Note that for any $m\in\Delta_{\mathbb{Z}}\backslash\{0\}$, if $\langle m,\log|z|\rangle+\log|a|\leq-\log|\Delta_{\mathbb{Z}}|$, then $|a_{m}z^{m}|\leq\frac{1}{|\Delta_{\mathbb{Z}}|}$. Hence the equation $\sum_{m\in\Delta_{\mathbb{Z}}\backslash\{0\}}a_{m}z^{m}=1$ cannot have solutions for $x\in\log^{-1}(\Delta^{\vee}_{\lambda-c})$. ∎

First of all note that since $X(U_{v}^{\beta})\cap Z_{a}\subset(\mathbb{C}\backslash\{0\})^{d}$, both $\log|z|$ and $\operatorname{Arg}(z)$ are well defined. Also by (2) of Lemma 3.2 the values of the vector field $\mathfrak{X}$ are in $(\operatorname{carrier}v)^{\vee}$. Hence, for any ${q\in U_{v}^{\beta}}$ the ray $\mathcal{F}_{q}$ is in $Q^{\lambda}_{(v|\{0\})}(\beta)$ and the standard estimates on values of the monomials at $z\in\log^{-1}(Q^{\lambda}_{(v|\{0\})}(\beta))$ apply:

$$|a_{m}z^{m}|\leq e^{-\beta}|a_{v}z^{v}|,\text{ all }m\neq v,\{0\}.$$

Or putting them all together we have

$$\displaystyle\left|\frac{1}{a_{v}z^{v}}\sum_{m\neq v,\{0\}}a_{m}z^{m}\right|\leq|\Delta_{\mathbb{Z}}|e^{-\beta}.$$

Hence,

$$\displaystyle\left|\log\left(1+\frac{1}{a_{v}z^{v}}\sum_{m\neq v,\{0\}}a_{m}z^{m}\right)\right|\leq C^{\prime}(\beta)\cdot|\Delta_{\mathbb{Z}}|e^{-\beta},$$

where $C^{\prime}(\beta)\to 1$ as $\beta\to\infty$. Writing the equation of $Z_{a}$ in $X(U_{v}^{\beta})$ as

$$a_{v}z^{v}\left(1+\frac{1}{a_{v}z^{v}}\sum_{m\neq v,\{0\}}a_{m}z^{m}\right)=1$$

or, equivalently,

$$\log(a_{v}z^{v})=-\log\left(1+\frac{1}{a_{v}z^{v}}\sum_{m\neq v,\{0\}}a_{m}z^{m}\right)$$

will give the claimed estimates. ∎

First we apply estimates similar to those in Lemma 4.2 to the differential $d\psi$, which we think of as an element in $(\mathbb{R}^{d}\oplus\mathbb{R}^{d})\otimes((\mathbb{R}^{d})^{*}\oplus(\mathbb{R}^{d})^{*})$.

$$d\psi=\left(\begin{array}[]{cc}\mathbbm{1}-\mathfrak{X}\otimes v-(\langle v,\log|z|\rangle+\lambda(v))\nabla\mathfrak{X}&0\\ -(\langle v,\operatorname{Arg}(z)\rangle+\theta_{v})\nabla\mathfrak{X}&\mathbbm{1}-\mathfrak{X}\otimes v\end{array}\right)\left(\begin{array}[]{c}d\log|z|\\ d\operatorname{Arg}(z)\end{array}\right)$$

Note that the complex structures $J_{Z_{a}}$ and $J_{W_{a}}$ would match exactly via $d\psi$ if there were no $\nabla\mathfrak{X}$ terms (this is what happens in the charts $V_{w}^{\beta^{\vee}}$ where $\mathfrak{X}$ is constant).

According to (2) of Lemma 3.2 $\langle v,\mathfrak{X}\rangle=1$ in $U_{v}^{\beta}$. Hence the projection operator

$$\left(\begin{array}[]{cc}\mathbbm{1}-\mathfrak{X}\otimes v&0\\ 0&\mathbbm{1}-\mathfrak{X}\otimes v\end{array}\right)$$

has a norm of order 1 when restricted to $Z_{a}$, and the desired bound on $d\psi\circ J_{Z_{a}}\circ(d\psi)^{-1}-J_{W_{a}}$ will follow from estimating the $\nabla\mathfrak{X}$ terms.

