Two-time spatial decorrelation for the flat KPZ fixed point

We establish quantitative two-time spatial decorrelation for the Kardar–Parisi–Zhang fixed point with flat initial data. For every $s,t>0$,there exist constants $C,c>0$ such that $$\mid \mathrm{C}\mathrm{o}\mathrm{v}(\mathfrak{h}(t,x),\mathfrak{h}(s,0))\mid \le C\exp \{-c\mathrm{\mid }x{\mathrm{\mid }}^3\},\mathrm{\mid }x\mathrm{\mid }\ge 1.$$ Unlike the fixed-time covariance, which is governed directly by the Airy${}_1$ process, the two-time covariance involves the nonlinear variational evolution of the entire earlier height profile. Our proof combines cubic-exponential mixing of the Airy${}_1$ process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, the centered spatial averages, normalized by $N^{1\mathrm{/}2}$, converge in finite-dimensional distributions to a centered Gaussian process whose covariance is the space-integrated two-time correlation of the flat KPZ fixed point.