that $S_{l}$ commutes with each $P_{k}$ and $Q_{jk}$.

### Choice of parameters {#choiceof}

Throughout the proof, we will fix some parameters as follows: $$\label{param}
\begin{aligned}
N&=1000, &\quad \delta&=1/990,&N_{0}&=10^{10};\\
D_{0}&\gg_{\mathcal{B}}1,&K_{0}&\gg_{D_{0}}1,&\varepsilon_{0}&\ll_{K_{0}}1,
\end{aligned}$$where $\mathcal{B}:=\{(b_{\alpha},c_{\alpha})\}$. Note that $\delta=1/(N-10)$. In addition, let $A$ denote any large absolute constant such that $A\ll N_{0}$ (say $A\sim 10^5$), whose exact value may vary at different places; in the same way, let $o$ denote any small constant, so for example $o\ll\delta^{4}$.

Description of the methods {#strategykg}
--------------------------

### Energy estimates and decay in $L^{\infty}$

Letting $v_{\sigma}=(\partial_{t}-i\Lambda_{\sigma})u_{\sigma}$, we can reduce (\[nlkg\]) to $$\label{diag}(\partial_{t}+i\Lambda_{\sigma})v_{\sigma}=\mathcal{N}_{\sigma}(v,v),\quad\nu\in\mathcal{P},$$ where $\mathcal{N}_{\sigma}$ is some quasilinear quadratic term. The starting point of our analysis is an energy estimate of form $$\label{energy0}\partial_{t}\mathcal{E}\approx\int_{\mathbb{R}^{3}}\partial^{N}v\cdot\partial^{N}v\cdot\partial^{2}v,\qquad\mathcal{E}\approx\|v\|_{H^{N}}^{2}.$$ To obtain this, which is needed even in local theory, one need to perform suitable symmetrization which relies on the assumption that the nonlinearities $\mathcal{Q}_{\alpha}$ are symmetric.

Next, from (\[energy0\]) and Gronwall we can estimate the energy $\mathcal{E}=\mathcal{E}(t)$ by$$\mathcal{E}(t)\lesssim \mathcal{E}(0)\exp\bigg(\int_{0}^{t}\|\partial^{2}v\|_{L^{\infty}}\bigg).$$ To bound the energy globally, it then suffices to study decay estimates of form $$\label{decay98}\|\partial^{2}v(t)\|_{L^{\infty}}\lesssim(1+|t|)^{-1-\delta}.$$

### Space localization and the $Z$ norm

The idea of obtaining (\[decay98\]) is to introduce the *profile* $$f_{\sigma}(t)=e^{it\Lambda_{\sigma}}v_{\sigma}(t)$$by taking back the linear flow. Under the scattering ansatz, $f_{\sigma}$ is expected to settle 