X}^{1/4,0,0}}\lesssim_{0}\|\partial_{t_{1}}\partial_{s_{1}}F\|_{L_{k}^{\infty}L^{1}(\mathbb{R}^{2})},\qquad\|\mathcal{V}_{\chi}F\|_{\mathfrak{X}^{1/4,0,0}}\lesssim_{0}\|\partial_{t_{2}}\partial_{s_{2}}F\|_{L_{k}^{\infty}L^{1}(\mathbb{R}^{2})}.$
(A.12)

###### Proof.

Assume
$\|\partial_{t_{j}}\partial_{s_{j}}F\|_{L_{k}^{\infty}L^{1}(\mathbb{R}^{2})}=1$,
where $j$ is $1$ or $2$ in the case of $\mathcal{U}_{\chi}$ or
$\mathcal{V}_{\chi}$. We may assume $F$ is compactly supported, then a simple
calculation yields that

$|\widehat{F}(\xi,\sigma,k)|\lesssim_{0}\langle\xi\rangle^{-1}\langle\sigma\rangle^{-1},\qquad|\mathscr{F}(F\cdot\mathbf{1}_{t_{j}>s_{j}})(\xi,\sigma,k)|\lesssim_{0}\langle\xi\rangle^{-1/2}\langle\sigma\rangle^{-1/2}$
(A.13)

uniformly in $k$, where $\mathscr{F}$ is the Fourier transform in $(t,s)$.

Consider first $\mathcal{U}_{\chi}$. By calculations similar to those for
$\mathcal{J}_{\chi}$ in the proof of Lemma A.2, we get

$\widehat{\mathcal{U}_{\chi}F}(\xi,\sigma,k)=\int_{t_{1}>s_{1}}\chi(t_{1})\chi(s_{1})e^{i\Gamma_{1}(k)(t_{1}-s_{1})}F(t_{1},s_{1},k)\,\mathrm{d}t_{1}\mathrm{d}s_{1}\cdot\int_{t_{1}}^{\infty}\chi(t)e^{-it\xi}\,\mathrm{d}t\cdot\int_{-\infty}^{s_{1}}\chi(s)e^{-is\sigma}\,\mathrm{d}s\\\
+(\mathrm{similar\ terms}).$ (A.14)

We may restrict to the case $|\xi|,|\sigma|\geq 1$ (otherwise the proof is
similar and much easier), and integrate by parts in the two integrals in $t$
and $s$. The resulting bulk term involving $\chi^{\prime}$ has the same form
but has better decay in $\xi$ and $\sigma$, and can be treated in the same
way; for convenience we will only consider the hardest case, which is the
boundary term. With this simplification we thus write

$\widehat{\mathcal{U}_{\chi}F}(\xi,\sigma,k)=\frac{1}{\xi\sigma}\int_{t_{1}>s_{1}}\chi^{2}(t_{1})\chi^{2}(s_{1})F(t_{1},s_{1},k)e^{i(\Gamma_{1}(k)-\xi)t_{1}}\cdot
e^{i(-\Gamma_{1}(k)-\sigma)s_{1}}\,\mathrm{d}t_{1}\mathrm{d}s_{1}\\\
+(\mathrm{similar\ or\ better\ terms}).$ (A.15)

Using (A.13) we get that

$|\widehat{\mathcal{U}_{\chi}F}(\xi,\sigma,k)|\lesssim_{0}\frac{1}{\langle\xi\rangle\langle\sigma\rangle}\cdot\frac{1}{\langle\Gamma_{1}(k)-\xi\rangle^{1/2}\langle\Gamma_{1}(k)+\sigma\rangle^{1/2}}\lesssim_{0}\frac{1}{\langle\xi\rangle\langle\sigma\rangle\langle\xi+\sigma\rangle^{1/2}}$
(A.16)

uniformly in $k$, which easily implies (A.12) for $\mathcal{U}_{\chi}$.

