(Local potential upper bound)

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  Inside $U^{s,*}_{w}\subset X_{s}$, for $\langle m,w\rangle=1$, the local potentials satisfy $\varphi_{m}-\psi_{m}\leq Cs^{-1/2},$ or equivalently $\varphi-\psi\leq Cs^{-1/2}$.

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  Inside $U^{s,face}_{w}$, the local potentials satisfies $\varphi_{0}-\psi_{0}\leq Cs^{-1/2}$, or equivalently $\varphi-\psi\leq Cs^{-1/2}$.