We know prove the continuity of the constant $K_{V_{G,\phi,t}}$ introduced in Theorem 2.3.

**Lemma B.10** The following map is continuous

$$t \mapsto K_{V_{G,\phi,t}} = 2P\|\mathcal{H}[\rho_{V_{G,\phi,t}}]\|_{\infty} + C + P\left|\iint_{\mathbb{R}^2} \log|x-y|d\mu_{V_{G,\phi,t}}(x)d\mu_{V_{G,\phi,t}}(y)\right|$$

for C some fixed constant.

**Proof** We already proved the continuity of the map $t \mapsto \|\mathcal{H}[\rho_{V_{G,\phi,t}}]\|_{L^{\infty}(\mathbb{R})}$ in Lemma B.1 so it just remains to show that the double integral is continuous with respect to $t$. We prove this by dominated convergence theorem. The function $(x,y) \mapsto \log|x-y|\rho_{V_{G,\phi,t}}(x)\rho_{V_{G,\phi,t}}(y)$ converges almost everywhere to

$$(x, y) \mapsto \log|x-y|\rho_{V_{G,\phi,t_0}}(x)\rho_{V_{G,\phi,t_0}}(y)$$

as $t$ goes to $t_0$. Furthermore we have the following domination $(x, y)$-almost everywhere

$$\left|\log|x-y|\right|\rho_{V_{G,\phi,t}}(x)\rho_{V_{G,\phi,t}}(y) \le \left|\log|x-y|\right|(1 + \max_{s \in [0,1]} \|u_s\|_{\infty})^2 \rho_{V_{G,\phi,t_0}}(x)\rho_{V_{G,\phi,t_0}}(y).$$

This allows us to conclude on the continuity of $t \mapsto K_{V_{G,\phi,t}}$. □

**Proposition B.11** The map $t \mapsto P_{K,k}^{V_{G,\phi,t}}$ where $P_{K,k}^V$ appears in (67) is integrable on $[0, 1]$.

**Proof** By the bounds on $\left\|\widetilde{\Xi}_1^{-1}\right\|_{H^i}$, $\left\|\widetilde{\Xi}_1^{-1}\right\|_{W_i^{\infty}}$ and $\left\|\Theta^{(a)} \circ \widetilde{\Xi}_1^{-1}\right\|_{W_i^{\infty}}$ for $0 \le i \le m_{K,k}$ in Theorem 4.14, 4.15 and 4.17 plus the continuity results of Appendix B, we conclude on the finiteness of $\int_0^1 P_{K,k}^{V_{G,\phi,t}} dt$. □

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