function that is localized in a fixed unit size box for each of its variables.
Fixing the integer parts of $\alpha_{\mathfrak{m}}$ as $\sigma_{\mathfrak{m}}$
(each of which having at most $L^{10d}$ possibilities), by summing over all
these integer parts, we can bound the
$(t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}})$ Fourier
transform of (7.18) by

$(C^{+}\delta)^{n(\mathscr{B})/2}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{1}}\sum_{\sigma[\mathcal{N}[\mathbb{V}]]}\sup_{|\alpha_{\mathfrak{m}}-\sigma_{\mathfrak{m}}|\leq
1}|\widehat{\mathcal{B}}(\tau_{\mathfrak{u}_{1}}+\widetilde{\lambda}_{\mathfrak{u}_{1}},\tau_{\mathfrak{u}_{21}}+\lambda_{\mathfrak{u}_{21}},\tau_{\mathfrak{u}_{22}}+\lambda_{\mathfrak{u}_{22}},\alpha[\mathcal{N}[\mathbb{V}]])|\\\
\times\sup_{\sigma[\mathcal{N}[\mathbb{V}]]}\sum_{\mathscr{E}}^{*}\widetilde{\mathcal{X}}(k[\mathcal{Q}_{0}[\mathbb{V}]])$
(7.19)

where the sum $\sum^{*}$ satisfies the additional localization that
$|\alpha_{\mathfrak{m}}-\sigma_{\mathfrak{m}}|\leq 1$ for each
$\mathfrak{m}\in\mathcal{N}[\mathbb{V}]$. The result (7.12) will follow from
the following two estimates:

$\displaystyle\bigg{\|}\max(\langle\tau_{\mathfrak{u}_{1}}\rangle,\langle\tau_{\mathfrak{u}_{21}}\rangle,\langle\tau_{\mathfrak{u}_{22}}\rangle)^{\eta^{5}}\sum_{\sigma[\mathcal{N}[\mathbb{V}]]}\sup_{|\alpha_{\mathfrak{m}}-\sigma_{\mathfrak{m}}|\leq
1}|\widehat{\mathcal{B}}(\tau_{\mathfrak{u}_{1}},\tau_{\mathfrak{u}_{21}},\tau_{\mathfrak{u}_{22}},\alpha[\mathcal{N}[\mathbb{V}]])|\bigg{\|}_{L_{\tau,\alpha}^{1}}\lesssim
C^{m}(\log L)^{2m+1}L^{4\eta^{5}},$ (7.20)
$\displaystyle\sup_{(k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}})}\sup_{\sigma[\mathcal{N}[\mathbb{V}]]}\sum_{\mathscr{E}}^{*}\widetilde{\mathcal{X}}(k[\mathcal{Q}_{0}[\mathbb{V}]])\leq(C^{+}\delta^{-1})^{m+1}L^{(2m+1)(d-\gamma)-5\eta}.$
(7.21)

Those two estimates are enough to give (7.12), since shifts of
$\widetilde{\lambda}_{\mathfrak{u}_{1}}$, $\lambda_{\mathfrak{u}_{21}}$ and
$\lambda_{\mathfrak{u}_{22}}$ can be absorbed by the weights
$\max(\langle\lambda_{\mathfrak{u}_{1}}\rangle,\langle\lambda_{\mathfrak{u}_{21}}\rangle,\langle\lambda_{\mathfrak{u}_{22}}\rangle,\langle\lambda_{\mathfrak{u}_{2}}\rangle)^{-\theta}$
and
$(\max_{\mathfrak{n}\in\mathcal{N}[\mathbb{V}]}\langle\vartheta_{\mathfrak{n}}\rangle)^{-\eta}$.

To prove (7.20), we note that
$\mathcal{B}(t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}})=\mathcal{C}(t_{\mathfrak{u}_{1}}-\max(\tau_{\mathfrak{u}_{21}},\tau_{\mathfrak{u}_{22}}))$
for some function $\mathcal{C}$ by changing time variables, and by Lemma A.6
(1) we can bound $\widehat{\mathcal{C}}(\tau,\alpha[\mathcal{N}[\mathbb{V}]])$
by

$|\widehat{\mathcal{C}}(\tau,\alpha[\mathcal{N}[\mathbb{V}]])|\lesssim\langle\tau-\gamma\rangle^{-10}\prod_{\mathfrak{m}\in\mathcal{N}[\mathbb{V}]}\langle
q_{\mathfrak{m}}\rangle^{-1},$

where $(q_{\mathfrak{m}})_{\mathfrak{m}\in\mathcal{N}[\mathbb{V}]}$ is
obtained from $\alpha[\mathcal{N}[\mathbb{V}]]$ by an invertible (actually
lower triangular) linear transformation with integer coefficients, and
$\gamma$ is the sum of at most two $q_{\mathfrak{m}}$ variables. Since each
$\sigma_{\mathfrak{m}}$ has at most $L^{10d}$ choices, we can easily sum over
all $(\sigma_{\mathfrak{m}})$, equivalently all $(q_{\mathfrak{m}})$, and
apply Lemma A.6 (2) to transform from $\mathcal{C}$ to $\mathcal{B}$, and get
(7.20).

To prove (7.21), we upper bound $\widetilde{\mathcal{X}}$ by 1 which reduces
the sum into a counting problem for $k[\mathcal{Q}_{0}[\mathbb{V}]]$
satisfying all the decoration and localization assumptions as before, and in
addition the condition that $\delta L^{2\gamma}\Omega_{\mathfrak{n}}$ for each
$\mathfrak{n}\in\mathcal{N}[\mathbb{V}]$ belongs to an interval of length $1$.
By the reparametrization in Step 2 above, this is a equivalent to counting the
number of choices for the variables $(x_{0},x_{j},y_{j})_{1\leq j\leq m}$,
each restricted to a fixed unit ball, such that each of $(r\cdot
x_{0},x_{j}\cdot y_{j},r\cdot\mu_{j})$ $(1\leq j\leq m$) is restricted to an
interval of length $O(\delta^{-1}L^{-2\gamma})$.

Now if $\gamma\geq\frac{4}{5}-\eta$, we can bound the number of choices of
$x_{0}$ by $O(\delta^{-1}L^{d-\gamma+(1-\gamma)})$ using Lemma A.3 (1), and
the number of choices for each $(x_{j},y_{j})$ by
$\delta^{-2}L^{2(d-\gamma)-(1-\gamma)-20\eta}$ using Lemma A.2 (2). This
implies (7.21), noticing that $m\geq 1$ for vines (II). If
$\gamma<\frac{4}{5}-\eta$ and $|r|\gtrsim L^{-\gamma-\eta}$, then we can bound
the number of choices of $x_{0}$ by $O(\delta^{-1}L^{d-\gamma+2\eta})$ using
Lemma A.3 (1), and the number of choices for each $(x_{j},y_{j})$ by
$\delta^{-2}L^{2(d-\gamma)-20\eta}$ using Lemma A.2 (2), which again implies
(7.21).

Step 4: The cancellation argument. We now treat the case where $\mathbb{V}$ is
vine (II), $\gamma\leq\frac{4}{5}-\eta$ and $|r|\leq L^{-\gamma-\eta}$. In
this case we need to rely on the cancellation happening in the sum over
$(\mathscr{B},\mathtt{ind})$ of the terms
$\widetilde{\mathcal{K}}_{(\mathtt{ind},\mathscr{B})}$ defined in (7.16). For
this cancellation to manifest itself, we need to utilize the couple structure
near $\mathfrak{u}_{2}$, which is depicted in Figure 17.

We start with the easiest case, which is that of vine (II-e). In this case, we
will estimate each element of the sum over $(\mathscr{B},\mathtt{ind})$
separately and use that there are $O(C^{n_{2}})$ elements in this sum. At this
point we can repeat the arguments in Step 3 to localize each of the
$(x_{0},x_{j},y_{j})$ and $k_{\mathfrak{n}}$ variables to a fixed unit ball,
and consequently expand $Z$ into (7.17) and each
$\mathcal{K}_{\mathcal{Q}^{(\mathfrak{l},\mathfrak{l}^{\prime})}}$ and
$\mathcal{K}_{\mathcal{T}^{(\mathfrak{m})}}^{*}$ as time Fourier integrals, to
reduce (7.16) to at most $O(C^{m})$ expressions of form

$\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{2m+1}(C^{+}\delta)^{n(\mathscr{B})/2}\sum_{|x_{0}-a_{0}|\leq
1}e^{\pi
i(\lambda_{\mathfrak{u}_{21}}t_{\mathfrak{u}_{21}}+\lambda_{\mathfrak{u}_{22}}t_{\mathfrak{u}_{22}}+(\lambda_{\mathfrak{u}_{1}}+\gamma_{1})t_{\mathfrak{u}_{1}})}\\\
\times\widetilde{Z}(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}})\cdot\mathcal{I}\left(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}},t_{*}\right)$

Here $a_{0}$ is a fixed vector, $\gamma_{1}$ is a linear combination of
$\lambda_{\mathfrak{u}_{2}}$ and the Fourier variables occurring in the
expansions of
$\mathcal{K}_{\mathcal{Q}}^{(\mathfrak{l},\mathfrak{l}^{\prime})}$ and
$\mathcal{K}^{*}_{\mathcal{T}^{(\mathfrak{m})}}$ as above, and $\mathcal{I}$
is an expression in the form (7.1), which can be obtained after defining the
new time variables $t_{j}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2j+1}}$,
$s_{j}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2j+2}}$ for $1\leq j\leq m$, and
$t_{0}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2}}$. The domain of integration
after the change of variables is the same as described in Lemma 7.1, with $t$
replaced by
$t_{*}:=t_{\mathfrak{u}_{1}}-\max(t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}})\\}$.
Moreover, due to the structure of the couple in vine (II-e) (see Figure 17),
namely the fact that $\mathfrak{u}_{2}$ is a parent of $\mathfrak{u}_{4}$ and
child of $\mathfrak{u}_{3}$, we have the additional condition (a) in Lemma
7.1. As a result, by (7.3), we have that

$\|\mathcal{I}\|_{X_{\mathrm{loc}}^{2\eta^{5},0}(t_{*})}\lesssim(C^{+}\delta^{-1})^{m}L^{2m(d-\gamma)-\gamma-\eta^{2}}.$

After summing over $x_{0}$, applying Lemma A.6 (2), and including the factors
$(\delta/(2L^{d-\gamma}))^{2m+1}$ and $(C^{+}\delta)^{n(\mathscr{B})/2}$ etc.,
this implies (7.12). Note that the shifts
$(\lambda_{\mathfrak{u}_{21}},\lambda_{\mathfrak{u}_{22}},\lambda_{\mathfrak{u}_{1}}+\gamma_{1})$
is again absorbed by the weights.

Now we turn to vines (II-a)–(II-d). We will consider the case of a pair of
couples with vines (II-a) and (II-b) that are twists of each other; the case
of vines (II-c) and (II-d) are similar. If $\mathscr{B}$ is fixed and we sum
over $\mathtt{ind}\in\\{0,1\\}$, then starting from (7.16), and recalling
Figure 17 (assuming without loss of generality that $\mathfrak{u}_{4}$ has
positive sign), we get the expression

$\displaystyle\sum_{\mathtt{ind}}\widetilde{\mathcal{K}}_{(\mathtt{ind},\mathscr{B})}$
$\displaystyle=\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{2m+1}\zeta[\mathbb{V}^{(a)}]\sum_{(x_{0},x_{j},y_{j}):1\leq
j\leq
m}\int_{\mathcal{E}^{*}[\mathbb{V}]}\prod_{j=2}^{2m+2}\mathrm{d}t_{\mathfrak{u}_{j}}$
(7.22) $\displaystyle\times\prod_{j=1}^{m}e^{\pi i\cdot\delta
L^{2\gamma}(t_{\mathfrak{u}_{2j+2}}-t_{\mathfrak{u}_{2j+1}})x_{j}\cdot
y_{j}}e^{\pi i\cdot\delta
L^{2\gamma}(t_{\mathfrak{u}_{2j+1}}-t_{\mathfrak{u}_{1}})(r\cdot\mu_{j})}\cdot
e^{\pi i\cdot\delta
L^{2\gamma}(t_{\mathfrak{u}_{2}}-t_{\mathfrak{u}_{1}})(r\cdot(k_{\mathfrak{u}_{22}}-x_{0}))}$
$\displaystyle\times
Z(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}},t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}},t_{\mathfrak{u}_{2}})\cdot\prod_{\mathfrak{l}\in\mathcal{L}[\mathbb{V}]\setminus\\{\mathfrak{l}^{*}\\}}^{(+)}\mathcal{K}_{\mathcal{Q}^{(\mathfrak{l},\mathfrak{l}^{\prime})}}(t_{\mathfrak{l}^{p}},t_{(\mathfrak{l}^{\prime})^{p}},k_{\mathfrak{l}})$
$\displaystyle\times\prod_{\mathfrak{m}\in\mathcal{N}[\mathbb{V}]\setminus\\{\mathfrak{u}_{2}\\}}\mathcal{K}_{\mathcal{T}^{(\mathfrak{m})}}^{*}(t_{\mathfrak{m}^{p}},t_{\mathfrak{m}},k_{\mathfrak{m}})\left[\mathcal{M}^{(a)}(t_{\mathfrak{u}_{2}},t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{4}},x_{0},y_{0})-\mathcal{M}^{(b)}(t_{\mathfrak{u}_{2}},t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{4}},x_{0},y_{0})\right].$

Here, we denote by $\zeta[\mathbb{V}^{(a)}]$ the $\zeta[\mathbb{V}]$ for the
couple with vine (II-a), which is the negative of the (II-b) couple, and by
$\mathcal{E}^{*}[\mathbb{V}]$ the domain of integration derived from
$\mathcal{E}[\mathbb{V}]$ by removing the condition
$t_{\mathfrak{u}_{4}}>t_{\mathfrak{u}_{2}}$ in case (II-a) and
$t_{\mathfrak{u}_{3}}>t_{\mathfrak{u}_{2}}$ in case (II-b). Moreover $x_{0}$
equals $k_{\mathfrak{u}_{23}}$ in case (II-a) and equals
$k_{\mathfrak{u}_{2}}$ in case (II-b), and $y_{0}$ is defined to be the other
element. We also denoted by $\mathfrak{l}^{*}$ the positive leaf in the leaf
pair $(\mathfrak{u}_{23},\mathfrak{u}_{0})$ (using the notation in Figure 17),
and introduced

$\displaystyle\mathcal{M}^{(a)}(t_{\mathfrak{u}_{2}},t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{4}},x_{0},y_{0})=\mathbf{1}_{t_{\mathfrak{u}_{2}}<t_{\mathfrak{u}_{4}}}(t_{\mathfrak{u}_{4}})\overline{\mathcal{K}_{\overline{\mathcal{Q}_{a}}}}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}},x_{0})\cdot\mathcal{K}_{\mathcal{T}_{a}}^{*}(t_{\mathfrak{u}_{4}},t_{\mathfrak{u}_{2}},y_{0})$
$\displaystyle\mathcal{M}^{(b)}(t_{\mathfrak{u}_{2}},t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{4}},x_{0},y_{0})=\mathbf{1}_{t_{\mathfrak{u}_{2}}<t_{\mathfrak{u}_{3}}}(t_{\mathfrak{u}_{3}})\cdot\mathcal{K}_{\mathcal{Q}_{b}}(t_{\mathfrak{u}_{4}},t_{\mathfrak{u}_{2}},y_{0})\cdot\overline{\mathcal{K}_{\overline{\mathcal{T}_{b}}}^{*}}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}},x_{0}),$

where we denoted by $\mathcal{Q}^{a}$ and $\mathcal{T}^{a}$ (resp.
$\mathcal{Q}^{b},\mathcal{T}^{b}$) the couples
$\mathcal{Q}^{(\mathfrak{u}_{23},\mathfrak{u}_{0})}$ and
$\mathcal{T}^{(\mathfrak{u}_{2})}$ (resp.
$\mathcal{Q}^{(\mathfrak{u}_{0},\mathfrak{u}_{23})}$ and
${\mathcal{T}^{(\mathfrak{u}_{2})}}$). Recall also that $y_{0}=x_{0}\pm r$
which holds for both cases (II-a) and (II-b). This puts us in the position to
apply Lemma 7.1 to conclude. In fact, the difference
$\mathcal{M}^{(a)}-\mathcal{M}^{(b)}$ leads to a sum involving one of the
following assumptions or terms:

  1. (1)

The assumption
$t_{\mathfrak{u}_{4}}<t_{\mathfrak{u}_{2}}<t_{\mathfrak{u}_{3}}$ or
$t_{\mathfrak{u}_{3}}<t_{\mathfrak{u}_{2}}<t_{\mathfrak{u}_{4}}$.

  2. (2)

Factors
$\mathscr{R}_{\mathcal{Q}}=\mathcal{K}_{\mathcal{Q}}-(\mathcal{K}_{\mathcal{Q}})_{\textrm{app}}$
or
$\mathscr{R}^{*}_{\mathcal{T}}=\mathcal{K}^{*}_{\mathcal{T}}-(\mathcal{K}^{*}_{\mathcal{T}})_{\textrm{app}}$
replacing at least one of the $\mathcal{K}_{\mathcal{Q}}$ or the
$\mathcal{K}^{*}_{\mathcal{T}}$ in (7.22), for
$\mathcal{Q}\in\\{\overline{\mathcal{Q}_{a}},\mathcal{Q}_{b}\\}$ and
$\mathcal{T}\in\\{\mathcal{T}_{a},\overline{\mathcal{T}_{b}}\\}$; here we use
Proposition 6.1.

  3. (3)

Factors

$\overline{\mathcal{J}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}})}\big{(}\mathcal{J}^{*}(t_{\mathfrak{u}_{4}},t_{\mathfrak{u}_{2}})-\mathcal{J}^{*}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}})\big{)}\quad\textrm{or}\quad\big{(}\mathcal{J}(t_{\mathfrak{u}_{4}},t_{\mathfrak{u}_{2}})-\mathcal{J}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}})\big{)}\overline{\mathcal{J}^{*}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}})},$

which equals $|t_{\mathfrak{u}_{3}}-t_{\mathfrak{u}_{4}}|^{1-\eta}$ multiplies
a weighted Fourier $L^{1}$ function; this comes from expanding
$(\mathcal{K}_{\mathcal{Q}})_{\mathrm{app}}$ and
$(\mathcal{K}_{\mathcal{T}}^{*})_{\mathrm{app}}$ as in (6.2).

  4. (4)

Factors
$(\mathcal{K}_{\mathcal{Q}})_{\mathrm{app}}(\cdot,\cdot,y_{0})-(\mathcal{K}_{\mathcal{Q}})_{\mathrm{app}}(\cdot,\cdot,x_{0})$
and
$(\mathcal{K}_{\mathcal{T}}^{*})_{\mathrm{app}}(\cdot,\cdot,x_{0})-(\mathcal{K}_{\mathcal{T}}^{*})_{\mathrm{app}}(\cdot,\cdot,y_{0})$,
for some $\mathcal{Q}$ and $\mathcal{T}$, with the time variables being the
same in both functions, which is bounded by $|r|$ using that
$|x_{0}-y_{0}|=r$.

