Suppose $$m\geq0, (j,k),(j_{1},k_{1}),(j_{2},k_{2})\in\mathcal{J};\qquad\max(m,j,|k|,j_{1},j_{2},|k_{1}|,|k_{2}|):=M.$$ Let $2^{m}-1\leq a\leq b\leq 2^{m}$, and define the quantity $J$ by $$\label{bilibili}\widehat{J}(\xi)=\int_{a}^{b}\int_{\mathbb{R}^{3}}e^{{i}s\Phi_{\sigma\mu\nu}(\xi,\eta)}{m}_{\sigma\mu\nu}(\xi,\eta)\mathcal{F}_{x}(f_{\mu})_{j_{1}k_{1}}^{*}(s,\xi-\eta)\mathcal{F}_{x}(f_{\nu})_{j_{2}k_{2}}^{*}(s,\eta)\,\mathrm{d}\eta\mathrm{d}s,$$ then we have$$\label{atomest}\|J_{jk}\|_{Z_{jk}}\lesssim \langle M\rangle^{-20}\varepsilon_{1}^{2}.$$

Proof of Proposition \[main01\]: The setup {#beginning}
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First, for each $|\beta|\leq N/2+2$ we have[^2] $$\label{bilibili2}\widehat{\Gamma^{\beta}J}(\xi)=\sum_{|\beta_{1}|+|\beta_{2}|\leq|\beta|}\int_{a}^{b}\int_{\mathbb{R}^{3}}e^{{i}s\Phi_{\sigma\mu\nu}(\xi,\eta)}{m}_{\sigma\mu\nu}^{\beta_{1}\beta_{2}}(\xi,\eta)\mathcal{F}_{x}(\Gamma^{\beta_{1}}f_{\mu})_{j_{1}k_{1}}^{*}(s,\xi-\eta)\mathcal{F}_{x}(\Gamma^{\beta_{2}}f_{\nu})_{j_{2}k_{2}}^{*}(s,\eta)\,\mathrm{d}\eta\mathrm{d}s,$$ where ${m}_{\sigma\mu\nu}^{\beta_{1}\beta_{2}}$ is obtained from ${m}_{\sigma\mu\nu}$ by applying $\Omega$, and has the same form as (\[weightm\]) with uniform bounds for each $|\mu|\leq N/2+2$. In (\[bilibili2\]), we further decompose $f_{\mu}$ into $S_{l_{1}}f_{\mu}$ and $f_{\nu}$ into $S_{l_{2}}f_{\nu}$, and reduce to estimating $I$, where $$\label{bilibili3}\widehat{I}(\xi)=\int_{a}^{b}\int_{\mathbb{R}^{3}}e^{{i}s\Phi(\xi,\eta)}{m}(\xi,\eta)\widehat{F}(s,\xi-\eta)\widehat{G}(s,\eta)\,\mathrm{d}\eta\mathrm{d}s,$$ where the $(\sigma,\mu,\nu)$ subindices are omitted, and $$F=(\Gamma^{\beta_{1}}S_{l_{1}}f_{\mu})_{j_{1}k_{1}}^{*},\qquad G=(\Gamma^{\beta_{2}}S_{l_{2}}f_{\nu})_{j_{2}k_{2}}^{*}.$$

Bounds for $F$, $G$ and their time derivatives
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\[linearbound0\]We have the following bounds for $F$ and $\partial_{t}F$; similar bounds will hold for $G$ and $\partial_{t}G$.

