Statistical Equivalence Does Not Determine Streaming Memory

Can asymptotically equivalent statistical experiments have different streaming-memory complexity? For every fixed $B>0$, we construct explicit finite realizable learning problems with $N$ concepts on $N$ uniformly sampled points, where $R:={\log }_{2}N$. Their labeled-sample experiments have Le Cam distance at most $R^{-B}$ uniformly over the sample count, while their base Gram matrices and robust-SQ profiles agree exactly up to radius $\Theta (\sqrt{R/\log R})$. Nevertheless, one problem admits exact one-pass learning with $R^{O_B(1)}$ samples and $O(R^2/\log R)$ bits, whereas for every fixed ordered pass count $p$, the other requires ${\Omega }_p(R^2)$ bits or $N^{{\Omega }_p(1)}$ samples, even for improper learners with error $1/64$. Rare field equations and binary parity observations have identical pairwise correlations but different storage costs, separating unrestricted statistical simulation from fixed-pass streaming simulation.