  5. (5)

The leading factor of the form

$\left[\overline{(\mathcal{K}_{\mathcal{Q}_{0}}})_{\mathrm{app}}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}},x_{0})(\mathcal{K}_{\mathcal{T}_{0}}^{*})_{\mathrm{app}}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}},x_{0})-(\mathcal{K}_{\mathcal{Q}_{0}^{\prime}})_{\mathrm{app}}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}},x_{0})\overline{(\mathcal{K}_{\mathcal{T}_{0}^{\prime}}^{*})_{\mathrm{app}}}(t_{\mathfrak{u}_{3}},t_{\mathfrak{u}_{2}},x_{0})\right].$

Note that the contribution of the last term (5) vanishes after summing over
$\mathscr{B}$, due to Lemma 6.3; therefore we are left with terms (1)–(4). For
each term, we may argue as in the case of vine (II-e) above, where we insert
them back into (7.22), then localize each vector to a fixed unit ball, expand
$Z$ into (7.17), and expand all the
$\mathcal{K}_{\mathcal{Q}^{(\mathfrak{l},\mathfrak{l}^{\prime})}}$ and
$\mathcal{K}_{\mathcal{T}^{(\mathfrak{m})}}$ (and $\mathscr{R}_{\mathcal{Q}}$
and $\mathscr{R}_{\mathcal{T}}^{*}$ if present) as time Fourier integrals
using Proposition 6.1, to reduce to at most $O(C^{m})$ expressions of form

$\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{2m+1}(C\delta)^{n(\mathscr{B})/2}\sum_{|k_{0}-a_{0}|\leq
2}e^{\pi i(\lambda_{\mathfrak{u}_{21}}t_{\mathfrak{u}_{21}}+\pi
i\lambda_{\mathfrak{u}_{22}}t_{\mathfrak{u}_{22}}+(\lambda_{\mathfrak{u}_{1}}+\gamma_{1})t_{\mathfrak{u}_{1}})}\\\
\times\widetilde{Z}(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}})\cdot\mathcal{I}\left(k_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}},t_{*}\right).$
(7.23)

Here $a_{0}$ is a fixed vector, $\gamma_{1}$ is a linear combination of
$\lambda_{\mathfrak{u}_{2}}$ and the Fourier variables occurring in the
expansions of
$\mathcal{K}_{\mathcal{Q}}^{(\mathfrak{l},\mathfrak{l}^{\prime})}$ and
$\mathcal{K}^{*}_{\mathcal{T}^{(\mathfrak{m})}}$ as above, and $\mathcal{I}$
is an expression in the form (7.1), which can be obtained after defining the
new time variables $t_{j}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2j+1}}$,
$s_{j}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2j+2}}$ for $1\leq j\leq m$, and
$t_{0}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2}}$. The domain of integration
after the change of variables is the same as described in Lemma 7.1, with $t$
replaced by
$t_{*}:=t_{\mathfrak{u}_{1}}-\max(t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}})\\}$.
Clearly, for each of the terms (1)–(4), one of the conditions (a)–(d) in Lemma
7.1 will be satisfied, so we can apply (7.3) to get that

$\|\mathcal{I}\|_{X_{\mathrm{loc}}^{2\eta^{5},0}(t_{*})}\lesssim(C^{+}\delta^{-1})^{m}L^{2m(d-\gamma)-\gamma-\eta^{2}}.$

After summing over $x_{0}$, applying Lemma A.6 (2), and including the factors
$(\delta/(2L^{d-\gamma}))^{2m+1}$ and $(C^{+}\delta)^{n(\mathscr{B})/2}$ etc.,
this implies (7.12); note that the shifts
$(\lambda_{\mathfrak{u}_{21}},\lambda_{\mathfrak{u}_{22}},\lambda_{\mathfrak{u}_{1}}+\gamma_{1})$
is again absorbed by the weights.

Step 5: The vine (I) case. We now treat the case where $\mathbb{V}$ is a (CL)
vine (I), i.e. one double bond. Here we will apply the cancellation structure
in Step 4, combined with the counting arguments in Step 3, but both will be in
this extremely simple setting of one double bond. In fact, in this case we
have $m=0$, so there is only one variable $x_{0}$ which we sum in. Moreover,
an easy examination of Figure 17 similar to Step 4 implies that, instead of
the $\mathcal{M}^{(a)}-\mathcal{M}^{(b)}$ factor in (7.22), we have the factor

$\overline{\mathcal{K}_{\overline{\mathcal{Q}_{a}}}}(t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{2}},x_{0})\mathcal{K}_{\mathcal{T}_{a}}^{*}(t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{2}},y_{0})-\mathcal{K}_{\mathcal{Q}_{b}}(t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{2}},y_{0})\overline{\mathcal{K}_{\overline{\mathcal{T}_{b}}}}(t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{2}},x_{0}),$

assuming $\mathfrak{u}_{1}$ has sign $+$ in Figure 17. Upon summing in
$\mathscr{B}$ and applying Lemma 6.3, this factor is bounded in
$X_{\mathrm{loc}}^{\eta,0}$ by $|r|$ since $|x_{0}-y_{0}|=r$.

Then, we reduce to the counting problem as in Step 3. Note that the result
corresponding to (7.20) is proved in the same way; as for the one
corresponding to (7.21), we are reduced to a counting problem with only one
variable $x_{0}$ involved, which is restricted to a unit ball and satisfies
that $r\cdot x_{0}$ belongs to a fixed interval of length
$\delta^{-1}L^{-2\gamma}$. By Lemma A.3 (1) and using the extra factor $|r|$
exhibited above, we get the total contribution factor

$(L^{d-1}+\delta^{-1}|r|^{-1}L^{d-2\gamma})\cdot|r|\lesssim
L^{d-\gamma-\gamma_{0}+O(\eta)},$

which suffices for the proof of (7.12) just as in Step 3.

The proof of Proposition 7.5 is now finished. ∎

######  Proposition 7.6.

Suppose $\mathbb{V}$ is a _normal_ (CL) vine. Let
$(\mathcal{Q}^{\mathrm{sp}},\mathtt{cod},\mathfrak{n},\mathscr{A}^{\mathrm{sp}})$
and $(Z,W)$ be fixed as in Section 7.2.1 and Proposition 7.5, and let
$(\mathtt{ind},\mathscr{B})$ also be fixed. Then, for any choice of
$\theta\in\\{\eta^{5},0\\}$ and in the same notations as in Proposition 7.5,
we have

$\big{\|}e^{-\pi i\cdot\delta
L^{2\gamma}t_{\mathfrak{u}_{1}}\Gamma}\cdot\mathcal{K}_{(\mathtt{sgn},\mathtt{ind},\mathscr{B})}^{(\mathbb{V},Z,W)}(x_{0}^{\prime},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}},t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}})\big{\|}_{Y_{\mathrm{loc}}^{\theta}}\lesssim(C^{+}\delta^{1/4})^{n_{2}}L^{\eta^{4}}\|Z\|_{Y_{\mathrm{loc}}^{\theta}}.$
(7.24)

Moreover, if $\mathbb{V}$ is replaced by a normal (CL) vine chain
$\mathbb{V}\mathbb{C}$, define $\mathcal{Q}_{0}[\mathbb{V}\mathbb{C}]$ as in
Proposition 4.12 for the block $\mathbb{V}\mathbb{C}$ and let
$\mathcal{Q}^{\mathrm{sp}}$ be the couple obtained by splicing the whole vine
chain $\mathbb{V}\mathbb{C}$. Fix
$(\mathcal{Q}^{\mathrm{sp}},\mathfrak{n},\mathscr{A}^{\mathrm{sp}})$ and
$(\mathtt{cod},\mathtt{ind},\mathscr{B})$ for each ingredient vine in the
chain, and define the expression
$\mathcal{K}_{(\mathtt{ind},\mathscr{B})}^{(\mathbb{V}\mathbb{C},Z,W)}$ in the
same way as in Section 7.2.1. Then, for $\theta\in\\{\eta^{5},0\\}$ we have

$\big{\|}e^{-\pi i\cdot\delta
L^{2\gamma}t_{\mathfrak{u}_{1}}\Gamma}\cdot\mathcal{K}_{(\mathtt{sgn},\mathtt{ind},\mathscr{B})}^{(\mathbb{V}\mathbb{C},Z,W)}(x_{0}^{\prime},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}},t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}})\big{\|}_{Y_{\mathrm{loc}}^{\theta}}\lesssim(C^{+}\delta^{1/4})^{n_{2}}L^{\eta^{4}}\|Z\|_{Y_{\mathrm{loc}}^{\theta}},$
(7.25)

where $n_{2}$ is the total number of branching nodes in
$\mathcal{Q}_{0}[\mathbb{V}\mathbb{C}]\setminus\\{\mathfrak{u}_{1}\\}$ plus
the sum of all the $n(\mathscr{B})$, as defined in Section 7.2.1.

###### Proof.

The proof of this proposition goes along the same lines as that of Proposition
7.5, so we will just sketch the similar arguments and only elaborate on the
differences, which are mainly in the reparametrization in Step 1 below. We
start with the case of a single normal (CL) vine. We replace
$Y_{\mathrm{loc}}^{\theta}$ by $Y^{\theta}$ and assume throughout
$\|Z\|_{Y^{\theta}}=1$.

Step 1: Reparametrization. Define the notations $\mathscr{E}[\mathbb{V}]$,
$\mathcal{N}[\mathbb{V}]$ and $\mathcal{L}[\mathbb{V}]$ etc. as in the proof
of Proposition 7.5. We first perform the reparametrization. By examining the
form of the normal vine $\mathbb{V}$ in Figure 16, we see that it contains
$n_{1}=2m+4$ atoms (excluding the joint atom $v_{1}$) for some $m$, which is
equal to the cardinality of
$\mathcal{Q}_{0}[\mathbb{V}]\setminus\\{\mathfrak{u}_{0}\\}$. These atoms are
split into three groups: (a) the joint $v_{2}$, (b) the $m$ pairs of atoms
connected by double bounds that belong to the ladders represented colored
dashed lines in Figure 16 (for Vine (VII) we also include here the pair of
atoms where all three ladders intersect), and (c) the three remaining atoms
$v_{1}^{\prime},v_{2}^{\prime},v_{3}^{\prime}$. At the level of the couple,
$\mathcal{Q}_{0}[\mathbb{V}]\backslash\\{\mathfrak{u}_{1}\\}$ has $2m+4$ leaf
pairs $(\mathfrak{l},\mathfrak{l}^{\prime})$, as well as $2m+4$ branching
nodes $\mathfrak{m}$. We will replace the variables $k_{\mathfrak{l}}$ and
$k_{\mathfrak{m}}$ (where $\mathfrak{l}\in\mathcal{L}[\mathbb{V}]$ and
$\mathfrak{m}\in\mathcal{N}[\mathbb{V}]$) occurring in the decoration
$\mathscr{E}$ by a new set of variables
$(x_{0},x_{j},y_{j},u_{1},u_{2},u_{3})_{1\leq j\leq m}$ as follows.

By the same argument and notation as in Step 2 of the proof of Proposition
7.5, we have that $\Omega_{\mathfrak{u}_{2}}=2r\cdot\mu_{0}$ where
$r=k_{\mathfrak{u}_{21}}-k_{\mathfrak{u}_{22}}$ is fixed and nonzero, and
$\mu_{0}=\alpha_{0}x_{0}+\beta_{0}y_{0}+\theta_{0}r$ for some
$\alpha_{0},\beta_{0},\theta_{0}\in\\{0,\pm 1\\}$ with
$y_{0}\in\\{k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}}\\}$, $x_{0}$ equals
$k_{\mathfrak{u}_{23}}$ if $\mathfrak{u}_{2}$ has sign $+$ and
$k_{\mathfrak{u}_{2}}$ otherwise, and $\alpha_{0}^{2}+\beta_{0}^{2}\neq 0$.
Next, for each pair of atoms connected by a double bond in a ladder, the same
argument as in Step 3 of the proof of Proposition 7.5 shows that if
$\mathfrak{u}$ and $\widetilde{\mathfrak{u}}$ are the branching nodes of
$\mathcal{Q}_{0}[\mathbb{V}]$ corresponding to those two atoms, then

$\zeta_{\mathfrak{u}}\Omega_{\mathfrak{u}}+\zeta_{\widetilde{\mathfrak{u}}}\Omega_{\widetilde{\mathfrak{u}}}=2\widetilde{r}\cdot\mu.$

Here $\mu$ is the difference of two $k_{\mathfrak{n}}$ vectors corresponding
to two of the four single bonds at these pair of atoms, and
$\widetilde{r}\in\\{0,r\\}$ is the the same for all pairs of atoms in the same
ladder and is equal to $r$ for the ladders attached to the joints and zero
otherwise. In Figure 16 the ladders whose $\widetilde{r}$ value is 0 are
colored in light blue, and we call such ladders _zero-gap ladders_ (note that
for the pair of atoms where all three ladders intersect in vine (VII), the
argument needs to be slightly adjusted but the result remains the same, with
$\widetilde{r}=r$ in this case). As such, we can define
$(x_{j},y_{j})\in\mathbb{Z}^{2d}$ such that
$\zeta_{\mathfrak{u}}\Omega_{\widetilde{\mathfrak{u}}}=2x_{j}\cdot y_{j}$, so
that
$\zeta_{\widetilde{\mathfrak{u}}}\Omega_{\widetilde{\mathfrak{u}}}=-2x_{j}\cdot
y_{j}+2\widetilde{r}\cdot\mu$, and $\mu$ can be written as
$a_{j}x_{j}+b_{j}y_{j}+c_{j}\widetilde{r}$ for $a_{j},b_{j},c_{j}\in\\{0,\pm
1\\}$ and $a_{j}^{2}+b_{j}^{2}\neq 0$. It now remains to define the variables
$(u_{1},u_{2},u_{3})$. Here we only discuss vine (III) in detail below, as
arguments in other cases are similar.

(1) For vine (III), using the notation in Figure 21 (B), we may denote
$\omega_{j}=\zeta_{\mathfrak{w}_{j}}\Omega_{\mathfrak{w}_{j}}$ where
$\mathfrak{w}_{j}$ is the node in $\mathcal{Q}_{0}[\mathbb{V}]$ corresponding
to the atom $w_{j}$. In Figure 21 (B) and Figure 16, note that $e-g=f-h=\pm
r$, and $(e,g)$ is determined by $x_{0}$ and some of the $(x_{j},y_{j})$
variables. Now, if the bonds decorated by $c$ and $d$ have opposite directions
(say $c$ goes from $w_{1}$ to $w_{3}$, and $d$ goes from $w_{3}$ to $w_{1}$),
we may define $(u_{1},u_{2},u_{3})=(c-d,d-a,d-b)$. If the bonds decorated by
$c$ and $d$ have the same direction (which has to go from $w_{1}$ to $w_{3}$),
then we may define $(u_{1},u_{2},u_{3})=(c-a,a-d,c-b)$. Then we have
$\omega_{1}=2u_{1}\cdot u_{2}$, and $\omega_{2}$ equals $2u_{1}\cdot u_{3}$ or
$2u_{3}\cdot(u_{1}+u_{2}-u_{3})$ in the first and second case. Moreover we
have
$\omega_{1}+\omega_{2}+\omega_{3}=\pm(|e|^{2}-|f|^{2}-|g|^{2}+|h|^{2})=2r\cdot\xi$
where $\xi=au_{1}+bu_{2}+cu_{3}+dr$ with $a,b,c,d\in\\{0,\pm 1\\}$. In any
case, the variables $(u_{1},u_{2},u_{3})$ determines $(a,b,c,d,f,h)$ and
allows one to proceed with parametrizing the next ladder starting from $(f,h)$
by the rest of $(x_{j},y_{j})$ variables. The argument for Vine (IV) is
similar, see Figure 21 (C).

(2) For Vines (V)–(VIII), the argument is again similar, and in fact much
easier. In Figure 21 (D) and Figure 16, note that the two bonds going in and
out the triangle are both decorated by $g$ (which is determined by the
$(x_{0},x_{j},y_{j})$ variables), which means that the vector $r$ for vine
(III) above is replaced by $\widetilde{r}=0$. In particular we have
$\omega_{1}+\omega_{2}+\omega_{3}=0$ where
$\omega_{j}=\zeta_{\mathfrak{w}_{j}}\Omega_{\mathfrak{w}_{j}}$. Then we argue
as above, with $(u_{1},u_{2},u_{3})=(a-e,b-e,a-c)$ if bonds decorated by $a$
and $b$ have the same direction, and $(u_{1},u_{2},u_{3})=(e-g,b-g,d-g)$ if
they have opposite directions, then the same results will hold.

As a result, in all cases we can define $(u_{1},u_{2},u_{3})$ so that the full
decoration of the vine $\mathbb{V}$ and hence that of
$\mathcal{Q}_{0}[\mathbb{V}]$ is completely determined by
$(x_{0},x_{j},y_{j},u_{1},u_{2},u_{3})_{1\leq j\leq m}$. The factors
$\zeta_{\mathfrak{u}_{2j+1}}\Omega_{\mathfrak{u}_{2j+1}}$ and
$\zeta_{\mathfrak{u}_{2j+2}}\Omega_{\mathfrak{u}_{2j+2}}$ for $1\leq j\leq m$
are given by $2x_{j}\cdot y_{j}$ and $-2x_{j}\cdot
y_{j}+2\widetilde{r}_{j}\cdot\mu_{j}$ where
$\mu_{j}=a_{j}x_{j}+b_{j}y_{j}+c_{j}r$ with $a_{j},b_{j},c_{j}\in\\{0,\pm
1\\}$ and $a_{j}^{2}+b_{j}^{2}\neq 0$, and $\widetilde{r}_{j}=r$ if $j\leq
m_{1}$ and $\widetilde{r}_{j}=0$ otherwise. The remaining three branching
nodes in $\mathcal{Q}_{0}[\mathbb{V}]$ have their resonance factors given by
$2u_{1}\cdot u_{2}$ and $\Lambda\in\\{2u_{1}\cdot
u_{3},2u_{3}\cdot(u_{1}+u_{2}-u_{3})\\}$, and $-2u_{1}\cdot
u_{2}-\Lambda+\widetilde{r}\cdot\xi$ where $\xi=au_{1}+bu_{2}+cu_{3}+dr$ with
$a,b,c,d\in\\{0,\pm 1\\}$ and $\widetilde{r}\in\\{r,0\\}$. In particular, the
change of variables from
$(k_{\mathfrak{n}})_{\mathfrak{n}\in\mathcal{Q}_{0}[\mathbb{V}]}$ into
$(x_{0},x_{j},y_{j},u_{1},u_{2},u_{3})$ satisfies the conditions stated in
Lemma 7.2.

Now, with the reparametrization, we can argue in the same way as in Step 2 of
the proof of Proposition 7.5 to restrict each of the
$(x_{0},x_{j},y_{j},u_{j})$ and $k_{\mathfrak{n}}$ variables to a fixed unit
ball, and consequently expand $Z$ into (7.17). Moreover, once we confirm
$r\neq 0$, we can get rid of the $\epsilon_{\mathscr{E}[\mathbb{V}]}$ factor
in the same way as in Step 2 of the proof of Proposition 7.5. Next, by using
the $(x_{0},x_{j},y_{j},u_{j})$ variables, we can reduce (7.24) to estimating
the expression

$\displaystyle\widetilde{\mathcal{K}}_{(\mathtt{ind},\mathscr{B})}$
$\displaystyle=e^{\pi
i(\lambda_{\mathfrak{u}_{21}}t_{\mathfrak{u}_{21}}+\lambda_{\mathfrak{u}_{22}}t_{\mathfrak{u}_{22}}+\lambda_{\mathfrak{u}_{1}}t_{\mathfrak{u}_{1}})}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{1}}\zeta[\mathbb{V}]\sum_{(x_{0},x_{j},y_{j}):1\leq
j\leq m}\sum_{(u_{1},u_{2},u_{3})}\int_{\mathcal{E}[\mathbb{V}]}$
$\displaystyle\times e^{2\pi
i\lambda_{\mathfrak{u}_{2}}t_{\mathfrak{u}_{2}}}e^{\pi i\cdot\delta
L^{2\gamma}(t_{\mathfrak{u}_{2}}-t_{\mathfrak{u}_{1}})(r\cdot\mu_{0})}\prod_{j=1}^{m}e^{\pi
i\cdot\delta
L^{2\gamma}(t_{\mathfrak{u}_{2j+2}}-t_{\mathfrak{u}_{2j+1}})x_{j}\cdot
y_{j}}\prod_{j=1}^{m_{1}}e^{\pi i\cdot\delta
L^{2\gamma}(t_{\mathfrak{u}_{2j+1}}-t_{\mathfrak{u}_{1}})(r\cdot\mu_{j})}$
$\displaystyle\times e^{\pi i\cdot\delta
L^{2\gamma}(t_{\mathfrak{w}_{1}}-t_{\mathfrak{w}_{3}})(u_{1}\cdot
u_{2})}e^{\pi i\cdot\delta
L^{2\gamma}(t_{\mathfrak{w}_{2}}-t_{\mathfrak{w}_{3}})\Lambda}e^{\pi
i\cdot\delta
L^{2\gamma}t_{\mathfrak{w}_{3}}(\widetilde{r}\cdot\xi)}{\prod_{\mathfrak{l}\in\mathcal{L}[\mathbb{V}]}^{(+)}\mathcal{K}_{\mathcal{Q}^{(\mathfrak{l},\mathfrak{l}^{\prime})}}(t_{\mathfrak{l}^{p}},t_{(\mathfrak{l}^{\prime})^{p}},k_{\mathfrak{l}})}$
$\displaystyle\times\prod_{\mathfrak{m}\in\mathcal{N}[\mathbb{V}]}\mathcal{K}_{\mathcal{T}^{(\mathfrak{m})}}^{*}(t_{\mathfrak{m}^{p}},t_{\mathfrak{m}},k_{\mathfrak{m}})\cdot\widetilde{Z}(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}})\prod_{j=1}^{m}\mathrm{d}t_{j}\mathrm{d}s_{j}\cdot\mathrm{d}t_{0}\mathrm{d}\tau_{1}\mathrm{d}\tau_{2}\mathrm{d}\tau_{3}.$
(7.26)

Step 2: Case $\gamma>\frac{1}{2}$: counting argument. The argument here is
very similar to Step 3 of the proof of Proposition 7.5, but with one
additional ingredient. After expanding all the $\mathcal{K}_{\mathcal{Q}}$ and
$\mathcal{K}^{*}_{\mathcal{T}}$ as time Fourier integrals, and localizing each
$k_{\mathfrak{n}}\,(\mathfrak{n}\in\mathcal{Q}_{0}[\mathbb{V}])$ to a unit
ball as in Step 3 of the proof of Proposition 7.5, we can reduce the estimate
of (7.26) to that of (7.19). Recall here that $m=m_{1}+m_{2}$ where $m_{2}$ is
the length of the zero-gap ladder in $\mathbb{V}$, and $m_{1}$ is the total
length of other ladders. Now it suffices to prove the following two estimates
(in fact they will allow us to gain a small power of $L$ in (7.24)):

$\bigg{\|}\max(\langle\tau_{\mathfrak{u}_{1}}\rangle,\langle\tau_{\mathfrak{u}_{21}}\rangle,\langle\tau_{\mathfrak{u}_{22}}\rangle)^{\eta^{5}}\sum_{\sigma[\mathcal{N}[\mathbb{V}]]}\sup_{|\alpha_{\mathfrak{m}}-\sigma_{\mathfrak{m}}|\leq
1}|\widehat{\mathcal{B}}(\tau_{\mathfrak{u}_{1}},\tau_{\mathfrak{u}_{21}},\tau_{\mathfrak{u}_{22}},\alpha[\mathcal{N}[\mathbb{V}]])|\bigg{\|}_{L_{\tau,\alpha}^{1}}\\\
\lesssim C^{m}\delta^{-(m+2)/2}L^{Cm_{1}\sqrt{\delta}}(\log
L)^{Cm_{1}}L^{4\eta^{5}},$ (7.27)
$\sup_{(k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}})}\sup_{\sigma[\mathcal{N}[\mathbb{V}]]}\sum_{\mathscr{E}}^{*}\mathcal{X}(k[\mathcal{Q}_{0}[\mathbb{V}]])\leq(C^{+}\delta^{-1})^{m+2}L^{(2m+4)(d-\gamma)-5\eta(1+m_{1})}.$
(7.28)

These two estimates are clearly enough to give (7.24) since the shifts can be
absorbed by the weights as in the proof of Proposition 7.5. Moreover, (7.27)
follows by combining the arguments in Step 3 of the proof of Proposition 7.5
with the ladder $L^{1}$ estimate proved in Proposition 10.1 of [14] (but only
to the zero-gap ladder, so the number of atoms not in this ladder is
$O(m_{1})$); we remark that while Proposition 10.1 of [14] is proved for
$L^{\infty}_{t}$ rather than $X^{\theta}$, but it extends directly to the
space $X^{\theta}$ by simply relying on Lemma A.6 instead of Lemma 10.2 in
[14].