1.  For $F$ we have$$\label{linearb1}
    \begin{aligned}\|F\|_{L^{2}}&\lesssim 2^{-(N-4)\max(k_{1},l_{1})/2}\varepsilon_{1};&\\\|\widehat{F}\|_{L^{\infty}}&\lesssim 2^{-j_{1}/4}\varepsilon_{1},&\mathrm{if\ }k_{1}\geq -K_{0}^{2};\\\|\widehat{F}\|_{L^{\infty}}&\lesssim 2^{l_{1}}2^{(-3k_{1}-j_{1})/2}\langle j_{1}\rangle^{-N_{0}}\varepsilon_{1},&\mathrm{if\ }k_{1}\leq-K_{0}^{2}.
    \end{aligned}$$

2.  If $k_{1}\leq -K_{0}$ we have $$\label{linearb2}
    \begin{aligned}\|F\|_{L^{2}}&\lesssim \langle j_{1}\rangle^{-N_{0}}2^{-j_{1}-k_{1}/2}\varepsilon_{1}\lesssim \langle j_{1}\rangle^{-N_{0}}2^{-j_{1}/2}\varepsilon_{1};\\\|e^{-it\Lambda_{\mu}}F\|_{L^{\infty}}&\lesssim \langle j_{1}\rangle^{-N_{0}}\min(2^{-j_{1}+k_{1}},2^{-(3m-j_{1}+k_{1})/2})\varepsilon_{1};\\\|e^{-it\Lambda_{\mu}}F\|_{L^{\infty}}&\lesssim\max(\langle m\rangle,\langle j_{1}\rangle)^{-N_{0}}2^{-\max(m,j_{1})}\varepsilon_{1}.
    \end{aligned}$$

3.  If $|k_{1}|<K_{0}$ we have $$\label{linearb3}
    \begin{aligned}\|F\|_{L^{2}}&\lesssim 2^{-5j_{1}/6}\langle j_{1}\rangle^{N_{0}}\epsilon_{1},&\mathrm{and\ }\|\widehat{F}\|_{L^{1}}&\lesssim 2^{-j_{1}}\langle j_{1}\rangle^{-N_{0}}\varepsilon_{1};\\
    \sup_{\theta\in \mathbb{S}^2}\|\widehat{F}(\rho\theta)\|_{L_{\rho}^2}&\lesssim 2^{l_1}2^{-5j_{1}/6}\langle j_{1}\rangle^{N_{0}}\epsilon_{1},&\sup_{\theta\in \mathbb{S}^2}\|\widehat{F}(\rho\theta)\|_{L_{\rho}^1}&\lesssim 2^{2l_1}2^{-j_{1}}\langle j_{1}\rangle^{-N_{0}}\epsilon_{1};\\\|e^{-it\Lambda_{\mu}}F\|_{L^{\infty}}&\lesssim2^{l_{1}}2^{-3m/2-j_{1}/3}\langle m\rangle^{N_{0}+A}\varepsilon_{1},&\mathrm{if\ }|j_{1}-m|&\geq A\log m;\\\|e^{-it\Lambda_{\mu}}F\|_{L^{\infty}}&\lesssim \langle m\rangle^{-N_{0}}2^{-j_{1}}\varepsilon_{1},&\mathrm{if\ }|j_{1}-m|&\leq A\log m.
    \end{aligned}$$

4.  If $k_{1}\geq K_{0}$ we have $$\label{linearb4}
    \begin{aligned}\|F\|_{L^{2}}&\lesssim  \langle j_{1}\rangle^{-N_{0}}2^{-j_{1}}\varepsilon_{1};&\\\|e^{-it\Lambda_{\mu}}F\|_{L^{\infty}}&\lesssim \langle j_{1}\rangle^{-N_{0}+A}2^{4k_1+l_{1}}2^{-(3m+j_{1})/2}\varepsilon_{1},&\mathrm{if\ }|j_{1}-m|\geq A\log m\\\|e^{-it\Lambda_{\mu}}F\|_{L^{\infty}}&\lesssim \langle m\rangle^{-N_{0}+A}2^{-j_{1}+3k_1/2}\varepsilon_{1},&\mathrm{if\ }|j_{1}-m|\leq A\log m.
    \end{aligned}$$

5.  For $\partial_{t}F$ we have$$\begin{aligned}\label{linearb5}\|\partial_{t}F\|_{L^{2}}&\lesssim\langle m\rangle^{A}\min(2^{-(3m+k_{1})/2},2^{3k_{1}/2})\varepsilon_{1}^{2}\lesssim\langle 