But according to the Lemma 4.2 we have the uniform bounds:

$$|\langle v,\log|z|\rangle+\lambda(v)|\leq Ce^{-\beta},\ |\langle v,\operatorname{Arg}(z)\rangle+\theta_{v}|\leq Ce^{-\beta}.$$

On the other hand, by (3) of Lemma 3.2 the gradient $\nabla\mathfrak{X}(\log|z|)$ is bounded by $O(\frac{1}{\beta^{\vee}})\cdot\left|g_{\lambda-c,\nu}(\log|z|)\right|$. ∎

Let $x\in X(U_{v}^{\beta})\cap Z_{a}$. Assuming $e^{-\beta}<\gamma/2$ we have

$$\left|\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma,h}(x)-\operatorname{Hess}\Phi_{\lambda+\gamma}(x)\right|<C_{3}(h).$$

The difference between $\operatorname{Hess}\Phi_{\lambda+\gamma}|_{T_{x}Z_{a}}$ and $\operatorname{Hess}\Phi_{\lambda+\gamma}|_{T_{\psi(x)}W_{a}}$ consists of two terms. The first term $C_{4}(\gamma)$ appears from comparing $\operatorname{Hess}\Phi_{\lambda+\gamma}|_{\mathbb{R}^{d}_{v}}$ at $q+t\mathfrak{X}(x)$ with $\operatorname{Hess}\Phi_{\lambda}|_{\mathbb{R}^{d}_{v}}$ at $q$. The other term $C_{1}(\gamma)\frac{C_{0}(\beta)}{\beta^{\vee}}e^{-\beta}$ reflects the error in the alignment of the tangent spaces via the map $d\psi$ in the proof of Lemma 4.3.

Now let $x\in X(V_{w}^{\beta^{\vee}})$. We will just need to check points in $\mathcal{F}_{q}$ for $q\in V_{w}^{\beta^{\vee}}\backslash\bigcup U_{v}^{\beta}=N^{\beta}(\partial\mathcal{U})\cap V_{w}^{\beta^{\vee}}$. Note that $\operatorname{Hess}\Phi_{\lambda}$ is bounded in $\mathcal{F}(V_{w}^{\beta^{\vee}})$ and is continuous at $\partial\mathcal{U}\backslash D$ (in particular, the Hessian vanishes into the $w$-direction at $\partial\mathcal{U}\backslash D$). Hence the $C_{5}$ bound from above are also valid for the regularization $\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma,h}$. Namely,

$$\left|\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma,h}(x)-g_{\lambda+\gamma,\nu}(q)\right|<C_{5}(\beta,c),$$

with possibly different function $C_{5}$. The discrepancy between $g_{\lambda+\gamma,\nu}$ and $g_{\lambda,\nu}$ in $N^{\beta}(\partial\mathcal{U})\cap V_{w}^{\beta^{\vee}}$ is encoded in the $C_{2}(\gamma)$ term.

Finally, the term $\epsilon\omega_{0,\nu}$ in $\omega$ can be bounded by $O(\epsilon)$. ∎

We consider the bi-PIKAS family in a neighborhood of $(\lambda_{0},\nu_{0})$ and extend it by rescaling to (a neighborhood of) the ray $(\lambda,\nu)=\left(\log|a|+\log|s|\cdot\lambda_{0},\frac{\nu_{0}}{\log|s|}\right)$ in $\operatorname{SC}(S)\times\operatorname{SC}(T)$.