To prove (7.28), we reduce it to a counting problem as in the proof of
Proposition 7.5. Here we are counting the number of choices for the variables
$(x_{0},x_{j},y_{j},u_{1},u_{2},u_{3})_{1\leq j\leq m}$, each of which is in a
fixed unit ball, such that each of $(r\cdot\mu_{0},x_{j}\cdot y_{j},u_{1}\cdot
u_{2},\Lambda)$ where $1\leq j\leq m$, and each of $r\cdot\mu_{j}$ where
$1\leq j\leq m_{1}$, belongs to a fixed interval of length
$O(\delta^{-1}L^{-2\gamma})$. Since $\gamma>1/2$, we know that the number of
choices for $x_{0}$ is $O(\delta^{-1}L^{d-\gamma-(1-\gamma)}$ by Lemma A.3
(1), that the number of choices for each $(x_{j},y_{j})\,(1\leq j\leq m_{1})$
is $O(\delta^{-2}L^{2(d-\gamma)-10\eta})$ by Lemma A.2 (1), the number of
choices for each $(x_{j},y_{j})\,(j>m_{1})$ is $O(\delta^{-1}L^{2(d-\gamma)})$
by Lemma A.2 (1), and the number of choices for $(u_{1},u_{2},u_{3})$ is
$O(\delta^{-2}L^{3(d-\gamma)-(1-\gamma)-10\eta})$ by Lemma A.2 (3). Putting
together, this proves (7.28).

Step 3: Case $\gamma\leq\frac{1}{2}$: Lemma 7.2. Here, the argument is
basically the same as in Step 4 of the proof of Proposition 7.5, except that
we rely on Lemma 7.2 instead of 7.1. We expand $\mathcal{K}_{\mathcal{Q}}$ and
$\mathcal{K}^{*}_{\mathcal{T}}$ as time Fourier integrals to obtain an
expression of the form

$\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{2m+4}(C\delta)^{n(\mathscr{B})/2}\sum_{|x_{0}-a_{0}|\leq
1}e^{\pi i(\lambda_{\mathfrak{u}_{21}}t_{\mathfrak{u}_{21}}+\pi
i\lambda_{\mathfrak{u}_{22}}t_{\mathfrak{u}_{22}}+(\lambda_{\mathfrak{u}_{1}}+\gamma_{1})t_{\mathfrak{u}_{1}})}\\\
\times\widetilde{Z}(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}})\mathcal{I}\left(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}},t_{*}\right).$
(7.29)

Here $a_{0}$ is a fixed vector, $\gamma_{1}$ is a linear combination of
$\lambda_{\mathfrak{u}_{2}}$ and the Fourier variables occurring in the
expansions of
$\mathcal{K}_{\mathcal{Q}}^{(\mathfrak{l},\mathfrak{l}^{\prime})}$ and
$\mathcal{K}^{*}_{\mathcal{T}^{(\mathfrak{m})}}$ as above, and $\mathcal{I}$
is an expression in the form (7.1) but modified as in Lemma 7.2, which can be
obtained after defining the new time variables
$t_{j}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2j+1}}$,
$s_{j}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2j+2}}$ for $1\leq j\leq m$,
$t_{0}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{u}_{2}}$, and
$\tau_{j}=t_{\mathfrak{u}_{1}}-t_{\mathfrak{w}_{j}}$. The domain of
integration after the change of variables is the same as described in Lemma
7.2, with $t$ replaced by
$t_{*}:=t_{\mathfrak{u}_{1}}-\max(t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}})\\}$.
As a result, by (7.7), we have that

$\|\mathcal{I}\|_{X_{\mathrm{loc}}^{2\eta^{5},0}(t_{*})}\lesssim(C^{+}\delta^{-1})^{m+2}L^{(2m+4)(d-\gamma)-d+\eta^{4}}.$

After summing over $x_{0}$, applying Lemma A.6 (2), and including the factors
$(\delta/(2L^{d-\gamma}))^{2m+4}$ and $(C^{+}\delta)^{n(\mathscr{B})/2}$ etc.,
this implies (7.24); note that the shifts
$(\lambda_{\mathfrak{u}_{21}},\lambda_{\mathfrak{u}_{22}},\lambda_{\mathfrak{u}_{1}}+\gamma_{1})$
is again absorbed by the weights.

Step 4: The Vine Chain case. The proof for the vine chain
$\mathbb{V}\mathbb{C}$ runs exactly as above, except that we apply Corollary
7.3 instead of Lemma 7.2 in Step 3. Indeed, by reparametrizing the whole vine
chain $\mathbb{V}\mathbb{C}$ by going from bottom to top and using the same
reparametrization in Step 1 for each ingredient vine, we can define the
variables $(x_{0}^{q},x_{j}^{q},y_{j}^{q},u_{1}^{q},u_{2}^{q},u_{3}^{q})$
where $0\leq q<Q$ and $Q$ is the number of ingredient vines in
$\mathbb{V}\mathbb{C}$. We then get an expression for
$\mathcal{K}_{(\mathtt{sgn},\mathtt{ind},\mathscr{B})}^{(\mathbb{V}\mathbb{C},W,Z)}$
generalizing that in (7.26). We treat the case when $\gamma>\frac{1}{2}$ by
reducing to a counting estimate as in Step 2 above, and treat the case
$\gamma\leq\frac{1}{2}$ using Corollary 7.3 as in Step 3 above. This then
completes the proof. ∎

######  Remark 7.7.

In the setting of Propositions 7.5 and 7.6 (described in Section 7.2.1) we
have assumed that $Z$ is a function of
$(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}},t_{\mathfrak{u}_{1}},t_{\mathfrak{u}_{21}},t_{\mathfrak{u}_{22}},t_{\mathfrak{u}_{2}})$.
In fact, we may allow $Z$ to depend on other variables (say denoted by
$k[\mathcal{U}]$ and $t[\mathcal{V}]$) provided that they do not appear in the
rest of the expression for
$\mathcal{K}_{(\mathtt{sgn},\mathtt{ind},\mathscr{B})}^{(\mathbb{V},Z,W)}$; in
this case, if we consider the norm $Y_{\mathrm{loc}}^{\theta}$ in all
variables including $k[\mathcal{U}]$ and $t[\mathcal{V}]$, then (7.12), (7.24)
and (7.25) still hold with the same implicit constants.

This is because, in the process of the proof of Propositions 7.5 and 7.6, we
have restricted each of the variables
$(x_{0},k_{\mathfrak{u}_{1}},k_{\mathfrak{u}_{11}},k_{\mathfrak{u}_{21}},k_{\mathfrak{u}_{22}})$
to a fixed unit ball. If $Z$ depends on $k[\mathcal{U}]$, then we may also
restrict each variable in $k[\mathcal{U}]$ to a unit ball, which reduces the
$Y_{\mathrm{loc}}^{\theta}$ bound to the $X_{\mathrm{loc}}^{\theta,0}$ bound.
Then we simply view $Z$ as a function with value in the Banach space
$L_{k[\mathcal{U}]}^{\infty}$, and apply Propositions 7.5 and 7.6 to handle
these extra $k[\mathcal{U}]$ variables. As for the extra time variables
$t[\mathcal{V}]$, note that
$\mathcal{K}_{(\mathtt{sgn},\mathtt{ind},\mathscr{B})}^{(\mathbb{V},Z,W)}$ is
linear in $Z$ and thus commutes with taking time Fourier transforms in
$t[\mathcal{V}]$. Let the Fourier dual of $t[\mathcal{V}]$ be
$\xi[\mathcal{V}]$, then

$\|\mathcal{Z}\|_{X^{\eta^{5},0}}\sim\int\big{(}\max_{\mathfrak{n}\in\mathcal{V}}\langle\xi_{\mathfrak{n}}\rangle\big{)}^{\eta^{5}}\|\mathcal{F}_{t[\mathcal{V}]}\mathcal{Z}(\cdot,\xi[\mathcal{V}])\|_{X^{0,0}}\,\mathrm{d}\xi[\mathcal{V}]+\int\|\mathcal{F}_{t[\mathcal{V}]}\mathcal{Z}(\cdot,\xi[\mathcal{V}])\|_{X^{\eta^{5},0}}\,\mathrm{d}\xi[\mathcal{V}]$
(7.30)

for any $\mathcal{Z}=\mathcal{Z}(\cdot,t[\mathcal{V}])$, where the $\cdot$
represents variables other than $k[\mathcal{U}]$ and $t[\mathcal{V}]$, and the
norm on the left hand side of (7.30) is the norm in all variables, while the
norms on the right hand side are in the $\cdot$ variables only. Therefore, we
can apply Propositions 7.5 and 7.6 for each fixed $\xi[\mathcal{V}]$, and then
integrate in these variables, to get the same results as in (7.12), (7.24) and
(7.25).

##  8\. Reduction to counting estimates

###  8.1. Preliminary setup

Recall the notions of vines and vine-chains (VC), hyper-vines (HV) and hyper-
vine-chains (HVC), and ladders in Definition 5.1.

######  Lemma 8.1.

Define a _double-vine (V)_ , or DV for short, to be the union of two vines
(V), see Figure 16, that share two common joints and no other common
atoms777This is not a vine-like object; see Definition 5.1.. Then, for any
molecule $\mathbb{M}$, there is a unique collection $\mathscr{C}$ of disjoint
atomic groups, such that each atomic group in $\mathscr{C}$ is an HV, VC, HVC
or DV, and any vine-like object in $\mathbb{V}$ is a subset of some atomic
group in $\mathscr{C}$.

###### Proof.

Consider all the maximal vine-like objects in $\mathbb{M}$, where maximal is
in the sense that it is not a subset of any other vine-like object; let this
collection be $\mathscr{C}_{1}$. We know that Lemma 4.11 applies to any two of
these objects. If $\mathbb{A},\mathbb{B}\in\mathscr{C}_{1}$ and
$\mathbb{A}\cap\mathbb{B}\neq\varnothing$, then we are in one of scenarios
(a)–(c) of Lemma 4.11. However scenarios (b) and (c) are impossible, because
$\mathbb{D}=\mathbb{A}\cup\mathbb{B}$ (in case (b)) or
$\mathbb{D}=\mathbb{C}_{0}\cup\mathbb{C}_{1}\cup\mathbb{C}_{2}$ (in case (c))
would be a larger vine-like object that contains $\mathbb{A}$ and
$\mathbb{B}$. We are then left with case (a), where
$\sigma(\mathbb{A})=\sigma(\mathbb{B})=1$. This means that each of
$\mathbb{A}$ and $\mathbb{B}$ must be one vine (V), and they share two common
joints and no other common atoms, so their union is a DV. Moreover, in this
case, neither $\mathbb{A}$ nor $\mathbb{B}$ can intersect with any other
maximal vine-like object $\mathbb{C}$ (otherwise $\mathbb{A}$ (for example)
would form another DV with $\mathbb{C}$, which leads to a $4$-regular
component). Therefore, let $\mathscr{C}$ be obtained from $\mathscr{C}_{1}$ by
replacing the two intersecting vines (V) with one DV, then it satisfies the
requirement and clearly is unique. ∎

######  Definition 8.2.

Let $\mathcal{Q}$ be a couple with skeleton $\mathcal{Q}_{\mathrm{sk}}$, and
let $\mathscr{C}$ be defined for the molecule
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ by Lemma 8.1. Define a collection
$\mathscr{V}$ of (CL) vines as follows: for each VC in $\mathscr{C}$ whose
joints do not both have degree $3$, we include into $\mathscr{V}$ all its vine
ingredients that are (CL) vines. For each HVC in $\mathscr{C}$ and each VC in
$\mathscr{C}$ whose joints both have degree $3$, we include into $\mathscr{V}$
all but one of its vine ingredients, such that (a) if there is a (CN) vine
then we only exclude this one, and (b) if all vines are (CL) vines then we
only exclude the “top” vine whose $\mathfrak{u}_{1}$ node is the ancestor of
all other $\mathfrak{u}_{1}$ nodes (as branching nodes of
$\mathcal{Q}_{\mathrm{sk}}$). We do not include anything in $\mathscr{C}$ from
any HV or DV.

Since $\mathscr{V}$ satisfies the assumptions in Definition 5.10, we shall
define any couple $\mathcal{Q}^{\prime}$ to be _congruent_ to $\mathcal{Q}$,
if $\mathcal{Q}^{\prime}$ is a full twist of $\mathcal{Q}$ with respect to
$\mathscr{V}$. Clearly, performing a full twist does not affect the molecule
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ nor the choice of vines in
$\mathscr{V}$, and congruence is an equivalence relation and preserves the
order of _each tree_ in a couple.

######  Definition 8.3.

Given any molecule $\mathbb{M}$ and block $\mathbb{B}\subset\mathbb{M}$, let
the four bonds in $\mathbb{B}$ at the two joints be
$\ell_{1},\ell_{2}\in\mathbb{B}$ at one joint, and
$\ell_{3},\ell_{4}\in\mathbb{B}$ at the other. Then for any decoration
$(k_{\ell})$ we have
$k_{\ell_{1}}-k_{\ell_{2}}=\pm(k_{\ell_{3}}-k_{\ell_{4}}):=r$. We call this
vector the _gap_ of $\mathbb{B}$ relative to this decoration. Note that once
the parameters $(c_{v})$ of a decoration are fixed as in Definition 4.6, then
this $r$ can be expressed as a function of the vectors $k_{\ell_{j}^{*}}$,
where $\ell_{j}^{*}$ runs over all bonds connecting a given joint of
$\mathbb{B}$ to atoms _not_ in $\mathbb{B}$. If $\mathbb{B}$ is concatenated
by blocks $\mathbb{B}_{j}$, then all $\mathbb{B}_{j}$ must have the same gap
as $\mathbb{B}$. For the hyper-block which is adjoint of $\mathbb{B}$, we
define its gap to be the gap of $\mathbb{B}$. Note that the gap of a block can
never be $0$ due to Remark 2.7.

More generally, if $v$ is an atom and $\ell_{1},\ell_{2}\sim v$ are two bonds
with opposite directions, then we define the _gap_ of the triple
$(v,\ell_{1},\ell_{2})$ relative to a given decoration as
$r:=k_{\ell_{1}}-k_{\ell_{2}}$. In particular the gap of any block or hyper-
block equals a suitable gap at either of its joints. Next, for any ladder of
length $\geq 1$ (see Definition 5.1 and Figure 16), the difference
$k_{\ell}-k_{\ell^{\prime}}$ for any pair of parallel single bonds
$(\ell,\ell^{\prime})$ must be equal (up to a sign change), which we also
define to be the _gap_ of the ladder. In particular, if $\mathbb{V}$ is a vine
(or VC) with gap $r$, then for any ladder contained in $\mathbb{V}$ that is
inserted between parallel dashed bonds of the same color in Figure 16, the gap
of this ladder is either $\pm r$ or $0$. Finally, for all the gaps defined
above, we say it is _small gap_ (or SG for short) if $|r|\leq
L^{-\gamma+\eta}$ (including $r=0$), and _large gap_ (or LG) if
$|r|>L^{-\gamma+\eta}$.

With the above preparations, we can reduce Propositions 2.4–2.5 to the
following

######  Proposition 8.4.

We can define a value $\rho=\rho(\mathcal{Q})$ associated to a non-regular
couple $\mathcal{Q}$, which is an integer and $1\leq\rho\leq n$ (where $n$ is
the order of $\mathcal{Q}$), such that (i) it takes the same value for
$\mathcal{Q}$ in the same congruence class, (ii) the number of couples
$\mathcal{Q}$ of order $n$ such that $\rho(\mathcal{Q})=\rho$ is at most
$(C\rho)!C^{n}$, and (iii) for any couple $\mathcal{Q}$ of order $n$, we have

$\bigg{|}\sum_{\mathcal{Q}^{\prime}}\mathcal{K}_{\mathcal{Q}^{\prime}}(t,t,k)\bigg{|}\lesssim\langle
k\rangle^{-20d}(C^{+}\delta^{1/4})^{n}\cdot
L^{-\eta^{7}\cdot\rho(\mathcal{Q})},$ (8.1)

where $\mathcal{Q}^{\prime}$ runs over all couples congruent to $\mathcal{Q}$.

###### Proof of Propositions 2.4–2.5 assuming Proposition 8.4.

Consider the sum on the left hand side of (2.24) and (2.25). The sum over all
regular couples $\mathcal{Q}$ is taken care of by Propositions 6.1 and 6.2 (in
particular by the bound (6.4) and the equality (6.8)), so we only need to
consider the sum over non-regular couples $\mathcal{Q}$.