The estimate for $\mathcal{V}_{\chi}$ is similar. We have

$\widehat{\mathcal{V}_{\chi}F}(\xi,\sigma,k)=\int_{t_{2}>s_{2}}\chi(t_{2})\chi(s_{2})F(t_{2},s_{2},k)\cdot
G(t_{2},\xi,k)\cdot
H(s_{2},\sigma,k)\,\mathrm{d}t_{2}\mathrm{d}s_{2}+(\mathrm{similar\ terms}),$
(A.17)

where

$\displaystyle G(t_{2},\xi,k)$
$\displaystyle=\int_{t>t_{1}>t_{2}}\chi(t)\chi(t_{1})e^{i\Gamma_{1}(k)(t_{1}-t_{2})}e^{-it\xi}\,\mathrm{d}t\mathrm{d}t_{1},$
(A.18) $\displaystyle H_{j}(s_{2},\sigma,k)$
$\displaystyle=\int_{s<s_{1}<s_{2}}\chi(s)\chi(s_{1})e^{i\Gamma_{2}(k)(s_{2}-s_{1})}e^{-is\sigma}\,\mathrm{d}s\mathrm{d}s_{1}.$

We will assume $|\xi|,|\sigma|\geq 1$. In the expression of $G$, we first
integrate in $t$ and may replace this integral by the boundary term
$(i\xi)^{-1}\chi(t_{1})e^{-it_{1}\xi}$ as above; this yields that

$G(t_{2},\xi,k)=(i\xi)^{-1}e^{-it_{2}\xi}\int_{t_{2}}^{\infty}\chi^{2}(t_{1})e^{i(\Gamma_{1}(k)-\xi)(t_{1}-t_{2})}\,\mathrm{d}t_{1}+(\mathrm{better\
terms}),$

and notice that this remaining integral is bounded in Schwartz class with
respect to $t_{2}$ (for example by a change of variable $t_{1}-t_{2}:=u$),
uniformly in $\xi$ and $k$. The same happens for $H$, and putting together
into (A.17) we get that

$|\widehat{\mathcal{V}_{\chi}F}(\xi,\sigma,k)|\lesssim_{0}\frac{1}{\langle\xi\rangle^{3/2}\langle\sigma\rangle^{3/2}}$
(A.19)

uniformly in $k$, which implies (A.12) for $\mathcal{V}_{\chi}$. ∎

###  A.3. Combinatorial lemmas

Next we present some combinatorial results needed in the proof. Most of them
are the same as in [15, 18], but we include them here for reader’s
convenience.

######  Lemma A.4.

Given a tree $\mathcal{T}$ of order $m$.

  1. (1)

We can order all the branching nodes of $\mathcal{T}$ as
$\mathfrak{n}_{1},\cdots,\mathfrak{n}_{m}$, such that the following holds for
any decoration $(k_{\mathfrak{n}})$: for each $j$ there exists $j^{\prime}<j$
such that $k_{\mathfrak{n}_{j^{\prime}}}-k_{\mathfrak{n}_{j}}$ is an algebraic
sum of at most $\Lambda_{j}$ distinct terms in $(k_{\mathfrak{l}})$ where
$\mathfrak{l}$ ranges over the leaves, and $\prod_{j=1}^{m}\Lambda_{j}\leq
C_{0}^{m}$.

  2. (2)

Moreover, consider the collections of all decorations $(k_{\mathfrak{n}})$ suc
that $|k_{\mathfrak{l}}-k_{\mathfrak{l}}^{0}|\leq 1$ for each leaf
$\mathfrak{l}$, where $k_{\mathfrak{l}}^{0}\in\mathbb{Z}_{L}^{d}$ are fixed
vectors for each leaf $\mathfrak{l}$. Then this collection can be divided into
at most $C_{0}^{m}$ sub-collections, such that for any decoration
$(k_{\mathfrak{n}})$ in a given sub-collection, each $k_{\mathfrak{n}}$
belongs to a fixed unit ball that depends only on $(k_{\mathfrak{n}}^{0})$,
$\mathfrak{n}$ and the sub-collection but not on $(k_{\mathfrak{n}})$ itself.

###### Proof.

See Lemma 6.6 of [15] and Lemma A.4 of [18]. ∎

######  Lemma A.5.

Consider a given molecule with a given decoration. Suppose $\mathbb{U}$ is
either a DV, or vine-like object that is LG or ZG, and assume $\mathbb{U}$
contains some disjoint SG vine-like objects $\mathbb{U}_{j}$ as subsets. Then
each $\mathbb{U}_{j}$ is either one double bond or two double bonds or Vine
(V), or the adjoint of a VC denoted by $\mathbb{U}_{j}^{\prime}$, which is
either Vine (V) or two double bonds. In any case, $\mathbb{U}$ still remains
connected after removing all the $\mathbb{U}_{j}$ (in case $\mathbb{U}_{j}$ is
a vine) or $\mathbb{U}_{j}^{\prime}$ (in case $\mathbb{U}_{j}^{\prime}$ is a
vine).

