Proposition: Ensemble mismatch

If $d_{\mathrm{TV}}(\mu ,{\mu }_{\mathrm{Haar}})\le \tau$, then at a fixed cut

These statements apply to fixed inputs independent of the sampled unitary.

Proof. Entropy ranges are $0\le S(R)\le {\log }_{2}m$ for a pure complete proxy and $0\le H(Y)\le {\log }_{2}r$. The expectation difference of a function in $[0,L]$ is at most $L{d}_{\mathrm{TV}}$, by integration of its level sets.