Note that by Definition 5.10, if
$\mathcal{Q}=(\mathcal{T}^{+},\mathcal{T}^{-})$ and
$\mathcal{Q}^{\prime}=((\mathcal{T}^{\prime})^{+},(\mathcal{T}^{\prime})^{-})$
are two congruent couples, then
$n(\mathcal{T}^{\pm})=n((\mathcal{T}^{\prime})^{\pm})$. Therefore both sums on
the left hand side of (2.24) and (2.25), over non-regular couples
$\mathcal{Q}$, can be written as a sum of subset sums, such that each subset
sum has the form of the left hand side of (8.1). We then classify these subset
sums according to the value $\rho(\mathcal{Q})$, and apply Proposition 8.4 to
get that

$\bigg{|}\sum_{\mathcal{Q}}\mathcal{K}_{\mathcal{Q}}(t,t,k)\bigg{|}\lesssim(C^{+}\delta^{1/4})^{m}\sum_{\rho=1}^{m}(C\rho)!C^{m}L^{-\eta^{7}\rho}.$
(8.2)

Here in (8.2) the value $m\in\\{2n,2n+1\\}$ is fixed, and the sum is either
taken over all non-regular couples $\mathcal{Q}$ of order $m$, or taken over
all non-regular couples $\mathcal{Q}=(\mathcal{T}^{+},\mathcal{T}-)$ with
$n(\mathcal{T}^{+})=n(\mathcal{T}^{-})=m/2=n$. In either case, since $\rho\leq
m$, we have

$(C\rho)^{C}\leq(Cm)^{C}\lesssim(\log L)^{C}\ll L^{\eta^{9}},$

so the $\rho$-sum on the right hand side of (8.2) is easly bounded by
$L^{-\eta^{8}}$, which then proves both (2.24) and (2.25). ∎

###  8.2. Stage 1 reduction: Small gap (CL) vines

We start the proof of Proposition 8.4. First, using (6.49), we can rewrite the
left hand side of (8.1) as

$\mathrm{LHS\ of\
}(\ref{kqmainest1-1})=\sum_{(\mathcal{Q}_{\mathrm{sk}},\mathscr{A})}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{0}}\zeta(\mathcal{Q}_{\mathrm{sk}})\sum_{\mathscr{E}_{\mathrm{sk}}}\int_{\mathcal{E}_{\mathrm{sk}}}\epsilon_{\mathscr{E}_{\mathrm{sk}}}\prod_{\mathfrak{n}\in\mathcal{N}_{\mathrm{sk}}}e^{\zeta_{\mathfrak{n}}\pi
i\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}}\\\
\times{\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sk}}}^{(+)}\mathcal{K}_{\mathcal{Q}^{(\mathfrak{l},\mathfrak{l}^{\prime})}}(t_{\mathfrak{l}^{p}},t_{(\mathfrak{l}^{\prime})^{p}},k_{\mathfrak{l}})}\prod_{\mathfrak{m}\in\mathcal{N}_{\mathrm{sk}}}\mathcal{K}_{\mathcal{T}^{(\mathfrak{m})}}^{*}(t_{\mathfrak{m}^{p}},t_{\mathfrak{m}},k_{\mathfrak{m}}).$
(8.3)

Here the $\mathcal{Q}_{\mathrm{sk}}$ runs over all twists of a given prime
couple, $\mathscr{A}$ runs over collections of regular trees and regular
couples that satisfy a certain set of assumptions (see Definition 5.10), and
$\mathscr{E}_{\mathrm{sk}}$ runs over all $k$-decorations of
$\mathcal{Q}_{\mathrm{sk}}$ (note that, different choices of
$\mathcal{Q}_{sk}$ are twists of each other, so their decorations are in one-
to-one correspondence as in Remark 5.8). In particular, if $\mathscr{C}$ is
defined by Lemma 8.1 for $\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$, then for
each vine-like object in $\mathscr{C}$, we can specify whether it has SG or LG
under the decoration (each such specification imposes a set of restrictions on
the decoration). This reduces the left hand side of (8.1) to a superposition
of at most $C^{n}$ terms, such that each vine-like object is specified to be
either SG or LG in each term. Let $\mathscr{V}_{0}$ be the collection of (CL)
vines in $\mathscr{V}$ (as Definition 8.2) that are SG. Note that
$\mathcal{Q}$ runs over a congruence class, which is defined by full twists at
all vines in $\mathscr{V}$; if we strengthen the equivalence relation by
allowing only full twists at all vines in $\mathscr{V}_{0}$, then the sum in
$\mathcal{Q}$ can again be written as a superposition of at most $C^{n}$
terms, each of which is a sum over the new equivalence class. As such, we only
need to consider one of these new terms, which we shall refer to as
$\mathscr{K}$ for below.

We define a new ordered collection $\mathscr{U}_{0}$ from $\mathscr{V}_{0}$,
whose elements are _bad (CL) vines and normal (CL) vine chains,_ as follows.
First organize vines in $\mathscr{V}_{0}$ into disjoint VC, then order these
VC arbitrarily; for each VC, we divide it into _units_ and order them from
bottom to top, where each unit is either a bad (CL) vine or a (longest) sub-VC
formed by consecutive normal (CL) vines.

Now let $\mathcal{Q}_{0}=\mathcal{Q}_{\mathrm{sk}}$ and
$\mathscr{A}_{0}=\mathscr{A}$ as in (8.3), then
$\mathcal{Q}\sim(\mathcal{Q}_{0},\mathscr{A}_{0})$. Denote $\mathscr{U}_{0}$,
with elements ordered as above, by
$\mathscr{U}_{0}=\\{\mathbb{U}_{0},\cdots,\mathbb{U}_{q-1}\\}$, and denote the
$\mathfrak{u}_{1},\mathfrak{u}_{23}$ nodes for $\mathbb{U}_{j}$ by
$\mathfrak{u}_{1}^{j},\mathfrak{u}_{23}^{j}$ etc. Define $\mathcal{Q}_{j+1}$
to be the result of splicing $\mathcal{Q}_{j}$ at $\mathbb{U}_{j}$ (so
$\mathbb{M}(\mathcal{Q}_{j+1})$ is the result of merging the $\mathbb{U}_{j}$
in $\mathbb{M}(\mathcal{Q}_{j})$ into a single atom), and let
$\mathscr{A}_{j}$ be the $\mathscr{A}$ collection corresponding to
$\mathcal{Q}_{j}$, and
$\mathscr{U}_{j}:=\\{\mathbb{U}_{j},\cdots,\mathbb{U}_{q-1}\\}$. As
$\mathcal{Q}$ runs over all full twists of a given couple at vines in
$\mathscr{V}_{0}$ (or $\mathscr{U}_{0}$), we know that $\mathcal{Q}_{j}$ runs
over all twists of a given couple at vines in $\mathscr{U}_{j}$. Moreover,
$\mathscr{A}_{j}$ runs over all collections of regular couples and regular
trees, such that the value of
$n(\mathcal{Q}^{(\mathfrak{u}_{23}^{i},\mathfrak{u}_{0}^{i})})+n(\mathcal{T}^{(\mathfrak{u}_{2}^{i})})$
is fixed for all $i\geq j$ such that $\mathbb{U}_{i}$ is a core vine, and all
other regular couples and regular trees are uniquely fixed.

By Remark 5.11, for each $j$ we have
$(\mathcal{Q}_{j},\mathscr{A}_{j})\leftrightarrow(\mathcal{Q}_{j+1},\texttt{cod}_{j},\mathfrak{n}_{j},\texttt{ind}_{j},\mathscr{B}_{j},\mathscr{A}_{j+1})$,
with $(\texttt{cod}_{j},\mathfrak{n}_{j})$ etc. corresponding to
$\mathbb{U}_{j}$ in $\mathcal{Q}_{j}$ (these matter only when $\mathbb{U}_{j}$
is a bad vine rather than a normal VC). By losing at most $C^{n}$ we may fix
the codes $\texttt{cod}_{j}$ for each $j$. Moreover, the branching node
$\mathfrak{n}_{j}$ for each $j$ is also uniquely fixed because it corresponds
to the atom formed by merging $\mathbb{U}_{i}\,(0\leq i\leq j)$ in
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$, and the molecule
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ (as a directed graph) does not vary
when $\mathcal{Q}_{\mathrm{sk}}$ is twisted (note however that
$\zeta_{\mathfrak{n}_{j}}$ may change if we twist at $\mathbb{U}_{j+1}$). As
such, we know that $(\mathcal{Q}_{j},\mathscr{A}_{j})$ uniquely corresponds to
the quadruple
$(\mathcal{Q}_{j+1},\mathscr{A}_{j+1},\texttt{ind}_{j},\mathscr{B}_{j})$. Let
$\mathcal{O}_{j}$ be the set of all nodes in $\mathcal{Q}_{j}$ that do _not_
belong to any
$\mathcal{Q}_{j}[\mathbb{U}_{i}]\backslash\\{\mathfrak{u}_{1}^{i}\\}$ for
$i\geq j$. Now we prove, by induction in $j$, the following result:

######  Proposition 8.5.

For each $j$, we have

$\mathscr{K}=(C^{+}\delta^{1/4})^{M_{j}}\sum_{(\mathcal{Q}_{j},\mathscr{A}_{j})}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{j}}\zeta(\mathcal{Q}_{j})\sum_{\mathscr{E}_{j}}\int_{\mathcal{E}_{j}}\epsilon_{\mathscr{E}_{j}}^{*}\prod_{\mathfrak{n}\in\mathcal{N}_{j}}e^{\zeta_{\mathfrak{n}}\pi
i\cdot\delta
L^{2\gamma}\Omega_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}}\\\
\times\mathcal{Z}_{j}\cdot{\prod_{\mathfrak{m}\in\mathcal{L}_{j}}^{(+)}\mathcal{K}_{\mathcal{Q}^{(\mathfrak{m},\mathfrak{m}_{-})}}(t_{\mathfrak{m}^{p}},t_{\mathfrak{m}_{-}^{p}},k_{\mathfrak{m}})}\prod_{\mathfrak{m}\in\mathcal{N}_{j}}\mathcal{K}_{\mathcal{T}^{(\mathfrak{m})}}^{*}(t_{\mathfrak{m}^{p}},t_{\mathfrak{m}},k_{\mathfrak{m}}).$
(8.4)

Here $n_{j}$ is the order of $\mathcal{Q}_{j}$,
$M_{j}=(n_{0}-n_{j})+n(\mathscr{A})-n(\mathscr{A}_{j})$, and all symbols with
subscript or superscript $j$ are associated with the couple $\mathcal{Q}_{j}$.
The summation in $(\mathcal{Q}_{j},\mathscr{A}_{j})$ is taken as above:
$\mathcal{Q}_{j}$ runs over all twists of a given couple at vines in
$\mathscr{U}_{j}$. Moreover, $\mathscr{A}_{j}$ runs over all collections of
regular couples and regular trees, such that the value of
$n(\mathcal{Q}^{(\mathfrak{u}_{23}^{i},\mathfrak{u}_{0}^{i})})+n(\mathcal{T}^{(\mathfrak{u}_{2}^{i})})$
is fixed for all $i\geq j$ such that $\mathbb{U}_{i}$ is a core vine, and all
other regular couples and regular trees are uniquely fixed.

In (8.4), the factor $\epsilon_{\mathscr{E}_{j}}^{*}$ is defined as in (2.10)
but with the product containing only $\mathfrak{n}\in\mathcal{W}_{j}$, where
$\mathcal{W}_{j}$ is a subset of $\mathcal{N}_{j}$; the function
$\mathcal{Z}_{j}=\mathcal{Z}_{j}(x_{0}^{j},k[\mathcal{X}_{j}],t[\mathcal{Y}_{j}],t_{\mathfrak{u}_{2}^{j}})$
where $\mathcal{X}_{j}$ and $\mathcal{Y}_{j}$ are two subsets of
$\mathcal{O}_{j}$ (when $j=q$ there is no $x_{0}^{j}$ and
$t_{\mathfrak{u}_{2}^{j}}$), and in the summation one replaces $x_{0}^{j}$ by
$k_{\mathfrak{u}_{2}^{j}}$ if $\zeta_{\mathfrak{u}_{2}^{j}}=-$, and by
$k_{\mathfrak{u}_{23}^{j}}$ otherwise. The objects
$\mathcal{W}_{j},\mathcal{X}_{j},\mathcal{Y}_{j},\mathcal{Z}_{j}$ etc. do not
depend on the choice of $\mathcal{Q}_{j}$ or $\mathscr{A}_{j}$, and the
function $\mathcal{Z}_{j}$ satisfies
$\|\mathcal{Z}_{j}\|_{Y_{\mathrm{loc}}^{\eta^{5}}}\lesssim(C^{+})^{j}L^{\Delta_{j}}$,
where $\Delta_{0}=0$ and $\Delta_{j+1}-\Delta_{j}$ equals $-\eta^{2}$ or
$\eta^{4}$ depending on whether $\mathscr{U}_{j}$ is a bad vine or a normal
vine chain. Finally, in the support of
$\mathcal{Z}_{j}\cdot\epsilon_{\mathscr{E}_{j}}^{*}$, the values of
$k_{\mathfrak{n}}$ variables must be inherited from a $k$-decoration of
$\mathcal{Q}_{\mathrm{sk}}$ that satisfies all the SG and LG assumptions
specified above, as well as the non-degeneracy condition
$\epsilon_{\mathscr{E}_{\mathrm{sk}}}\neq 0$.

###### Proof.

For $j=0$, recall that we have fixed the term $\mathscr{K}$ by specifying
whether each vine-like object in $\mathscr{C}$ has LG or SG; this is done by
attaching factors to the expression in (8.3) which are indicator functions of
differences of various $k_{\mathfrak{m}}$ in the decoration. Let the product
of these functions be $\mathcal{Z}_{0}$, then it only depends on the
$k_{\mathfrak{m}}$ variables for $\mathfrak{m}\in\mathcal{O}_{0}$ (see
Definition 8.3), and does not depend on any time variables $t_{\mathfrak{m}}$.
Let $\mathcal{W}_{0}=\mathcal{N}_{\mathrm{sk}}$, then (8.4) is true for $j=0$.

Suppose (8.4) is true for $j$, we will prove it for $j+1$. Start with the
expression $\mathscr{K}$ as in (8.4), note that the summation

$\sum_{(\mathcal{Q}_{j},\mathscr{A}_{j})}=\sum_{(\mathcal{Q}_{j+1},\mathscr{A}_{j+1})}\sum_{(\mathtt{ind}_{j},\mathscr{B}_{j})}$

as described above (recall that $\mathtt{cod}$ has been fixed). We now fix
$(\mathcal{Q}_{j+1},\mathscr{A}_{j+1})$, and consider the part of the sum and
integral in (8.4) that involves only $(\mathtt{ind}_{j},\mathscr{B}_{j})$ and
the $k_{\mathfrak{n}}$ and $t_{\mathfrak{n}}$ variables for
$\mathfrak{n}\in\mathcal{Q}_{j}[\mathbb{U}_{j}]\backslash\\{\mathfrak{u}_{1}^{j}\\}$.
Note also that
$\mathcal{Z}_{j}=\mathcal{Z}_{j}(x_{0}^{j},k[\mathcal{X}_{j}],t[\mathcal{Y}_{j}],t_{\mathfrak{u}_{2}^{j}})$;
by inserting finitely many smooth time cutoff functions, we may formally write

$\mathcal{Z}_{j}=\mathcal{Z}_{j}(x_{0}^{j},k_{\mathfrak{u}_{1}^{j}},k_{\mathfrak{u}_{11}^{j}},k_{\mathfrak{u}_{21}^{j}},k_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{1}^{j}},t_{\mathfrak{u}_{21}^{j}},t_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{2}^{j}},k[\mathcal{U}],t[\mathcal{V}]),$

for some
$\mathcal{U}\subset\mathcal{X}_{j}\subset\mathcal{O}_{j},\,\mathcal{V}\subset\mathcal{Y}_{j}\subset\mathcal{O}_{j}$
where
$\\{\mathfrak{u}_{1}^{j},\mathfrak{u}_{11}^{j},\mathfrak{u}_{21}^{j},\mathfrak{u}_{22}^{j}\\}\cap\mathcal{U}=\varnothing$
and
$\\{\mathfrak{u}_{1}^{j},\mathfrak{u}_{21}^{j},\mathfrak{u}_{22}^{j}\\}\cap\mathcal{V}=\varnothing$.

Then, this part of summation and integration is exactly of the form

$\sum_{(\mathtt{ind}_{j},\mathscr{B}_{j})}\mathcal{K}_{(\mathtt{ind}_{j},\mathscr{B}_{j})}^{(\mathbb{U}_{j},Z,W)}$
(8.5)

defined as in (7.12) in Section 7.2.1, where the set
$W=\mathcal{W}_{j}\cap\mathcal{Q}_{j}[\mathbb{U}_{j}]$, the function

$\qquad
Z(x_{0}^{j},k_{\mathfrak{u}_{1}^{j}},k_{\mathfrak{u}_{11}^{j}},k_{\mathfrak{u}_{21}^{j}},k_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{1}^{j}},t_{\mathfrak{u}_{21}^{j}},t_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{2}^{j}})=\mathcal{Z}_{j}(x_{0}^{j},k_{\mathfrak{u}_{1}^{j}},k_{\mathfrak{u}_{11}^{j}},k_{\mathfrak{u}_{21}^{j}},k_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{1}^{j}},t_{\mathfrak{u}_{21}^{j}},t_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{2}^{j}},k[\mathcal{U}],t[\mathcal{V}])$

with $k[\mathcal{U}]$ and $t[\mathcal{V}]$ viewed as parameters, and the sum
over $(\mathtt{ind}_{j},\mathscr{B}_{j})$ is exactly as in Propositions 7.5
and 7.6. Here again we plug in $x_{0}^{j}=k_{\mathfrak{u}_{2}^{j}}$ if
$\mathtt{ind}_{j}=\zeta_{\mathfrak{u}_{2}^{j}}=-$, and
$x_{0}^{j}=k_{\mathfrak{u}_{23}^{j}}$ otherwise.

Now define
$\mathcal{W}_{j+1}=\mathcal{W}_{j}\backslash\mathcal{Q}_{j}[\mathbb{U}_{j}]$
(where we identify branching nodes in $\mathcal{Q}_{j+1}$ with the
corresponding ones in $\mathcal{Q}_{j}$), and

$\displaystyle\mathcal{Z}_{j+1}$
$\displaystyle=\mathcal{Z}_{j+1}(k_{\mathfrak{u}_{1}^{j}},k_{\mathfrak{u}_{11}^{j}},k_{\mathfrak{u}_{21}^{j}},k_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{1}^{j}},t_{\mathfrak{u}_{21}^{j}},t_{\mathfrak{u}_{22}^{j}},k[\mathcal{U}],t[\mathcal{V}])$
(8.6) $\displaystyle:=(C^{+}\delta^{1/4})^{-n_{*}}\exp(-\pi i\cdot\delta
L^{2\gamma}t_{\mathfrak{u}_{1}^{j}}\Gamma)\sum_{(\mathtt{ind}_{j},\mathscr{B}_{j})}\mathcal{K}_{(\mathtt{sgn}_{j},\mathtt{ind}_{j},\mathscr{B}_{j})}^{(\mathbb{U}_{j},Z,W)}(k_{\mathfrak{u}_{1}^{j}},k_{\mathfrak{u}_{11}^{j}},k_{\mathfrak{u}_{21}^{j}},k_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{1}^{j}},t_{\mathfrak{u}_{21}^{j}},t_{\mathfrak{u}_{22}^{j}})$

as in (7.12), where $\mathtt{sgn}_{j}=\zeta_{\mathfrak{u}_{1}^{j}}$, and
$n_{*}$ equals the number of branching nodes in
$\mathcal{Q}_{j}[\mathbb{U}_{j}]\backslash\\{\mathfrak{u}_{1}^{j}\\}$ plus
$n(\mathscr{B}_{j})$. Then, the expression $\mathscr{K}$ in (8.4) can be
reduced to the same expression (8.4) with $j$ replaced by $j+1$ (note that
$M_{j+1}=M_{j}+n_{*}$, and similarly $n(\mathcal{Q}_{j})-n(\mathcal{Q}_{j+1})$
equals the number of branching nodes in
$\mathcal{Q}_{j}[\mathbb{U}_{j}]\backslash\\{\mathfrak{u}_{1}^{j}\\}$), with
the new quantities $\mathcal{W}_{j+1}$ and $\mathcal{Z}_{j+1}$.

It remains to prove that the new expression $\mathcal{Z}_{j+1}$ verifies our
assumptions. In addition to $k[\mathcal{U}]$ and $t[\mathcal{V}]$, the
function $\mathcal{Z}_{j+1}$ also depends on the new variables
$(k_{\mathfrak{u}_{1}^{j}},k_{\mathfrak{u}_{11}^{j}},k_{\mathfrak{u}_{21}^{j}},k_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{1}^{j}},t_{\mathfrak{u}_{21}^{j}},t_{\mathfrak{u}_{22}^{j}})$;
therefore we may define
$\mathcal{X}_{j+1}=(\mathcal{X}_{j}\cup\\{\mathfrak{u}_{1}^{j},\mathfrak{u}_{11}^{j},\mathfrak{u}_{21}^{j},\mathfrak{u}_{22}^{j}\\})\cap\mathcal{O}_{j+1}$
and
$\mathcal{Y}_{j+1}=(\mathcal{Y}_{j}\cup\\{\mathfrak{u}_{1}^{j},\mathfrak{u}_{21}^{j},\mathfrak{u}_{22}^{j}\\})\cap\mathcal{O}_{j+1}$.
Note that with such recursive definition of $\mathcal{X}_{j}$ and
$\mathcal{Y}_{j}$, it is easy to verify that, if $\mathbb{U}_{j}$ is
concatenated with $\mathbb{U}_{j+1}$, then
$\\{\mathfrak{u}_{1}^{j},\mathfrak{u}_{11}^{j}\\}\cap\mathcal{X}_{j}=\varnothing$;
this is because $\mathfrak{u}_{1}^{j}=\mathfrak{u}_{2}^{j+1}$ and
$\mathfrak{u}_{11}^{j}=\mathfrak{u}_{23}^{j+1}$, so these nodes belong to
$\mathbb{U}_{j+1}$, and are thus not involved in any vine merged _before_
$\mathbb{U}_{j}$.

