Quantitative Brownian regularity of the KPZ fixed point with arbitrary initial data

We show that the spatial increments of the KPZ fixed point starting from arbitrary initial data, exhibit strong quantitative comparison against rate two Brownian motion on compacts. The above estimates are uniform in the initial data supported in some compact set. As applications, we obtain a one-sided large deviation inequality for spatial increments of the KPZ fixed point and show that the Wiener density of the centred KPZ fixed point started from arbitrary initial data has finite entropy.