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

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,

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 $O(\epsilon)$. ∎