Now, if any of these new variables
$(k_{\mathfrak{u}_{1}^{j}},k_{\mathfrak{u}_{11}^{j}},k_{\mathfrak{u}_{21}^{j}},k_{\mathfrak{u}_{22}^{j}},t_{\mathfrak{u}_{1}^{j}},t_{\mathfrak{u}_{21}^{j}},t_{\mathfrak{u}_{22}^{j}})$
coincides with $k_{\mathfrak{n}}$ or $t_{\mathfrak{n}}$ for any
$\mathbb{U}\in\mathscr{U}_{j+1}$ and
$\mathfrak{n}\in\mathcal{Q}_{j+1}[\mathbb{U}]\backslash\\{\mathfrak{u}_{1}\\}$
(where $\mathfrak{u}_{1}$ is associated with $\mathbb{U}$ as in Proposition
4.12)–or otherwise the assumptions for $\mathcal{Z}_{j+1}$ is already
verified–then $\mathbb{U}_{j}$ and $\mathbb{U}$ must share a common joint in
$\mathbb{M}(\mathcal{Q}_{j})$, so $\mathbb{U}=\mathbb{U}_{j+1}$ and it is
concatenated with $\mathbb{U}_{j}$ as above. Consequently the function
$\mathcal{Z}_{j}$, as well as $Z$ defined above, does not depend on the
variables $k_{\mathfrak{u}_{1}^{j}}=k_{\mathfrak{u}_{2}^{j+1}}$ and
$k_{\mathfrak{u}_{11}^{j}}=k_{\mathfrak{u}_{23}^{j+1}}$. By applying Remark
7.4 to (8.6), we see that $\mathcal{Z}_{j+1}$ depends only on the vector
variables $k[\mathcal{X}_{j+1}]$ and $x_{0}^{j+1}$ (which is
$k_{\mathfrak{u}_{1}^{j}}=k_{\mathfrak{u}_{2}^{j+1}}$ if
$\mathtt{ind}_{j+1}=\zeta_{\mathfrak{u}_{2}^{j+1}}=-$ and is
$k_{\mathfrak{u}_{11}^{j}}=k_{\mathfrak{u}_{23}^{j+1}}$ otherwise), and that
it does not depend on the choice of $\mathtt{ind}_{j+1}$ when regarded as a
function. The $t_{\mathfrak{n}}$ variables are similar, as in this case
$t_{\mathfrak{u}_{1}^{j}}=t_{\mathfrak{u}_{2}^{j+1}}$, so $\mathcal{Z}_{j+1}$
is also allowed to depend on $t_{\mathfrak{u}_{2}^{j+1}}$ and
$t[\mathcal{Y}_{j+1}]$, as desired.

Finally, it is also clear by definition that the support of
$\mathcal{Z}_{j+1}\cdot\epsilon_{\mathscr{E}_{j+1}}^{*}$ contains only those
decorations that are inherited from decorations in the support of
$\mathcal{Z}_{j}\cdot\epsilon_{\mathscr{E}_{j}}^{*}$, so it remains to prove
that

$\|\mathcal{Z}_{j+1}\|_{Y_{\mathrm{loc}}^{\eta^{5}}}\lesssim
L^{\Delta_{j+1}-\Delta_{j}}\|\mathcal{Z}_{j}\|_{Y_{\mathrm{loc}}^{\eta^{5}}},$
(8.7)

but this follows from Propositions 7.5, 7.6 and Remark 7.7. ∎

Define the couple $\mathcal{Q}_{\mathrm{sub}}$ to be the result of doing
splicing at all vines in $\mathscr{V}_{0}$ (equivalently, all ingredient vines
of all vine-like objects in $\mathscr{U}_{0}$) from
$\mathcal{Q}_{\mathrm{sk}}$, so the molecule
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ is obtained by merging all the vines
in $\mathscr{V}_{0}$ from $\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$. This whole
splicing process will be called _stage 1 reduction_. Using Proposition 8.5, we
can reduce Proposition 8.4 to the following counting estimate:

######  Proposition 8.6.

Suppose we fix $k\in\mathbb{Z}_{L}^{d}$ and
$k_{\ell}^{0}\in\mathbb{Z}_{L}^{d}$ for each bond $\ell$ of
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$, and $\beta_{v}\in\mathbb{R}$ for
each atom $v$ of $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$. Consider all the
maximal ladders in $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$; assume they are
$\mathcal{L}_{j}\,(1\leq j\leq q_{\mathrm{sub}})$ with length $z_{j}$, and fix
a dyadic number $P_{j}\in[L^{-1},1]\cup\\{0\\}$ for each $j$. Define
$m_{\mathrm{sub}}^{\prime}$ as the number of atoms not in any of these maximal
ladders, and $\rho_{\mathrm{sub}}=q_{\mathrm{sub}}+m_{\mathrm{sub}}^{\prime}$.

Now consider all the $k$-decorations $(k_{\ell})$ of
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$, with the following additional
requirements:

  1. (i)

This $k$-decoration $(k_{\ell})$ is inherited from a $k$-decoration
$\mathscr{E}_{\mathrm{sk}}$ of $\mathcal{Q}_{\mathrm{sk}}$ (and
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$) that satisfies all the SG and LG
assumptions specified in the proof above, as well as the non-degeneracy
condition $\epsilon_{\mathscr{E}_{\mathrm{sk}}}\neq 0$.

  2. (ii)

The decoration is restricted by $(\beta_{v})$ and $(k_{\ell}^{0})$, i.e.
$|\Gamma_{v}-\beta_{v}|\leq\delta^{-1}L^{-2\gamma}$ and
$|k_{\ell}-k_{\ell}^{0}|\leq 1$ (see Definition 4.6), and the gap $r_{j}$ of
each ladder $\mathcal{L}_{j}$ satisfies that $|r_{j}|\sim P_{j}$ (or
$|r_{j}|\gtrsim P_{j}$ if $P_{j}=1$).

Let $n_{\mathrm{sub}}$ be the order of $\mathcal{Q}_{\mathrm{sub}}$,
$\Delta_{\mathrm{sub}}$ is the end value $\Delta_{j}\,(j=q)$ in Proposition
8.5, and define $\mathfrak{X}_{j}=\min((\log L)^{2},1+\delta
L^{2\gamma}P_{j})$ (so $1\leq\mathfrak{X}_{j}\lesssim(\log L)^{2}$). Then, the
number $\mathfrak{C}$ of such restricted $k$-decorations is bounded by

$\mathfrak{C}\leq(C^{+}\delta^{-1/2})^{n_{\mathrm{sub}}}L^{(d-\gamma)n_{\mathrm{sub}}}\cdot
L^{-\Delta_{\mathrm{sub}}}L^{-\eta^{6}\rho_{\mathrm{sub}}}\prod_{j=1}^{q_{\mathrm{sub}}}\mathfrak{X}_{j}^{-z_{j}}.$
(8.8)

###### Proof of Proposition 8.4 assuming Proposition 8.6.

We start by defining
$\rho(\mathcal{Q})=\rho_{\mathrm{sub}}=q_{\mathrm{sub}}+m_{\mathrm{sub}}^{\prime}$
where $q_{\mathrm{sub}}$ and $m_{\mathrm{sub}}^{\prime}$ are defined as above.
Clearly its value does not depend on the congruence class of $\mathcal{Q}$ (as
$\mathcal{Q}_{\mathrm{sub}}$ does not), and for non-regular couples
$\mathcal{Q}$, the couple $\mathcal{Q}_{\mathrm{sk}}$ is nontrivial, and so is
$\mathcal{Q}_{\mathrm{sub}}$. This means that $1\leq\rho(\mathcal{Q})\leq n$
(the latter inequality is trivial as
$\rho_{\mathrm{sub}}=q_{\mathrm{sub}}+m_{\mathrm{sub}}^{\prime}$ does not
exceed $n_{\mathrm{sub}}\leq n$).

Next, once $\rho_{\mathrm{sub}}=\rho$ is fixed, the number of choices for
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ is clearly $\lesssim(C\rho)!C^{n}$
because this molecule must equal at most $\rho$ ladders of total length $\leq
n$, plus at most $\rho$ extra atoms. Then, going back from
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ to
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ again involves inserting finitely many
VC with total number of atoms $\leq n$, which again has $\lesssim C^{n}$
choices, as a VC of given length $m$ has $\lesssim C^{m}$ possibilities, thus
the number of choices for $\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ is also
$\lesssim(C\rho)!C^{n}$. By Propositions 4.5 and 4.9, we know that the same
bound holds for $\mathcal{Q}_{\mathrm{sk}}$ and $\mathcal{Q}$, and certainly
also for congruence classed of $\mathcal{Q}$.

It suffices to prove (8.1). We only need to control a single term
$\mathscr{K}$ as defined above, and using Proposition 8.5, we can start with
the expression $\mathscr{K}$ in (8.4) with $\mathcal{Q}_{j}$ replaced by
$\mathcal{Q}_{\mathrm{sub}}$ (which is the final result of all the splicing
described above), and all subscripts $(\cdots)_{j}$ replaced by
$(\cdots)_{\mathrm{sub}}$. There is now no summation in
$(\mathcal{Q}_{\mathrm{sub}},\mathscr{A}_{\mathrm{sub}})$, as the
$\mathscr{U}_{0}$ is reduced to empty set in the end, and the choice of
$\mathcal{Q}_{\mathrm{sub}}$ and $\mathscr{A}_{\mathrm{sub}}$ are fixed. For
simplicity we will denote
$(\rho_{\mathrm{sub}},q_{\mathrm{sub}},m_{\mathrm{sub}}^{\prime})$ by
$(\rho,q,m^{\prime})$.

Due to the $\langle k_{\mathfrak{l}}\rangle^{-40d}$ decay for each leaf
$\mathfrak{l}$, we may also restrict $k_{\mathfrak{l}}$ for each leaf
$\mathfrak{l}$ to a unit ball in $\mathbb{R}^{d}$, say
$|k_{\mathfrak{l}}-k_{\mathfrak{l}}^{0}|\leq 1$. By Lemma A.4, we can
decompose the full expression into $\lesssim C^{n}$ terms, such that in each
term, the variable $k_{\mathfrak{m}}$ belongs to a fixed unit ball not only
for _leaves_ $\mathfrak{m}=\mathfrak{l}$, but also for _all nodes_
$\mathfrak{m}$. With this localization, we may use the
$X_{\mathrm{loc}}^{\eta,40d}$, $X_{\mathrm{loc}}^{\eta,0}$ and
$Y_{\mathrm{loc}}^{\eta^{5}}$ bounds for the factors
$\mathcal{K}_{\mathcal{Q}^{(\mathfrak{l},\mathfrak{l}^{\prime})}}$,
$\mathcal{K}_{\mathcal{T}^{(\mathfrak{m})}}$ and $\mathcal{Z}$, see
Propositions 6.1 and 8.5, to reduce the product

$\mathcal{Z}_{\mathrm{sub}}\cdot{\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sub}}}^{(+)}\mathcal{K}_{\mathcal{Q}^{(\mathfrak{l},\mathfrak{l}^{\prime})}}(t_{\mathfrak{l}^{p}},t_{(\mathfrak{l}^{\prime})^{p}},k_{\mathfrak{l}})}\prod_{\mathfrak{m}\in\mathcal{N}_{\mathrm{sub}}}\mathcal{K}_{\mathcal{T}^{(\mathfrak{m})}}^{*}(t_{\mathfrak{m}^{p}},t_{\mathfrak{m}},k_{\mathfrak{m}})$

in (8.4) to a linear combination of functions

$L^{\Delta_{\mathrm{sub}}}(C^{+}\delta)^{n(\mathscr{A}_{\mathrm{sub}})/2}\cdot
e^{\pi i\mu t}\cdot\prod_{\mathfrak{n}\in\mathcal{N}_{\mathrm{sub}}}e^{\pi
i\lambda_{\mathfrak{n}}t_{\mathfrak{n}}}\cdot\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sub}}}^{(+)}\langle
k_{\mathfrak{l}}\rangle^{-40d}\cdot\mathcal{X}(k[\mathcal{Q}_{\mathrm{sub}}])$

for different choices of $(\mu,\lambda[\mathcal{N}_{\mathrm{sub}}])$, with the
coefficient being an $L^{1}$ integrable function of
$(\mu,\lambda[\mathcal{N}_{\mathrm{sub}}])$, where $\mathcal{X}$ is a bounded
function of all the $k_{\mathfrak{m}}$ variables. Below we may fix one choice
of $(\mu,\lambda[\mathcal{N}_{\mathrm{sub}}])$ (in which our estimates will be
uniform). To match Proposition 8.6, we also identify the $q$ maximal ladders
in $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ and fix a dyadic number
$P_{j}\in[L^{-1},1]\cup\\{0\\}$ such that the gap of the $j$-th ladder
$\mathcal{L}_{j}$ is $|r_{j}|\sim P_{j}$ (or $|r_{j}|\gtrsim P_{j}$ if
$P_{j}=1$). This causes a loss of $(\log L)^{q}$ by summing over all choices
of $P_{j}$, which will be ignored in view of the $L^{-\eta^{6}\rho}$ gain
expected in (8.8).

Now consider such a term (we call it $\mathscr{M}$), rewrite it as

$\mathscr{M}=e^{\pi i\mu
t}\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sub}}}^{(+)}\langle
k_{\mathfrak{l}}^{0}\rangle^{-40d}\cdot(C^{+}\delta^{1/4})^{M_{\mathrm{sub}}}(C^{+}\delta)^{n(\mathscr{A}_{\mathrm{sub}})/2}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{\mathrm{sub}}}\\\
\times
L^{\Delta_{\mathrm{sub}}}\sum_{\mathscr{E}_{\mathrm{sub}}}\mathcal{B}(t,t,\alpha[\mathcal{N}_{\mathrm{sub}}])\mathcal{Y}(k[\mathcal{Q}_{\mathrm{sub}}]),$
(8.9)

where in (8.9), the sum is taken over all $k$-decorations
$\mathscr{E}_{\mathrm{sub}}$ of $\mathcal{Q}_{\mathrm{sub}}$ restricted by
some fixed $(k_{\ell}^{0})$, $n_{\mathrm{sub}}$ is the order of
$\mathcal{Q}_{\mathrm{sub}}$ and
$M_{\mathrm{sub}}=(n_{0}-n_{\mathrm{sub}})+n(\mathscr{A})-n(\mathscr{A}_{\mathrm{sub}})$
where $n_{0}$ is the order of $\mathcal{Q}_{\mathrm{sk}}$ and $\mathscr{A}$
and $\mathscr{A}_{\mathrm{sub}}$ are as in (8.3) and (8.4). The function
$\mathcal{Y}$ is uniformly bounded, and is supported on the $k$-decorations of
$\mathcal{Q}_{\mathrm{sub}}$ that satisfy the above gap assumptions for
ladders, and are inherited from $k$-decorations of $\mathcal{Q}_{\mathrm{sk}}$
that satisfy all the SG, LG and non-degeneracy assumptions specified above,
due to Proposition 8.5. Finally, the $\mathcal{B}$ expression is defined as

$\mathcal{B}(t,t,\alpha[\mathcal{N}_{\mathrm{sub}}]):=\int_{\mathcal{E}_{\mathrm{sub}}}\prod_{\mathfrak{n}\in\mathcal{N}_{\mathrm{sub}}}e^{\pi
i\alpha_{\mathfrak{n}}t_{\mathfrak{n}}}\,\mathrm{d}t_{\mathfrak{n}}$ (8.10)

and $\alpha_{\mathfrak{n}}:=\zeta_{\mathfrak{n}}\delta
L^{2\gamma}\Omega_{\mathfrak{n}}+\lambda_{\mathfrak{n}}$. From (8.9), by first
fixing the values of
$\lfloor\alpha_{\mathfrak{n}}\rfloor:=\sigma_{\mathfrak{n}}\in\mathbb{Z}$
(where each $\sigma_{\mathfrak{n}}$ belongs to a fixed set of $\leq L^{10d}$
elements) and summing over all choices of $\sigma_{\mathfrak{n}}$, we can
bound

$|\mathscr{M}|\lesssim\langle
k\rangle^{-20d}\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sub}}}^{(+)}\langle
k_{\mathfrak{l}}^{0}\rangle^{-20d}\cdot(C^{+}\delta^{1/4})^{M_{\mathrm{sub}}}(C^{+}\delta)^{n(\mathscr{A}_{\mathrm{sub}})/2}\bigg{(}\frac{\delta}{2L^{d-\gamma}}\bigg{)}^{n_{\mathrm{sub}}}\\\
\times
L^{\Delta_{\mathrm{sub}}}\sum_{\sigma[\mathcal{N}_{\mathrm{sub}}]}\sup_{\alpha[\mathcal{N}_{\mathrm{sub}}]:|\alpha_{\mathfrak{n}}-\sigma_{\mathfrak{n}}|\leq
1}|\mathcal{B}(t,t,\alpha[\mathcal{N}_{\mathrm{sub}}])|\cdot\sup_{\sigma[\mathcal{N}_{\mathrm{sub}}]}\sum_{\mathscr{E}_{\mathrm{sub}}}1,$
(8.11)

where the first summation in (8.11) is taken over all
$\sigma[\mathcal{N}_{\mathrm{sub}}]$, and the second summation is taken over
all $k$-decorations $\mathscr{E}_{\mathrm{sub}}$ that satisfies all the above
assumptions, as well as $|\delta
L^{2\gamma}\Omega_{\mathfrak{n}}\pm(\sigma_{\mathfrak{n}}-\lambda_{\mathfrak{n}})|\leq
1$ for each node $\mathfrak{n}$.

Now, with fixed $\sigma[\mathcal{N}_{\mathrm{sub}}]$, consider the
$k$-decoration $\mathscr{E}_{\mathrm{sub}}$ in the second summation in (8.11),
and the corresponding $k$-decoration of
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ defined by Definition 4.6, then the
latter decoration will have to satisfy all the requirements made in the
statement of Proposition 8.6 (for some choice of $k_{\ell}^{0}$ and
$\beta_{v}$). Thus, by (8.8), the second summation in (8.11) is bounded by

$\sup_{\sigma[\mathcal{N}_{\mathrm{sub}}]}\sum_{\mathscr{E}_{\mathrm{sub}}}1\lesssim(C^{+}\delta^{-1/2})^{n_{\mathrm{sub}}}L^{(d-\gamma)n_{\mathrm{sub}}}\cdot
L^{-\Delta_{\mathrm{sub}}}L^{-\eta^{6}\rho}\prod_{j=1}^{q}\mathfrak{X}_{j}^{-z_{j}}.$
(8.12)

Then, if we can prove that the first summation is bounded by

$\sum_{\sigma[\mathcal{N}_{\mathrm{sub}}]}\sup_{\alpha[\mathcal{N}_{\mathrm{sub}}]:|\alpha_{\mathfrak{n}}-\sigma_{\mathfrak{n}}|\leq
1}|\mathcal{B}(t,t,\alpha[\mathcal{N}_{\mathrm{sub}}])|\lesssim(C^{+}\delta^{-1/4})^{n_{\mathrm{sub}}}\cdot
L^{C\rho\sqrt{\delta}}\prod_{j=1}^{q}\mathfrak{X}_{j}^{z_{j}}\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sub}}}^{(+)}\langle
k_{\mathfrak{l}}^{0}\rangle^{20d},$ (8.13)

then putting together (8.11), (8.12) and (8.13) (noticing also equalities like
$n=n_{0}+n(\mathscr{A})$ etc.). we can get

$|\mathscr{M}|\lesssim(C^{+}\delta^{1/4})^{n}\langle
k\rangle^{-20d}L^{-\eta^{7}\rho},$

which then proves Proposition 8.4.

Finally let us prove (8.13); of course we only need to prove it under the
restriction on $(\sigma_{\mathfrak{n}})$ that, there exists some
$k$-decoration $\mathscr{E}_{\mathrm{sub}}$ satisfying all the previously
specified assumptions, such that the corresponding quantities
$\alpha_{\mathfrak{n}}=\zeta_{\mathfrak{n}}\delta
L^{2\gamma}\Omega_{\mathfrak{n}}+\lambda_{\mathfrak{n}}$ satisfies
$|\alpha_{\mathfrak{n}}-\sigma_{\mathfrak{n}}|\leq 1$. Then, this is basically
a consequence of Proposition 10.1 of [14], but with some additional twists.
Consider all the ladders $\mathcal{L}_{j}$ such that $1+\delta
L^{2\gamma}P_{j}\leq(\log L)^{2}$, so $\mathfrak{X}_{j}=1+\delta
L^{2\gamma}P_{j}$; assume these correspond to $1\leq j\leq p$ for some $1\leq
p\leq q$. For each ladder $\mathcal{L}_{j}\,(j\leq p)$ and each pair of atoms
$(v_{1},v_{2})$ connected by a double bond, we consider them together, as well
as the corresponding branching nodes $\mathfrak{n}_{j}=\mathfrak{n}(v_{j})$ in
$\mathcal{Q}_{\mathrm{sub}}$. If the gap of this ladder is $|r_{j}|\sim
P_{j}$, then it is easy to see that for some choice of $\pm$, we have that
$\Omega_{\mathfrak{n}_{1}}\pm\Omega_{\mathfrak{n}_{2}}=r_{j}\cdot(k_{\mathfrak{m}}\pm
k_{\mathfrak{m}^{\prime}})$ for some fixed nodes $\mathfrak{m}$ and
$\mathfrak{m}^{\prime}$ depending on $v_{1}$ and $v_{2}$.