The $C_{i}$-estimates for the $(\lambda,\nu)$ bi-PIKAS considered in the bi-PIKAS metric (rather than in Euclidean) are equivalent to the corresponding estimates for the rescaled structure $((\log|s|)^{-1}\lambda,\log|s|\nu)$ made in the Euclidean metric as before. This is because the Euclidean metric on $\partial\Delta_{\lambda_{0}}$ is equivalent to the bi-PIKAS metric on $\Sigma$. But to pass from $(\lambda,\nu)$ to $((\log|s|)^{-1}\lambda,\log|s|\nu)$ we will need to rescale all the parameters as well:

	$$\displaystyle\lambda\sim\lambda_{0}\log|s|,\ \beta\sim\beta_{0}\log|s|,\ \gamma\sim\gamma_{0}\log|s|,\ h\sim h_{0}\log|s|,\ c\sim c_{0}\log|s|,$$
	$$\displaystyle\nu\sim\frac{\nu_{0}}{\log|s|},\ \beta^{\vee}\sim\frac{\beta^{\vee}_{0}}{\log|s|},\ \operatorname{Hess}\Phi\sim\frac{\operatorname{Hess}\Phi_{0}}{(\log|s|)^{2}}.$$

Or equivalently, we could apply the log map with the base $|s|$ as in [HZ02].

We saw in the proof of Lemma 4.4 that $\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda_{0}+\gamma_{0}}$ is degenerate along $\mathcal{F}$ outside $\Delta^{\vee}_{\lambda_{0}+\gamma_{0}+h_{0}}$. This argument extended to the entire toric variety shows that up to terms of order $O(\epsilon)$, the set $X_{T}\backslash\log^{-1}(\Delta^{\vee}_{\lambda_{0}+\gamma_{0}+h_{0}})$ (which contains $Z_{s}$) has distance from $\partial\Delta^{\vee}_{\lambda}$ bounded by the diameters of the torus fibers $\mathbb{T}$. The size of the tori $\mathbb{T}$ is determined by the norm of $\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma}$ at the corresponding point, which is bounded by $\frac{C_{1}(\gamma_{0})}{(\log|s|)^{2}}$.

The rest of the proof consists of careful picks for asymptotics of the rescaled parameters $\beta_{0},\beta^{\vee}_{0},h,\gamma_{0},c_{0},\epsilon$ to ensure that the following expressions

	$$\displaystyle\beta_{0},\quad\beta^{\vee}_{0},\quad\gamma_{0},\quad c_{0},\quad h_{0},$$
	$$\displaystyle C_{1}(\gamma_{0})(\log|s|)^{-2},$$
	$$\displaystyle\log|s|\frac{1}{\beta^{\vee}_{0}}e^{-\beta_{0}\log|s|}\cdot C_{0}(\beta_{0}),$$
	$$\displaystyle\log|s|\frac{C_{0}(\beta_{0})C_{1}(\gamma_{0})}{\beta^{\vee}_{0}}e^{-\beta_{0}\log|s|}+C_{2}(\gamma_{0})+C_{3}(h_{0})$$
	$$\displaystyle\hskip 144.54pt+C_{4}(\gamma_{0})+C_{5}(\beta_{0},c_{0})+O(\epsilon)(\log|s|)^{2}$$

go to 0 as $\log|s|\to\infty$, where the first two lines take care of the Hausdorff convergence, and the last two give matching of the complex structure and the metric under the embedding $\psi:Z^{\mathrm{sm}}_{s}\to W_{s}$

For instance, we can choose

$$\beta^{\vee}_{0}\sim\frac{1}{\log|s|},\quad\gamma_{0}\sim C_{1}^{-1}(\log|s|),\quad c_{0}\sim\frac{1}{\log|s|},\quad h_{0}\sim\frac{1}{\log|s|}.$$

And $\beta_{0}(\log|s|)$ has satisfy $C_{1}(\frac{1}{\log|s|}e^{-\beta_{0}\log|s|})<\log|s|$ (i.e. $e^{-\beta}<\gamma$, which is needed for the proof of Lemma 4.4) and

$$(\log|s|)^{3}C_{0}(\beta_{0})e^{-\beta_{0}\log|s|}\to 0,$$

which is possible due to the fast decreasing factor of $e^{-\beta_{0}\log|s|}$ when $\beta_{0}(\log|s|)$ is changing slowly.

Finally, notice that the bi-PIKAS of type $((\log|s|)^{-1}\cdot\lambda,\log|s|\cdot\nu)=(\lambda_{0}+(\log|s|)^{-1}\log|a|,\nu_{0})$ converges to the $(\lambda_{0},\nu_{0})$-bi-PIKAS. ∎