Posterior replicas and conditional information in Gaussian regression

For a signal with density bounded by L relative to uniform probability on $S^{d-1}$, we bound the information in a finite message W formed from exact Gaussian measurements, conditional on an independent projection revealed only to the analyst. For explicit row counts proportional to d, the bound is $O(H(W)/d+d+\log (2+\log L))$. Consequently, a finite-state learner with $o(d^2)$ persistent bits and a deterministic sample horizon needs $\Omega (d\log (1/ϵ))$ fresh noiseless Gaussian measurements for constant-probability angular accuracy $0<ϵ\le 1/10$ under the uniform spherical prior.