By Lemma A.4, we can decompose the set of all $k$-decorations into at most
$C^{n_{\mathrm{sub}}}\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sub}}}^{(+)}\langle
k_{\mathfrak{l}}^{0}\rangle^{10d}$ subsets such that whenever a $k$-decoration
belongs to a fixed subset, and whenever $|r_{j}|\sim P_{j}$, the value of
$r_{j}\cdot(k_{\mathfrak{m}}\pm k_{\mathfrak{m}^{\prime}})$ must belong to an
fixed interval of length $P_{j}$, which does not depend on the choice of
$r_{j}$ or the decoration. This implies that
$\sigma_{\mathfrak{n}_{1}}\pm\sigma_{\mathfrak{n}_{2}}$ belongs to a fixed
interval of length $1+\delta L^{2\gamma}P_{j}$, and thus it has at most
$O(1+\delta L^{2\gamma}P_{j})$ choices. This means that, after losing a factor

$C^{n_{\mathrm{sub}}}\prod_{j=1}^{p}(1+\delta
L^{2\gamma}P_{j})^{z_{j}+1}=C^{n_{\mathrm{sub}}}\prod_{j=1}^{p}\mathfrak{X}_{j}^{z_{j}+1},$

we can assume the value of
$\sigma_{\mathfrak{n}_{1}}\pm\sigma_{\mathfrak{n}_{2}}$ is fixed for each pair
$(\mathfrak{n}_{1},\mathfrak{n}_{2})$ as above. Note that we may replace the
above power $z_{j}+1$ of $\mathfrak{X}_{j}$ by $z_{j}$, as
$\mathfrak{X}_{j}\leq(\log L)^{2}$ and $p\leq\rho_{\mathrm{sub}}$, so the loss
$(\log L)^{Cp}$ here can be covered by the gain
$L^{-\eta^{6}\rho_{\mathrm{sub}}}$ in (8.8).

Notice that the ladders are the same as type II chains defined in [14], we can
now run the same arguments in the proof of Proposition 10.1 in [14], to get
that

$\sum_{\sigma[\mathcal{N}_{\mathrm{sub}}]}\sup_{\alpha[\mathcal{N}_{\mathrm{sub}}]:|\alpha_{\mathfrak{n}}-\sigma_{\mathfrak{n}}|\leq
1}|\mathcal{B}(t,t,\alpha[\mathcal{N}_{\mathrm{sub}}])|\\\ \lesssim
C^{n_{\mathrm{sub}}}\prod_{j=1}^{p}\mathfrak{X}_{j}^{z_{j}}\prod_{\mathfrak{l}\in\mathcal{L}_{\mathrm{sub}}}^{(+)}\langle
k_{\mathfrak{l}}^{0}\rangle^{10d}\cdot(C^{+})^{n_{\mathrm{sub}}}\delta^{-n_{\mathrm{sub}}/4}L^{Cp\sqrt{\delta}}(\log
L)^{Cp+m^{\prime\prime}},$ (8.14)

where $m^{\prime\prime}$ is the number of atoms not in the ladders
$\mathcal{L}_{j}\,(1\leq j\leq p)$, so in particular
$m^{\prime\prime}=m^{\prime}+2(z_{p+1}+\cdots+z_{q})$; namely each ladder
$\mathcal{L}_{j}\,(1\leq j\leq p)$ causes a loss of $L^{C\sqrt{\delta}}(\log
L)^{C}$ and each remaining atom causes a loss of $\log L$, see for example
(10.10) in Section 10.2 of [14]. It is easily seen that (8.14) is stronger
than (8.13), using the fact that $\rho=q+m^{\prime}\geq p+m^{\prime}$. This
completes the proof. ∎

##  9\. Reduction to large gap molecules

###  9.1. Preliminary setup

We now start the proof of Proposition 8.6. The idea is to perform operations
to further reduce the molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ to
simpler molecules (this reduction will be called stage 2, compared to stage 1
reduction in Section 8.2). In this process we will keep track of the
corresponding counting estimate, as well as certain parameters of the
molecule, especially the characteristics $\chi=E-V+F$, where $E$, $V$ and $F$
are the number of bonds, atoms, and components (Definition 4.1).

Before introducing the stage 2 reduction operations, we first need to setup
the properties of the molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$,
compared to the original molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$
before stage 1 reduction, as well as the corresponding decorations. These are
summarized in Proposition 9.1 below.

######  Proposition 9.1.

Suppose the molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ is reduced to
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ via stage 1 reduction, as described
in Section 8.2. Let $\mathscr{C}$ be defined for $\mathcal{Q}_{\mathrm{sk}}$
as in Lemma 8.1. Consider $k$-decoration $(k_{\ell})$ of
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ that is inherited from a
$k$-decoration $\mathscr{E}_{\mathrm{sk}}$ of $\mathcal{Q}_{\mathrm{sk}}$ (and
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$), as in Proposition 8.6. Then we have
the following properties:

  1. (a)

Each SGHVC in $\mathscr{C}$ is reduced to an SGHV, and each SGHV remains
unchanged. Each SGVC in $\mathscr{C}$ is reduced to an SG vine that is also a
(CN) vine, or an SG vine that is also a root (CL) vine (see Proposition 4.12),
or a single atom. Each DV in $\mathscr{C}$ remains unchanged, and still
contains two vines (V).

  2. (b)

The molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ is connected, there are
only two atoms of degree $3$, and all other atoms have degree $4$. We define
an atom $v$ in $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ to be a _hinge atom_ ,
if it is the atom that results from merging a SGVC in
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$. These include the single atoms
defined in (a), as well as some of the joints of the SGHV and SG vines defined
in (a).

  3. (c)

Any degenerate atom (where there are two bonds $(\ell_{1},\ell_{2})$ at $v$ of
opposite directions and $k_{\ell_{1}}=k_{\ell_{2}}$ in the decoration) must be
a hinge atom, and any triple bond must have an endpoint that is a hinge atom.

  4. (d)

For any hinge atom $v$, we can find two bonds $(\ell_{1},\ell_{2})$ at $v$, so
that in the original molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ they are
the two bonds at one joint of an SGVC in $\mathscr{C}$ that do _not_ belong to
this VC (this VC is merged in Stage 1). Also the triple
$(v,\ell_{1},\ell_{2})$ has SG and $k_{\ell_{1}}\neq k_{\ell_{2}}$. Moreover,
if $v$ has degree $4$, then the other two bonds $(\ell_{3},\ell_{4})$ at $v$
satisfy the same properties as $(\ell_{1},\ell_{2})$ above.

  5. (e)

Any SG vine-like object that is not a subset of an object in (a), which also
does _not_ contain a hinge atom, must be a subset of a LG vine-like object in
$\mathscr{C}$. Note that it must be in one of the cases defined in Proposition
A.5.

  6. (f)

Any SG vine-like object (say $\mathbb{U}$) that is not a subset of an object
in (a), which 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 as in (d).

###### Proof.

First, (a) directly follows from the definition of $\mathscr{V}_{0}$ and
$\mathscr{U}_{0}$ in Section 8.2 and the definition of splicing operation in
Proposition 4.12, using also Corollary 4.13. Also (b) follows from Proposition
4.4 and Remark 4.7 (note that degree $2$ atoms cannot be generated by splicing
due to our treatment for root (CL) vines), and (c) follows from the fact that
there is no degenerate atoms nor triple bonds in
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$. As for (d), if $v$ is a hinge atom,
since $v$ has degree $3$ or $4$, we must have in
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ an SGVC (say $\mathbb{V}\mathbb{C}$)
with one of its joints, say $v_{1}$, having degree $4$. Let the two bonds
$(\ell_{1}^{\prime},\ell_{2}^{\prime})$ at $v_{1}$ belong to
$\mathbb{V}\mathbb{C}$ and the other two bonds $(\ell_{1},\ell_{2})$ at
$v_{1}$ not belong to $\mathbb{V}\mathbb{C}$, then the triple
$(v,\ell_{1},\ell_{2})$ in $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ is just
the triple $(v_{1},\ell_{1},\ell_{2})$ in
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ and must have SG as
$\mathbb{V}\mathbb{C}$ has SG. If $d(v)=4$, then the other joint $v_{2}$ of
$\mathbb{V}\mathbb{C}$ also has degree $4$ and we can repeat the above
argument to show that $(\ell_{3},\ell_{4})$ satisfy the same properties as
$(\ell_{1},\ell_{2})$.

Now let an SG vine-like object, say $\mathbb{U}$, be in
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ which is not a subset of an object in
(a). If $\mathbb{U}$ does not contain a hinge atom, then the same $\mathbb{U}$
must exist in the original molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ as
a vine-like object, and is not changed in the process. By Lemma 8.1,
$\mathbb{U}$ must be the subset of an object in $\mathscr{C}$, and this object
must be an LG vine-like object, since otherwise it will be involved in the
splicing process and then $\mathbb{U}$ would be a subset of an object in
(a)–(b). This proves (e).

Next, suppose $\mathbb{U}$ contains a hinge atom. If (f) does not hold, then
$\mathbb{U}$ must contain one hinge atom as a joint, such that the bonds
$(\ell_{1},\ell_{2})$ defined in (d) both belong to $\mathbb{U}$ (if not, then
replace $(\ell_{1},\ell_{2})$ by $(\ell_{3},\ell_{4})$ defined in (d)). The
other joint $v_{2}$ of $\mathbb{V}$ may also be a hinge atom; if so then again
the bonds $(\ell_{1}^{\prime},\ell_{2}^{\prime})$ defined in (d)
(corresponding to $v_{2}$) both belong to $\mathbb{U}$. Then the same
$\mathbb{U}$ must exist in the original molecule
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ as an SGVC, and is not changed in the
process. Suppose $v_{1}$ is formed by merging an SGVC called
$\mathbb{V}\mathbb{C}_{1}$, and $v_{2}$ (if applicable) is formed by merging
an SGVC called $\mathbb{V}\mathbb{C}_{2}$. Then in
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$ one can concatenate $\mathbb{U}$ with
$\mathbb{V}\mathbb{C}_{1}$ (and $\mathbb{V}\mathbb{C}_{2}$ if applicable) to
form a larger SG vine-like object, which must be a subset of an SG vine-like
object in $\mathscr{C}$. This implies that $\mathbb{U}$ must be a subset of an
object in (a), which is a contradiction and proves (f). ∎

######  Remark 9.2.

Consider any $k$-decoration $(k_{\ell})$ of
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$, restricted by $(\beta_{v})$ and
$(k_{\ell}^{0})$, that is inherited from a $k$-decoration
$\mathscr{E}_{\mathrm{sk}}$ of $\mathcal{Q}_{\mathrm{sk}}$ (and
$\mathbb{M}(\mathcal{Q}_{\mathrm{sk}})$), as in Proposition 8.6. Then, the
following conditions hold for this decoration:

  1. (i)

Those imposed by Proposition 9.1,

  2. (ii)

The gaps $|r_{j}|\sim P_{j}$ for each maximal ladder $\mathcal{L}_{j}$, where
$P_{j}$ are fixed as in Proposition 8.6.

  3. (iii)

By losing at most $C^{n_{\mathrm{sub}}}$, we may assume that each atom that is
not an interior atom of an SG vine-like object in (a) and (e) of Proposition
9.1 is fixed to be either SG or LG. For any SG atom $v$ we fix the
corresponding bonds $(\ell_{1},\ell_{2})$ according to the following rules:
for any hinge atom we must fix the bonds as in (d) of Proposition 9.1, for any
joint of any SG vine-like object defined in (a) and (e) of Proposition 9.1 we
must fix the bonds $(\ell_{1},\ell_{2})$ to belong to the corresponding vine-
like object (if both vines (V) in one DV are SG, then $(\ell_{1},\ell_{2})$ is
chosen from only one them), and for any endpoint of any triple bond in (c) we
must fix the bonds using the endpoint that is a hinge atom. For all other SG
atoms we may fix the bonds $(\ell_{1},\ell_{2})$ arbitrarily.

  4. (iv)

Finally, for each SG atom $v$, we fix its gap as $|r|\sim R_{v}$, where
$R_{v}\in[L^{-1},L^{-\gamma+\eta}]$ is a dyadic number. Note that this
condition leads to a loss of $(\log L)^{p}$, where $p$ is the number of SG
atoms, which we will also treat in the proof below.

###  9.2. The cutting operation

Now we define the basic operation in the stage 2 reduction, namely the
_cutting_ operation, which involves dividing an atom of degree $3$ or $4$ into
an atom of degree $2$ and another atom of degree $1$ or $2$.

######  Definition 9.3.

Given a molecule $\mathbb{M}$ and an atom $v$ of degree $3$ or $4$. 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 22. We also call this cut an
$\alpha$\- (resp. $\beta$-) cut, if it does not (resp. does) generate a new
connected component, and accordingly we call the resulting atoms $\alpha$\- or
$\beta$\- atoms. If we are also given a decoration, then we may define the gap
of this cut to be $r:=k_{\ell_{1}}-k_{\ell_{2}}$.

Figure 22. A cutting operation executed at a degree $4$ atom $v$, see
Definition 9.3.

The stage 2 reduction, which will be applied to
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ involves cutting various atoms as in
Definition 9.3, 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 a decoration
$(k_{\ell})$ of $\mathbb{M}_{\mathrm{pre}}$ naturally leads to a decoration of
$\mathbb{M}_{\mathrm{pos}}$.

For any molecule $\mathbb{M}$ (which could be $\mathbb{M}_{\mathrm{pre}}$ or
$\mathbb{M}_{\mathrm{pos}}$), consider the possible $(c_{v})$-decorations of
$\mathbb{M}$, also restricted by $(\beta_{v})$ and $(k_{\ell}^{0})$; we also
assume that this decoration is inherited from a $k$-decoration of
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ that satisfies all assumptions in
Remark 9.2. 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 $\mathfrak{C}$. In view of the right hand side of (8.8) and the
logarithmic loss in Remark 9.2, we also define

$\mathfrak{A}:=\mathfrak{C}\cdot
L^{-(d-\gamma)\chi(\mathbb{M})}(C^{+}\delta^{-1/2})^{-\chi(\mathbb{M})}\prod_{j=1}^{q}\mathfrak{X}_{j}^{z_{j}}\cdot(\log
L)^{Cp},$ (9.1)

where $\chi(\mathbb{M})$ is the characteristic of $\mathbb{M}$, $p$ is the
number of remaining SG atoms in $\mathbb{M}$, and the other notations are
under the setting of Proposition 8.6 but adapted to $\mathbb{M}$ instead of
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$, for example $\mathcal{L}_{j}$ are
the maximal ladders in $\mathbb{M}$. Define also $m^{\prime}$ to be the number
of atoms not in the maximal ladders, and $\rho=q+m^{\prime}$ as in Proposition
8.6 before, and denote the product $\prod_{j=1}^{q}\mathfrak{X}_{j}^{z_{j}}$
in (9.1) as $\mathfrak{P}$. The notations
$(\mathfrak{A},\mathfrak{C},\mathfrak{P})$ and $(\rho,q,m^{\prime})$ will
apply to all the molecules appearing in the rest of this paper, with possible
subscripts matching those of $\mathbb{M}$ (so $\mathfrak{A}_{\mathrm{pre}}$ is
defined for the molecule $\mathbb{M}_{\mathrm{pre}}$ etc.; sometimes for
emphasis we may also write $\mathfrak{A}(\mathbb{M})$ or
$\mathfrak{A}(\mathbb{M}_{\mathrm{pre}})$ instead of $\mathfrak{A}$ and
$\mathfrak{A}_{\mathrm{pre}}$).

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

$\mathfrak{A}_{\mathrm{pre}}\lesssim\mathfrak{D}\cdot\mathfrak{A}_{\mathrm{pos}}$
(9.2)

for some quantity $\mathfrak{D}$, which we define to be the _deviation_ of
this operation. Clearly, if we know the deviation in each operation step, and
an upper bound for the counting problem associated with the final molecule
$\mathbb{M}_{\mathrm{fin}}$, then we can deduce from this information an upper
bound for the counting problem associated with
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ using (9.1) and (9.2).

In Section 9.3 we define the stage 2 reduction process, and study the
deviation in each step; in Section 9.4 and Section 10 we solve the counting
problem associated with the final molecule $\mathbb{M}_{\mathrm{fin}}$, and
completes the proof of Proposition 8.6.

###  9.3. Stage 2 reduction: Cutting degree $3$ and $4$ atoms

Start with the molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ and a
restricted decoration $(k_{\ell})$ satisfying the assumptions in Remark 9.2.
Recall the notion of $V,E,F$ and $\chi$ as in Definition 4.1; define also
$V_{\alpha}$ and $V_{\beta}$ to be the number of $\alpha$\- and $\beta$-atoms,
and use $\Delta$ to denote increments.

_Step 1: removing SG vine-like objects._ In Step 1, we collect all the SG
vine-like objects defined in (a) and (e) of Proposition 9.1, including also
the triple bonds in (c). Note that if both vines (V) in one DV in (a) are SG
then we only collect one of them. For each of these objects $\mathbb{U}$, we
cut the molecule at each joint, along the two bonds $(\ell_{1},\ell_{2})$
fixed as in Remark 9.2 (i.e. along the two bonds that belong to $\mathbb{U}$).
This disconnects a VC, say $\mathbb{V}\mathbb{C}$ (which is $\mathbb{U}$ or
$\mathbb{U}$ minus a bond in case of HV or HVC), from the rest of the
molecule, and then we remove $\mathbb{V}\mathbb{C}$. The two joints then
become degree $1$ or $2$ atoms, and we _define them as $\alpha$-atoms_; we
also label each of them by the dyadic number $R\in[L^{-1},L^{-\gamma+\eta}]$
such that $|r|\sim R$ for the gap $r$ of $\mathbb{V}\mathbb{C}$ (note that we
must have $r\neq 0$).

_Step 2: removing triangles._ In Step 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
as fixed in Remark 9.2 for $j\in\\{1,2\\}$. Let
$|k_{\ell_{j+1}}-k_{\ell_{j+2}}|\sim R_{j}$ for $j\in\\{1,2\\}$ with
$R_{j}\in[L^{-1},L^{-\gamma+\eta}]$. If $\mathrm{deg}\,v_{j}\geq 3$, then we
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 $1$ or $2$. We call $v_{3}$ a $\beta$-atom, and call $v_{j}\,(1\leq
j\leq 2)$ 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 call $v_{1}$ an
$\alpha$-atom and $v_{2}$ a $\beta$-atom. For any $\alpha$-atom $v_{j}$ we
label it by the corresponding $R_{j}$.

_Step 3: remaining SG cuts._ In Step 3, we select each of the remaining SG
atoms $v$ of degree $\geq 3$, and cut them along the designated bonds
$(\ell_{1},\ell_{2})$ in Remark 9.2, in arbitrary order. Note that each cut
may be $\alpha$\- or $\beta$-cut; we call the resulting atoms $\alpha$\- or
$\beta$-atoms, and label any $\alpha$-atom by the dyadic number
$R\in[L^{-1},L^{-\gamma+\eta}]$, as in Definition 9.3 (again $R\neq 0$ due to
our choice of $(\ell_{1},\ell_{2})$).

_Step 4: remaining $\beta$-cuts._ After Step 3, there is now no more SG atoms
left in the molecule. In Step 4, we look for all the possible degree $3$ or
$4$ atoms where a $\beta$-cut is possible, and perform the corresponding
$\beta$-cut, until this can no longer be done.

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 happened before the cutting at the atom $v$, such that the gaps
$r,r^{\prime}$ of these cuttings satisfy $0<|r\pm r^{\prime}|\leq
L^{-50\eta}R_{v}$; if no such cutting operation exists then define
$\iota_{v}=0$. Note that the number of choices for all $(\iota_{v})$ is at
most $C^{p}$, which can be absorbed into the last factor on the right hand
side of (9.1). Therefore we may assume a choice of $(\iota_{v})$ is fixed in
the proof below.

