Localization costs and information growth for exact Gaussian observations

For the image of a uniform cube under a spherical coordinate map, we prove that finite messages from t blocks of $\Theta (d)$ exact Gaussian measurements reveal only $O_A(dt)$ information when each message has at most $\exp (A{d}^2)$ values, for fixed A. The same bound holds when each message is supplemented with a nested cell that restores the required geometric spread.