###### Proof.

First assume $\mathbb{U}$ is a VC or HVC, which is concatenated by one or
multiple vine ingredients. Then each $\mathbb{U}_{j}$ must be a subset of a
single vine ingredient, because the gap of $\mathbb{U}_{j}$ is not the same as
the gap of $\mathbb{U}$, hence $\mathbb{U}_{j}$ cannot be concatenated by any
of the vine ingredients of $\mathbb{U}$. Therefore we may focus on a single
vine ingredient, which is Vine (I)–(VIII) as in Figure 15. The result is then
self-evident by examining the structures of these vines. The case of DV is
similar, since each $\mathbb{U}_{j}$ must be a subset of one of the two vines
(V) forming $\mathbb{U}$. ∎

######  Lemma A.6.

Fix $M\geq L$. Then for any $k=(k^{1},\cdots,k^{d})\in\mathbb{Z}_{L}^{d}$,
there must exist $0\leq q\leq d$ and nonzero orthogonal vectors
$v_{j}\in\mathbb{Z}_{L}^{d}\,(1\leq j\leq q)$, and vector
$y=(y^{1},\cdots,y^{q})\in(L^{-2}\mathbb{Z})^{q}$, such that $|v_{j}|\leq
M^{C_{0}}$ and $|y|\leq M^{C_{0}}$, where $C_{0}$ is a constant depending only
on $d$. Moreover the system $(q,v_{1},\cdots,v_{q},y)$ _represents_ $k$, in
the following sense:

  1. (1)

If $z=(z^{1},\cdots,z^{d})\in\mathbb{Z}_{L}^{d}$, $|z|\leq M$ and $|\langle
k,z\rangle|\leq M$, then $z$ is a linear combination of
$\\{v_{1},\cdots,v_{q}\\}$;

  2. (2)

If $z=\gamma^{1}v_{1}+\cdots+\gamma^{q}v_{q}$, then we have $\langle
k,z\rangle=y^{1}\gamma^{1}+\cdots+y^{q}\gamma^{q}$.

In particular, there exists a subset $\mathfrak{Rep}\subset\mathbb{Z}_{L}^{d}$
with at most $M^{C_{0}}$ elements, such that for any $k\in\mathbb{Z}_{L}^{d}$
there exists $k_{*}\in\mathfrak{Rep}$ and

$\mathrm{For\ any\ }z\in\mathbb{Z}_{L}^{d},\,|z|\leq M,\mathrm{\ we\ have\
that\ }|\langle k,z\rangle|\leq M\Leftrightarrow|\langle k_{*},z\rangle|\leq
M;\\\ \mathrm{in\ this\ case\ we\ also\ have\ }\langle k,z\rangle=\langle
k_{*},z\rangle.$ (A.20)

###### Proof.

Upon multiplying everything by $L$, we may replace $\mathbb{Z}_{L}$ by
$\mathbb{Z}$. Fix $k\in\mathbb{Z}^{d}$, consider the set $H$ of
$z\in\mathbb{Z}^{d}$ such that $|z|\leq M$ and $|\langle k,z\rangle|\leq M$
(clearly $0\in H$). Let $q$ be the maximal number of linearly independent
vectors in $H$, we may fix a maximum independent set
$\\{w_{1},\cdots,w_{q}\\}\subset H$, and apply Gram-Schmidt process to get
orthogonal vectors $(v_{1},\cdots,v_{q})$. Since each $w_{j}\in\mathbb{Z}^{d}$
and $|w_{j}|\leq M$, we can easily make $v_{j}\in\mathbb{Z}^{d}$ and
$|v_{j}|\leq M^{C_{0}}$. Moreover, since $|\langle k,w_{j}\rangle|\leq M$ for
$1\leq j\leq q$, we know that also $|y|\leq M^{C_{0}}$ where
$y=(y^{1},\cdots,y^{q})$ with $y^{j}=\langle k,v_{j}\rangle$. This means that
$k$ is represented by the system $(q,v_{1},\cdots,v_{q},y)$, as desired.
Clearly, if both $k$ and $k_{*}$ are represented by the same system
$(q,v_{1},\cdots,v_{q},y)$, then (A.20) holds by definition; this completes
the proof as the number of choices of $(q,v_{1},\cdots,v_{q},y)$ is at most
$M^{C_{0}}$. ∎

######  Remark A.7.

Lemma A.6 is adapted to the current setting of square torus. A similar (and
slightly more complicated) version was proved in [15], which is adapted to
generic irrational tori. If we consider more general (partly rational and
partly irrational) tori, then the corresponding result can be stated and
proved by combining the arguments here and in [15]. We omit the details.