######  Proposition 9.4.

After each operation in Steps 1–4, we have that

$\mathfrak{D}\lesssim
L^{-\eta^{3}}\cdot\prod_{v}^{(\alpha)}L^{(\gamma+3\eta)/2-\kappa_{v}\eta}R_{v}\cdot
L^{(2\gamma_{0}+5\eta^{2})\Delta F-(\gamma_{0}+2\eta^{2})\Delta V_{\beta}}$
(9.3)

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

###### Proof.

First assume $\iota_{v}=0$ for all $\alpha$-atoms $v$. We start with the
simplest case (in Steps 3–4) where we cut at a single atom $v$. Note that, if
this is an $\alpha$\- (resp. $\beta$-) cut then we have $(\Delta\chi,\Delta
F,\Delta V_{\beta})=(-1,0,0)$ (resp. $(0,1,2)$). In the case of $\beta$-cut at
$v$ along $(\ell_{1},\ell_{2})$, the value of $k_{\ell_{1}}-k_{\ell_{2}}$ is
uniquely fixed (by summing the equation (4.1) over all atoms $v^{\prime}\neq
v$ that belongs to one component after making the cut). Similarly, using
(4.2), we know that the number of choices of
$|k_{\ell_{1}}|^{2}-|k_{\ell_{2}}|^{2}$, up to distance
$\delta^{-1}L^{-2\gamma}$, is $\lesssim n_{\mathrm{sub}}\leq(\log L)^{C}$.
This means that $\mathfrak{C}_{\mathrm{pre}}\lesssim(\log
L)^{C}\mathfrak{C}_{\mathrm{pos}}$, hence $\mathfrak{D}\lesssim(\log L)^{C}$
using (9.1) and (9.2), and noticing that the cut only changes the length of at
most one maximal ladder by $O(1)$, using also $1\leq\mathfrak{X}_{j}\leq(\log
L)^{2}$. Therefore (9.3) is true in this case. As for $\alpha$-cuts, let the
corresponding gap $R$ be fixed, then the number of choices for
$k_{\ell_{1}}-k_{\ell_{2}}$ is $\lesssim(RL)^{d}$; once
$k_{\ell_{1}}-k_{\ell_{2}}$ is fixed (and $|k_{\ell_{1}}-k_{\ell_{2}}|\sim
R$), the number of choices of $|k_{\ell_{1}}|^{2}-|k_{\ell_{2}}|^{2}$, up to
distance $\delta^{-1}L^{-2\gamma}$, is $\lesssim 1+\delta RL^{2\gamma}$
because $k_{\ell_{1}}+k_{\ell_{2}}$ belongs to a fixed ball of radius $\sim
1$. This implies that

$\mathfrak{D}\lesssim(RL)^{d}(1+RL^{2\gamma})L^{-(d-\gamma)}(\log
L)^{C}\lesssim L^{-\eta^{3}}\cdot L^{\gamma+3\eta}R^{2},$

where the $(\log L)^{C}$ factor is similar to above, and the last inequality
can be verified using $R\in[L^{-1},L^{-\gamma+\eta}]$ and $d\geq 3$. Since
both new $\alpha$-atoms are labeled by $R$, this proves (9.3).

Next consider Step 2, for which we know $\Delta F\in\\{0,1,2\\}$, $\Delta
V_{\beta}=\Delta F+1$, and $\Delta\chi=\Delta F-3$; moreover
$\mathfrak{D}\lesssim L^{(d-\gamma)\Delta\chi}(\log L)^{C}\mathfrak{F}$, where
$\mathfrak{F}$ is the number of choices for
$(k_{\ell_{1}},k_{\ell_{2}},k_{\ell_{3}})$. If $\Delta F=2$, then similar to
above, for _each_ $j$ we know that

$k_{\ell_{j}}-k_{\ell_{j+1}}\textrm{\ is uniquely fixed,\ and\ the\ number\
of\ choices}\\\ \textrm{for\
}|k_{\ell_{j}}|^{2}-|k_{\ell_{j+1}}|^{2},\textrm{\ up\ to\ distance\
}\delta^{-1}L^{-2\gamma},\textrm{\ is\ }\lesssim(\log L)^{C}.$ (9.4)

By Lemma A.3 (1) (where we also use that $R\geq L^{-1}$), we have

$\mathfrak{D}\lesssim\delta^{-1}\min(L^{d-2\gamma+1},L^{d})\cdot
L^{-(d-\gamma)}(\log L)^{C}\lesssim L^{-\eta^{3}}\cdot
L^{\gamma_{0}+4\eta^{2}}$

using that $\gamma_{0}=\min(\gamma,1-\gamma)$, which proves (9.3). If $\Delta
F=1$, then (9.4) still holds for _some_ $j$. Moreover, once
$(k_{\ell_{j}},k_{\ell_{j+1}})$ is fixed, the number of choices for
$k_{\ell_{j+2}}$ is at most $(RL)^{d}$ where $R$ is the label of the unique
$\alpha$-atom. This implies that

$\mathfrak{D}\lesssim\delta^{-1}\min(L^{d-2\gamma+1},L^{d})\cdot
L^{-2(d-\gamma)}(RL)^{d}(\log L)^{C}\lesssim L^{-\eta^{3}}\cdot
L^{\eta^{2}}L^{(\gamma+3\eta)/2}R$

using that $R\in[L^{-1},L^{-\gamma+\eta}]$ and $d\geq 3$, so (9.3) is again
true. Now if $\Delta F=0$, then simply by using the gap assumptions we have
$\mathfrak{F}\lesssim L^{d}(R_{1}L)^{d}(R_{2}L)^{d}$, hence

$\mathfrak{D}\lesssim L^{3d}(R_{1}R_{2})^{d}L^{-3(d-\gamma)}(\log
L)^{C}\lesssim L^{-\eta^{3}}\cdot
L^{-\gamma_{0}-2\eta^{2}}L^{\gamma+3\eta}R_{1}R_{2},$

again using that $R_{1},R_{2}\in[L^{-1},L^{-\gamma+\eta}]$ and $d\geq 3$,
which proves (9.3).

Now consider Step 1. By Proposition A.5, we know that $\mathbb{V}\mathbb{C}$
is either a single vine, or is formed by two double bonds. First assume it is
a single vine. For the operation we have $\Delta F=\Delta V_{\beta}=0$ by
combining Proposition 4.12, Proposition 9.1 (a) (e), and Proposition A.5 (and
an easy verification for DV). If $\mathbb{V}\mathbb{C}$ contains $m$ atoms
(including the joints) then it is easy to verify that $\Delta\chi=-m$; let the
the number of choices for $(k_{\ell})$ for all bonds
$\ell\in\mathbb{V}\mathbb{C}$ be $\mathfrak{F}$, then

$\mathfrak{D}\lesssim\mathfrak{F}\cdot L^{-m(d-\gamma)}\cdot(\log
L)^{C}\delta^{m/2}\prod_{1\leq j\leq
q:\mathcal{L}_{j}\subset\mathbb{V}\mathbb{C}}\mathfrak{X}_{j}^{z_{j}},$ (9.5)

where the product is taken over all $j$ such that the maximal ladder
$\mathcal{L}_{j}$ (see Proposition 8.6) is a subset of $\mathbb{V}\mathbb{C}$
(so there are at most three such $j$, see Figure 16), and $z_{j}$ is the
length of $\mathcal{L}_{j}$.

To calculate $\mathfrak{F}$, first fix $x_{0}=k_{\ell_{1}}$ and
$y_{0}=k_{\ell_{2}}$, where $\ell_{1}$ and $\ell_{2}$ are the two bonds at one
joint of $\mathbb{V}\mathbb{C}$ that belong to $\mathbb{V}\mathbb{C}$. These
have $L^{d}(RL)^{d}$ choices since the gap of $\mathbb{V}\mathbb{C}$ is
$|r|\sim R$. Then, we invoke the reparametrization introduced in the proof of
Propositions 7.5–7.6, and define the new variables $(x_{j},y_{j})\,(1\leq
j\leq m_{1})$ if $\mathbb{V}\mathbb{C}$ is bad vine, or $(x_{j},y_{j})\,(1\leq
j\leq\widetilde{m})$ and $(u_{1},u_{2},u_{3})$ if $\mathbb{V}\mathbb{C}$ 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.3), and $(u_{1},u_{2},u_{3})$
satisfies a system of form (A.4) with $r=x_{0}-y_{0}$ or $r=0$ (so $P=R$ or
$P=0$) and $v=0$. Therefore, by applying Lemma A.2, we get in either case that

$\mathfrak{F}\lesssim L^{d}(RL)^{d}L^{(m-2)(d-\gamma)}\delta^{-m/2}(\log
L)^{C}\prod_{1\leq j\leq
q:\mathcal{L}_{j}\subset\mathbb{V}\mathbb{C}}\mathfrak{X}_{j}^{-z_{j}},$

and hence $\mathfrak{D}\lesssim R^{d}L^{2\gamma}(\log L)^{C}\lesssim
L^{-\eta^{3}}\cdot R^{2}L^{\gamma+3\eta}$, so (9.3) is true. Finally, if
$\mathbb{V}\mathbb{C}$ is formed by two double bonds, then $\Delta\chi=-3$. We
again first fix $(x_{0},y_{0})$, and then apply Lemma A.3 (1) to get

$\mathfrak{D}\lesssim
L^{-3(d-\gamma)}L^{d}(RL)^{d}(L^{d-1}+\delta^{-1}\min(R^{-1}L^{d-2\gamma},L^{d}))(\log
L)^{C}\lesssim L^{-\eta^{3}}\cdot R^{2}L^{\gamma+3\eta},$

using that $R\in[L^{-1},L^{-\gamma+\eta}]$ and $d\geq 3$.

In the case where $\iota_{v}\neq 0$ for some $\alpha$-atom $v$, it is clear
that the corresponding gap $r$ must belong to a fixed ball of radius
$L^{-50\eta}R$ where $|r|\sim R$, and the number of possible choices of such
balls is bounded by $(\log L)^{C}$. As such, the factor $(RL)^{d}$ in the
above proof, which indicates the number of choices for this gap $r$, is
replaced by $(\log L)^{C}(RL^{1-50\eta})^{d}$, and the other factors remain
the same, therefore (9.3) is true with the improved $\kappa_{v}$. ∎

###  9.4. The molecule $\mathbb{M}_{\mathrm{fin}}$

Let the result of stage 2 reduction be $\mathbb{M}_{\mathrm{fin}}$. If it is
not connected, let $\mathbb{M}_{1}$ be any of its components. Consider also a
decoration of $\mathbb{M}_{\mathrm{fin}}$ inherited from a decoration
$(k_{\ell})$ of $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$ as in Remark 9.2.
Then they have the following properties:

######  Proposition 9.5.

There is only one component, which we call the _odd component_ , that contain
two atoms of degree $1$ and $3$. All the other atoms have degree $2$ and $4$.
The atoms of degree $1$ and $2$ are classified as $\alpha$-, or $\beta$-atoms.
Each $\alpha$-atom $v$ is labeled by a dyadic number
$R_{v}\in[L^{-1},L^{-\gamma+\eta}]$, such that if $v$ has two bonds
$(\ell_{1},\ell_{2})$ then $|k_{\ell_{1}}-k_{\ell_{2}}|\sim R_{v}$ in the
decoration; recall also $\iota_{v}$ and $\kappa_{v}$ introduced in Section 9.3
and Proposition 9.4. Atoms that are neither $\alpha$\- nor $\beta$-atoms are
called _$\varepsilon$ -atoms_; any $\varepsilon$-atom in
$\mathbb{M}_{\mathrm{fin}}$ _must_ be $LG$ in the decoration. Moreover
$\mathbb{M}_{\mathrm{fin}}$ contains no triple bond.

Any even (i.e. non-odd) component has at least one $\beta$-atom. If an even
component $\mathbb{M}_{0}$ is a cycle, 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 $\mathbb{M}_{0}$ 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, if any
component $\mathbb{M}_{0}$ is a _vine_ (with two joints having degree $2$),
then it _must_ be LG in the decoration.

###### Proof.

The total number of odd degree atoms is not changed under the cutting
operation and removing of isolated components (which only contain even degree
atoms). This value is $2$ initially (Proposition 9.1), so it will remain $2$.
The two odd degree atoms have to be in the same component as any component
must have an even number of odd degree atoms. The degree $1$ and $2$ atoms are
classified as $\alpha$-, or $\beta$-atoms, and $(R_{v},\iota_{v},\kappa_{v})$
for $\alpha$-atoms $v$ are defined as before. $\mathbb{M}_{\mathrm{fin}}$
contains no triple bond because any triple bond is destroyed in Step 1.

Now consider an even component $\mathbb{M}_{0}$ with only degree $2$ and $4$
atoms. Such a component can only be formed after a $\beta$-cut, so it will
contain at least one $\beta$-atom. If it is a cycle, then it is either a
double bond, or a triangle, or has at least length $4$. If it is a triangle,
then it cannot have at least two $\alpha$-atoms, since any $\alpha$-atom must
have SG in the original molecule $\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$, and
any triangle with at least two SG atoms will be removed in Step 2. If
$\mathbb{M}_{0}$ is not a cycle, then it has at least one degree $4$ (i.e.
$\varepsilon$-) atom. For any degree $2$ atom, consider the longest chain of
degree $2$ atoms containing it, which ends at two degree $4$ atoms $v_{1}$ and
$v_{2}$ in both directions; they cannot coincide, otherwise we can perform a
$\beta$-cut at this common atom according to Step 4 above.

Finally, each $\varepsilon$-atom must have LG by definition, and
$|k_{\ell_{1}}-k_{\ell_{2}}|\sim R_{v}$ for any $\alpha$-atom $v$ of degree
$2$ labeled by $R_{v}$. If any component $\mathbb{M}_{1}$ is an SG vine
$\mathbb{V}$, then this vine $\mathbb{V}$ must exist in the original molecule
$\mathbb{M}(\mathcal{Q}_{\mathrm{sub}})$, and is not changed in the process.
But the SG vine $\mathbb{V}$ cannot be any vine occurring in (a) or (e) of
Proposition 9.1 (or any triple bond in (c)), because then it would be removed
in Step 1. Therefore, $\mathbb{V}$ has to be a vine occurring in (f) of
Proposition 9.1. However, since our reduction involves cutting the molecule at
the hinge atom along the two bonds $(\ell_{1},\ell_{2})$, by (f) of
Proposition 9.1, the vine $\mathbb{V}$ cannot remain intact after this cutting
operation. This completes the proof. ∎

With all the above preparations, we can now reduce Proposition 8.6 to the
following

######  Proposition 9.6.

For each component $\mathbb{M}_{0}$ of the final molecule
$\mathbb{M}_{\mathrm{fin}}$, define
$(\mathfrak{C}_{0},\mathfrak{A}_{0},\mathfrak{P}_{0})$ and $\rho_{0}$ as in
Section 9.2 and (9.1), but associated with $\mathbb{M}_{0}$ (here $p$ should
be replaced by $0$ in (9.1) as there is no more SG atoms in $\mathbb{M}_{0}$).
Let also each $\alpha$-atom $v$ be labeled by the dyadic number $R_{v}$, then
we have

$\mathfrak{A}_{0}\lesssim\prod_{v}^{(\alpha)}L^{-(\gamma+3\eta)/2+\kappa_{v}\eta}R_{v}^{-1}\cdot
L^{(\gamma_{0}+2\eta^{2})V_{\beta}-(2\gamma_{0}+5\eta^{2})G}\cdot
L^{-\eta^{5}\rho_{0}},$ (9.6)

where the product is taken over all $\alpha$-atoms $v$, and $V_{\beta}$ is the
number of $\beta$-atoms in $\mathbb{M}_{0}$; moreover $G$ is $0$ or $1$
depending on whether $\mathbb{M}_{0}$ is odd or even component.

###### Proof of Proposition 8.6 assuming Proposition 9.6.

Since (9.6) holds for each component $\mathbb{M}_{0}$, clearly it also holds
for the union $\mathbb{M}_{\mathrm{fin}}$, if all the expressions and
quantities on the right hand side are replaced by the ones corresponding to
$\mathbb{M}_{\mathrm{fin}}$ (and $G$ replaced by $F-1$ where $F$ is the number
of components of $\mathbb{M}_{\mathrm{fin}}$). Using (9.6) and the deviation
bound (9.3) for each operation, we get that

$\mathfrak{A}_{\mathrm{sub}}\leq(C^{+})^{n_{\mathrm{sub}}}\cdot
L^{-\eta^{5}(\sigma+\rho_{\mathrm{fin}})-\eta^{3}\sigma/2},$ (9.7)

where $\mathfrak{A}_{\mathrm{sub}}$ and $\rho_{\mathrm{fin}}$ are defined as
before, and $\sigma$ is the total number of operations that are done in Steps
1–4. It is easy to see that $\Delta_{\mathrm{sub}}\leq C\eta^{4}\sigma$ for
the value $\Delta_{\mathrm{sub}}$ in Proposition 8.6. This is because each
SGVC described in Proposition 9.1 (a) leads to a hinge atom as in Proposition
9.1 (b) at which we perform a cut, so the number of these SGVC is at most
$\sigma$, while by the definition of $\Delta_{j}$ in Proposition 8.5 the
contribution of each such SGVC to $\Delta_{\mathrm{sub}}$ is at most
$\eta^{4}$ (or negative). Therefore, the term $L^{\eta^{3}\sigma/2}$ in (9.7)
takes care of the $L^{-\Delta_{\mathrm{sub}}}$ term in (8.8).

Comparing (9.7) with (8.8), using also the definition (9.1) and noticing the
logarithmic loss in Remark 9.2, it now suffices to prove that
$\rho_{\mathrm{sub}}\leq C(\sigma+\rho_{\mathrm{fin}})$ for the quantity
$\rho$ defined in Proposition 8.6. However, the value of $\rho_{\mathrm{sub}}$
becomes $\rho_{\mathrm{fin}}$ after all the operations, and each operation
changes the value of $\rho$ by at most $O(1)$ because each vine contains at
most $O(1)$ maximal ladders and at most $O(1)$ atoms apart from these maximal
ladders (which is clear from Figure 16), so it is clear that
$|\rho_{\mathrm{sub}}-\rho_{\mathrm{fin}}|\leq C\sigma$, as desired. ∎

##  10\. Counting problem for large gap molecules

###  10.1. Preliminary setup

We now start the proof of Proposition 9.6. We will first get rid of the
expression on the right hand side of (9.6), and reduce Proposition 9.6 to the
following Propositions 10.1–10.4. In these propositions, we always consider a
connected molecule or pseudomolecule $\mathbb{M}$ (cf. Definition 4.1), and a
$(c_{v})$-decoration $(k_{\ell})$ of $\mathbb{M}$ which is also restricted by
$(\beta_{v})$ and $(k_{\ell}^{0})$; however, _in Propositions 10.1–10.4 only_,
we will relax the definition of decorations by _not_ requiring $c_{v}=0$ for
degree $4$ atoms $v$ as in Definition 4.6. For each atom $v\in\mathbb{M}$ we
also assume the decoration is LG at $v$ (i.e. for any bonds
$(\ell_{1},\ell_{2})$ of opposite directions at $v$ we have
$|k_{\ell_{1}}-k_{\ell_{2}}|\geq L^{-\gamma+\eta}$).

Let $(V,E,\chi)$ etc. be associated with $\mathbb{M}$, and let $\mathfrak{C}$
and $(\rho,q,m^{\prime})$ be defined for $\mathbb{M}$ as before. Note that,
for a given ladder, the differences $k_{\ell}-k_{\ell^{\prime}}$ for different
pairs $(\ell,\ell^{\prime})$ of parallel single bonds may not be the same as
in Definition 8.3, due to the relaxation of the assumption $c_{v}=0$ for
degree $4$ atoms $v$. Therefore a ladder of length $z_{j}\geq 1$ will have
$z_{j}$ different gaps $|r_{ji}|\sim P_{ji}\,(1\leq i\leq z_{j})$;
consequently we define the quantity $\mathfrak{A}$ as in (9.1) but with $p$
replaced by $0$ and the $\mathfrak{X}_{j}^{z_{j}}$ factor in $\mathfrak{P}$
replaced by $\prod_{i=1}^{z_{j}}\min((\log L)^{2},1+\delta L^{2\gamma}P_{ji})$
where $|r_{ji}|\sim P_{ji}$ for the $i$-th gap of the ladder $\mathcal{L}_{j}$
with the notations of Proposition 8.6. Note that we may drop any $C^{n}$
factor below, since they can be absorbed into the definition of
$\mathfrak{A}$.

######  Proposition 10.1.

If $\mathbb{M}$ is a molecule as above, then we always have
$\mathfrak{A}\lesssim 1$.

######  Proposition 10.2.

If $\mathbb{M}$ is a molecule that contains no triple bond and $E=2V-1$, then
$\mathfrak{A}\lesssim L^{-3\gamma_{0}/5}$; if $\mathbb{M}$ contains no triple
bond and $E=2V-2>0$, then $\mathfrak{A}\lesssim L^{-\eta/3}$. Furthermore, if
the number of atoms in $\mathbb{M}$ not of degree $4$ is $w$, _and we also
allow at most $w$ atoms to be SG in the decoration_, then we have
$\mathfrak{A}\lesssim L^{-\eta^{2}\cdot\rho+C(w+1)}$.

######  Proposition 10.3.

If $\mathbb{M}$ is a molecule that contains only single bonds, and has two
degree $3$ atoms with all other atoms having degree $4$, then
$\mathfrak{A}\lesssim L^{-\gamma_{0}-\eta/2}$.

######  Proposition 10.4.

(1) If $\mathbb{M}$ is molecule that has at most one triple bond and $E=2V-1$,
and one cannot make a $\beta$-cut in $\mathbb{M}$ such that one of the new
components has all atoms of degree $4$ except the newly formed $\beta$-atom
which has degree $2$, and $\mathbb{M}$ is not formed by removing the two
joints (with their bonds) of a vine (II) and adding one new bond between the
two atoms connected to one of the joints, then $\mathfrak{A}\lesssim
L^{-4\gamma_{0}/7}$.

(2) Suppose $\mathbb{M}$ is $4$-regular pseudomolecule that contains at most
two triple bonds, and is _not_ formed by removing the two joints of a vine and
adding one new bond between the pair of atoms connected to each joint. Fix any
bond $\ell$ in $\mathbb{M}$, then either $\mathfrak{A}(\mathbb{M})\lesssim
L^{-\eta/4}$ for $\mathbb{M}$, or
$\mathfrak{A}(\mathbb{M}\backslash\\{\ell\\})\lesssim L^{-\gamma_{0}-\eta/4}$
for the molecule after removing $\ell$ (note that the exact case may depend on
assumptions we impose on the decoration, which will be made clear in the
proof).

We start with Proposition 10.1 as it is the simplest, and some ingredients in
its proof will be reused later.

###### Proof of Proposition 10.1.

We will reduce the molecule $\mathbb{M}$ to the empty set by the following
operations: Operation (a) which consists of removing one atom (and all the
bonds), and Operation (b) which consists of removing two atoms $v_{1}$ of
degree 3, and $v_{2}$ of degree 3 or 4, that are connected by a double bond,
and all the bonds attached to them. Note that $\mathbb{M}$ is a molecule which
does not have components of only degree $4$ atoms, so we can always assume
that the removed atom has degree $\leq 3$ in any operation (a); assume also
that in the whole process, we perform operation (a) only when (b) is not
possible.

For each operation we will consider the deviation, i.e. the quantity
$\mathfrak{D}$ such as
$\mathfrak{A}_{\mathrm{pre}}\lesssim\mathfrak{D}\cdot\mathfrak{A}_{\mathrm{pos}}$
as in (9.2), where the meaning of the quantities should be obvious. Note that
each operation (a) does not affect any ladder of length at least one (since
the removed atom cannot belong to such a ladder, or otherwise we should
perform operation (b)), and hence does not affect the factor $\mathfrak{P}$.
Moreover, if the removed atom $v$ has degree $r\in\\{1,2,3\\}$, then
$0\leq\Delta F\leq r-1$ and $\Delta\chi=\Delta F-r+1$. By Lemma A.3 (1)–(2),
we see that

$\mathfrak{D}\lesssim\delta^{-1}L^{2(d-\gamma)}\cdot
L^{-2(d-\gamma)}(C^{+}\delta^{-1/2})^{-2}\lesssim 1$

if $\Delta\chi=-2$ (so $(r,\Delta F)=(3,0)$), and that

$\mathfrak{D}\lesssim\delta^{-1}L^{d-\gamma-\eta}\cdot L^{-(d-\gamma)}\lesssim
L^{-\eta/2}$

if $\Delta\chi=-1$ (so $(r,\Delta F)\in\\{(2,0),(3,1)\\}$), and
$\mathfrak{D}\lesssim 1$ if $\Delta\chi=0$ (so $(r,\Delta
F)\in\\{(1,0),(2,1),(3,2)\\}$). Here we have also used the fact that $R\gtrsim
L^{-\gamma+\eta}$ in Lemma A.3 (1) due to LG assumption; moreover if $\Delta
F\geq 1$ then one of the values $k_{\ell}$ for bonds $\ell$ at $v$ will be
uniquely fixed (if $\Delta F\geq 2$ then all $k_{\ell}$ will be uniquely
fixed).

Now consider operation (b). Here, we have two cases. Either the operation does
not affect any ladder of length at least one, or it can reduce the length of
one such ladder by one. Now, in the former case we clearly has
$\mathfrak{D}\lesssim 1$ as the operation (b) can be split into two operations
(a). In the latter case, the operation removes a factor $\min((\log
L)^{2},1+\delta L^{2\gamma}P)$ from the product $\mathfrak{P}$, where
$P\sim|k_{\ell}-k_{\ell^{\prime}}|$ for the two single bonds
$(\ell,\ell^{\prime})$ at the two removed atoms. If both atoms have degree 3,
then we have $\Delta\chi=-2$, and by Lemma A.3 (3) we have

$\mathfrak{D}\lesssim\delta^{-1}L^{2(d-\gamma)}\cdot\max(\delta^{-1}L^{-\gamma_{0}},(\log
L)^{-2},(1+\delta L^{2\gamma}P)^{-1})\\\ \times
L^{-2(d-\gamma)}(C^{+}\delta^{-1/2})^{-2}\cdot\min((\log L)^{2},1+\delta
L^{2\gamma}P)\lesssim 1.$ (10.1)

If $v_{2}$ has degree 4, then either $\Delta\chi=-2$ and we have the same
bound as in (10.1), or $\Delta\chi=-3$ and a better bound holds using Lemma
A.3 (4). Therefore, in all cases we will have
$\mathfrak{A}_{\mathrm{pre}}\lesssim\mathfrak{A}_{\mathrm{pos}}$. By choosing
the constant $C^{+}$ in (9.1) large enough we can assume
$\mathfrak{A}_{\mathrm{pre}}\leq\mathfrak{A}_{\mathrm{pos}}$; this implies
that we must have $\mathfrak{A}\leq 1$ in the beginning, since it trivially
holds in the end when $\mathbb{M}$ has no bonds. This completes the proof. ∎

In the rest of this subsection we prove Proposition 9.6 assuming Propositions
10.2–10.4.

###### Proof of Proposition 9.6 assuming Propositions 10.2–10.4.

Let $\mathbb{M}_{0}$ be a component of $\mathbb{M}_{\mathrm{fin}}$ as in
Proposition 9.6. If $\mathbb{M}_{0}$ is a double bond, then it must be LG and
both atoms must be $\beta$-atoms, due to Proposition 9.5. In this case we have
$\mathfrak{A}_{0}\lesssim\delta^{-1}L^{d-\gamma-\eta}\cdot
L^{-(d-\gamma)}\lesssim L^{-\eta/2}\lesssim L^{-\eta^{5}}\cdot
L^{2(\gamma_{0}+2\eta^{2})-(2\gamma_{0}+5\eta^{2})}$ using Lemma A.3 (1) and
the LG condition, so (9.6) is true. If $\mathbb{M}_{0}$ is a triangle, then
there are at least two $\beta$-atoms due to Proposition 9.5. Moreover by Lemma
A.3 (1) we see that

$\mathfrak{A}_{0}\lesssim\delta^{-1}L^{d-\gamma+\gamma_{0}}\cdot
L^{-(d-\gamma)}\lesssim L^{-3\eta^{5}}\cdot
L^{3(\gamma_{0}+2\eta^{2})-(2\gamma_{0}+5\eta^{2})}$

if there is no $\alpha$-atom, and that

$\mathfrak{A}_{1}\lesssim(L^{d-1}+\delta^{-1}R^{-1}L^{d-2\gamma})L^{-(d-\gamma)}\lesssim
L^{-3\eta^{5}}\cdot L^{-(\gamma+3\eta)/2}R^{-1}\cdot
L^{2(\gamma_{0}+2\eta^{2})-(2\gamma_{0}+5\eta^{2})}$

if there is one $\alpha$-atom labeled by $R\in[L^{-1},L^{-\gamma+\eta}]$. In
either case (9.6) is true. If $\mathbb{M}_{1}$ is a cycle of length $s\geq 4$,
then there is at least one $\beta$-atom. If there is no $\alpha$-atom the
proof is same as the triangle case; if there is at least one $\alpha$-atom
labeled by $R$, then

$\mathfrak{A}_{1}\lesssim(L^{d-1}+\delta^{-1}R^{-1}L^{d-2\gamma})L^{-(d-\gamma)}\lesssim
L^{-s\cdot\eta^{5}}L^{-(\gamma+3\eta)/2}R^{-1}\cdot(L^{(\gamma-5\eta)/2})^{s-2}L^{\gamma_{0}+2\eta^{2}}\cdot
L^{-(2\gamma_{0}+5\eta^{2})},$

which proves (9.6) since $L^{-(\gamma+3\eta)/2}R^{-1}\gtrsim
L^{(\gamma-5\eta)/2}$ and $L^{\gamma_{0}+2\eta^{2}}\gtrsim
L^{(\gamma-5\eta)/2}$ when $R\leq L^{-\gamma+\eta}$. This completes case when
$\mathbb{M}_{0}$ is a cycle.

Now assume $\mathbb{M}_{0}$ has at least one $\varepsilon$-atom. Thanks to
Proposition 10.2 we always have

$\mathfrak{A}_{0}\lesssim L^{-\eta^{2}\rho_{0}+C(V_{\alpha}+V_{\beta}+1)}.$
(10.2)

Then we need to prove some other estimates to interpolate with (10.2). To
achieve this we need to perform some operations on $\mathbb{M}_{0}$; these
operations include Operation (a) and Operation (b) defined in the proof of
Proposition 10.1 above, as well as the new ones defined below.

Operation (c): Remove a chain of $\alpha$\- and $\beta$-atoms (as in
Proposition 9.5), which has two distinct $\varepsilon$-atoms at both its ends,
and all bonds. By Lemma A.3 (1) we have

$\mathfrak{D}\lesssim 1+\delta^{-1}R^{-1}L^{-\gamma}$ (10.3)

for both $\alpha$\- and $\beta$-atoms, where $R$ is such that $\max|r|\sim R$
for all the gaps $r$ at the $\alpha$\- and $\beta$-atoms in this chain.

Operation (d): Remove a chain of $\alpha$\- and $\beta$-atoms, which has two
distinct $\varepsilon$-atoms $v_{1}$ and $v_{2}$ at both its ends, and all
bonds, and then add a new bond between $v_{1}$ and $v_{2}$ in the same
direction as the chain. This operation does not change $\chi$, and we will
show that $\mathfrak{D}\lesssim(\log L)^{C}$.

To see this, fix a decoration of the molecule $\mathbb{M}_{\mathrm{pre}}$
before operation. Let the two bonds in the chain at the two
$\varepsilon$-atoms be $\ell_{1}$ and $\ell_{2}$ respectively, and the new
bond added be $\ell_{3}$. Then $k_{\ell_{1}}-k_{\ell_{2}}$ and
$|k_{\ell_{1}}|^{2}-|k_{\ell_{2}}|^{2}$ are fixed (the latter up to distance
$\delta^{-1}L^{-2\gamma}$) in the decoration. For any integer $|g|\leq 3$, the
value is also fixed of
$|k_{\ell_{1}}|^{2}-|k_{\ell_{2}}+g(k_{\ell_{1}}-k_{\ell_{2}})|^{2}$ up to
distance $O(1)\delta^{-1}L^{-2\gamma}$, thus we obtain a decoration for the
molecule $\mathbb{M}_{\mathrm{pos}}$ after operation, by setting
$k_{\ell_{3}}=k_{\ell_{2}}+g(k_{\ell_{1}}-k_{\ell_{2}})$. The LG condition for
the new decoration is still satisfied, if we choose a suitable $g$, and weaken
LG condition to $|r|\geq L^{-\gamma+\eta}/10$ (if not, then by pigeonhole
principle we must have $|k_{\ell_{1}}-k_{\ell_{2}}|\leq L^{-\gamma+\eta}/10$,
and thus the LG condition for $\mathbb{M}_{\mathrm{pre}}$ implies the weakened
LG condition for $\mathbb{M}_{\mathrm{pos}}$). This implies that
$\mathfrak{C}_{\mathrm{pre}}\lesssim\mathfrak{C}_{\mathrm{pos}}$ and hence
$\mathfrak{D}\lesssim(\log L)^{C}$, because this operation affects at most
$O(1)$ nodes in $O(1)$ ladders, and modifies $\mathfrak{P}$ by at most a
$(\log L)^{C}$ factor.

Operation (e): Suppose after operation (d), a triple bond forms between
$v_{1}$ and $v_{2}$. If these two atoms have at most one external bond then we
remove them and all bonds; otherwise, if they are connected to two
$\varepsilon$-atoms $v_{3}\neq v_{4}$ by two single bonds, then we remove
$(v_{1},v_{2})$ and all bonds, then add a new bond between $v_{3}$ and
$v_{4}$, matching the directions of the removed single bonds.

By using the same arguments above (assigning a suitable
$k_{\ell_{2}}+g(k_{\ell_{1}}-k_{\ell_{2}})$ to the new bond $\ell_{3}$, where
$\ell_{1}$ and $\ell_{2}$ are the two single bonds connecting to $v_{3}$ and
$v_{4}$ to $v_{1}$ and $v_{2}$) and also using Lemma A.3 (3) we can show that
as a result of applying Operations (d) and (e) consecutively,

$\mathfrak{D}\lesssim L^{-\gamma_{0}+200\eta},$ (10.4)

provided that $\max|r|\gtrsim L^{-\gamma-100\eta}$ for all the gaps $r$ at the
$\alpha$\- and $\beta$-atoms in this chain. Note that, after operations (d)
and (e), we no longer require $c_{v}=0$ for degree $4$ atoms $v$ as in
Definition 4.6.

We treat the remaining cases of Proposition 9.6. For even component, note that
$V_{\beta}\geq 1$ because the last cut that separate $\mathbb{M}_{0}$ from the
other components must be $\beta$-cut; moreover if $V_{\beta}=1$ then we must
have $V_{\alpha}\geq 1$, since otherwise the only $\beta$-atom will have gap
$r=0$ which is not possible.

(1) If $\mathbb{M}_{0}$ is the odd component, then we perform the same
operations (a) and (b) as in the proof of Proposition 10.1 above, but we only
remove $\varepsilon$-atoms (even if they may become degree $1$ or $2$ in the
process). Note that the value of $\mathfrak{A}$ becomes $1$ after removing all
$\varepsilon$-atoms (because the remaining $\alpha$\- and $\beta$-atoms can
only form finitely many chains for which $\chi=0$), and $\mathfrak{D}\lesssim
1$ for each step in the same way as in the proof of Proposition 10.1, we
conclude that $\mathfrak{A}_{0}\lesssim 1$. Moreover, let $V_{\alpha}$ and
$V_{\beta}$ be the number of $\alpha$\- and $\beta$-atoms respectively; if
$w:=V_{\alpha}+V_{\beta}=0$ then $E=2V-1$ and $\mathfrak{A}_{0}\lesssim
L^{-3\gamma_{0}/5}$ by Proposition 10.2, and if $w>0$ then
$\mathfrak{A}_{0}\lesssim 1$ and the right hand side of (9.6) is at least
$L^{(\gamma_{0}-5\eta)w/2-\eta^{5}\rho_{0}}$ (note also that $G=0$ for the odd
component). In either case (9.6) follows from an interpolation with (10.2).

(2) From now on we assume $\mathbb{M}_{0}$ is an even component. If
$V_{\beta}\geq 3$, then we first choose any chain of $\alpha$\- and
$\beta$-atoms and perform operation (c). After this, the molecule will no
longer have any component such that all $\varepsilon$-atoms have degree $4$,
so we can perform operations (a) and (b) as in (1) above, to prove that
$\mathfrak{A}_{0}\lesssim\delta^{-1}L^{\gamma_{0}}$ using also (10.3). As the
right hand side of (9.6) is at least

$L^{\eta^{2}V_{\alpha}}\cdot
L^{V_{\beta}(\gamma_{0}+2\eta^{2})-2\gamma_{0}-5\eta^{2}-\eta^{5}\rho_{0}}\gtrsim
L^{\gamma_{0}+\eta^{2}/2-\eta^{5}\rho_{0}}\cdot
L^{(V_{\alpha}+V_{\beta})\eta^{2}/10},$

we can interpolate (10.2) with the bound
$\mathfrak{A}_{0}\lesssim\delta^{-1}L^{\gamma_{0}}$ to prove (9.6).

(3) If $V_{\beta}=2$ and $V_{\alpha}\geq 1$, or if $V_{\beta}=1$ and
$V_{\alpha}\geq 3$, then we choose any chain containing at least one
$\alpha$-atom and perform operation (c), and proceed as in (2) above. Let the
label of the $\alpha$-atom in this chain be $R$, then by (10.3) we have
$\mathfrak{A}_{0}\lesssim 1+\delta^{-1}R^{-1}L^{-\gamma}$. Interpolating with
(10.2) we get

$\mathfrak{A}_{0}\lesssim(1+\delta^{-1}R^{-1}L^{-\gamma})\cdot
L^{-\eta^{5}\rho_{0}+C\eta^{3}(V_{\alpha}+1)}\\\ \lesssim
L^{-\eta^{5}\rho_{0}}\cdot L^{-(\gamma+3\eta)/2}R^{-1}\cdot
L^{V_{\beta}(\gamma_{0}+2\eta^{2})-2\gamma_{0}-5\eta^{2}}\cdot
L^{(V_{\alpha}-1)(\gamma-5\eta)/2}$ (10.5)

which is better than (9.6) and is easily verified when $V_{\beta}=2$ and
$V_{\alpha}\geq 1$ or $V_{\beta}=1$ and $V_{\alpha}\geq 3$.

(4) If $V_{\alpha}=0$ and $V_{\beta}=2$, then choose all the chains of
$\beta$-atoms and perform operation (d) to them. The resulting molecule
$\mathbb{M}_{1}$ is $4$-regular and has at most two triple bonds. Therefore,
by Proposition 10.4 (2), we know that either $\mathfrak{A}_{1}\lesssim
L^{-\eta/5}$ or $\mathbb{M}_{1}$ is formed by removing the two joints of a
vine and adding one new bond between the pair of atoms connected to each
joint. (note that, the number of choices for vector $k_{\ell}$ for a newly
added bond $\ell$ is bounded by $\delta^{-1}L^{d-\gamma+\gamma_{0}}$ due to
definition of operation (d) and Lemma A.3 (1), so if $\mathfrak{A}\lesssim
L^{-\gamma_{0}-\eta/4}$ for the molecule $\mathbb{M}_{1}\backslash\\{\ell\\}$
we also have $\mathfrak{A}_{1}\lesssim L^{-\eta/5}$). In the former case
interpolating with (10.2) yields $\mathfrak{A}_{0}\lesssim
L^{-\eta/5-\eta^{5}\rho_{0}}\lesssim L^{\eta^{2}-\eta^{5}\rho_{0}}$ which
implies (9.6).

In the latter case, an enumeration shows that $\mathbb{M}_{0}$ must be a vine
and is thus LG due to Proposition 9.5, so we may now perform operation (c) to
the chains of $\beta$-atoms in $\mathbb{M}_{0}$. For the resulting molecule
$\mathbb{M}_{1}$ we have $\mathfrak{A}_{1}\lesssim 1$ by Proposition 10.1, and
using also (10.3) and the LG assumption we get that $\mathfrak{A}_{0}\lesssim
L^{-\eta/2}$, so (9.6) again follows by interpolation.

(5) If $V_{\alpha}=V_{\beta}=1$, then we choose the chain containing the
$\alpha$-atom and perform operation (c), and choose any other possible chain
and perform operation (d), to get a molecule $\mathbb{M}_{1}$ with only
$\varepsilon$-atoms. Note that $\mathbb{M}_{1}$ has at most one triple bond,
satisfies the $\beta$-cut assumption in Proposition 10.4 (1) (which follows
because one cannot make any $\beta$-cuts in $\mathbb{M}_{0}$) as well as
$E=2V-1$, so by Proposition 10.4 (1) we know either $\mathfrak{A}_{1}\lesssim
L^{-4\gamma_{0}/7}$, or $\mathbb{M}_{1}$ is formed by removing the two joints
of a vine (II) and adding one new bond between the two atoms connected to one
of the joints. But the latter case is impossible, because then