######  Lemma A.8.

Let $S$ be any finite set of vectors in $\mathbb{R}^{d}$. Then there exist
$q\leq d$ vectors $X_{1},\cdots X_{q}\in S$ such that any $Y\in S$ is a linear
combination of $(X_{k})_{1\leq k\leq q}$ with coefficients bounded by
$2^{d-1}$.

###### Proof.

Let $X_{1}\in S$ be such that $|X_{1}|$ is maximal. For $k\geq 2$, pick
$X_{k}\in S$ successively such that $|X_{1}\wedge X_{2}\cdots\wedge X_{k}|$ is
positive and maximal, if such an $X_{k}$ exists. This process must terminate
after at most $q\leq d$ steps, which results in a subset
$\\{X_{1},\cdots,X_{q}\\}$. Clearly they are linearly independent, and any
$Y\in S$ can be written as a unique linear combination
$Y=\lambda_{1}X_{1}+\cdots+\lambda_{q}X_{q}$.

Now we have $|X_{1}\wedge\cdots\wedge X_{q-1}\wedge
Y|=|\lambda_{q}|\cdot|X_{1}\wedge\cdots\wedge X_{q-1}\wedge X_{q}|$; by the
choice of $X_{q}$ we have $|\lambda_{q}|\leq 1$. Next consider
$|X_{1}\wedge\cdots\wedge
X_{q-2}\wedge(Y-\lambda_{q}X_{q})|=|\lambda_{q-1}||X_{1}\wedge\cdots\wedge
X_{q-2}\wedge X_{q-1}|$; since both $Y$ and $X_{q}$ belong to $S$, by the
choice of $X_{q-1}$ we have $|\lambda_{q-1}|\leq|\lambda_{q}|+1$. In the same
way we get that $|\lambda_{k}|\leq|\lambda_{k+1}|+\cdots+|\lambda_{q}|+1$ for
all $1\leq k\leq q-1$, which gives the result. ∎

###  A.4. Counting lemmas

Finally we recall some counting estimates from [18].

######  Lemma A.9.

Fix $\alpha,\beta\in\mathbb{R}$ and $r,v\in\mathbb{Z}_{L}^{d}$, such that
$|r|\sim_{0}P$ with $P\in[L^{-1},1]\cup\\{0\\}$ (with $|r|\gtrsim_{0}1$ if
$P=1$). Let each of $(x,y,z)$ belong to $\mathbb{Z}_{L}^{d}$ intersecting a
fixed unit ball, assume $\xi\in\\{x,y,x+y\\}$ and
$\zeta\in\\{x-v,x+y-v,x+y-v-z\\}$, and define $\mathfrak{X}:=\min((\log
L)^{2},1+\delta L^{2\gamma}P)$ as in Proposition 10.5. Then we have the
following counting bounds

$\displaystyle\big{\\{}(x,y):|x\cdot
y-\alpha|\leq\delta^{-1}L^{-2\gamma},\,\,|r\cdot\xi-\beta|\leq\delta^{-1}L^{-2\gamma}\big{\\}}$
$\displaystyle\lesssim_{1}\delta^{-1}L^{2(d-\gamma)}\mathfrak{X}^{-1},$ (A.21)
$\displaystyle\big{\\{}(x,y,z):|x\cdot
y-\alpha|\leq\delta^{-1}L^{-2\gamma},\,\,|\zeta\cdot
z-\beta|\leq\delta^{-1}L^{-2\gamma}\big{\\}}$
$\displaystyle\lesssim_{1}\delta^{-2}L^{3(d-\gamma)-\gamma_{0}}.$ (A.22)

###### Proof.

This follows from Lemma A.2 of [18] (the bounds (A.21)–(A.22) needed here are
much weaker than in [18]). ∎

######  Lemma A.10.

Let $r\in\\{2,3\\}$ and $\epsilon_{j}\in\\{\pm 1\\}$, and $k_{j}\,(1\leq j\leq
r)$ be in $\mathbb{Z}_{L}^{d}$ intersecting a fixed unit ball, that solve the
following system