Theorem: One unitary, one threshold, all later retained cuts

Fix $\zeta >0$, $0<p<1$, and a positive-residual cutoff $K$. If a retained cut $k_{\ast }\le K$ satisfies

then, with Haar probability at least $1-p$, one sampled unitary has, simultaneously for every $k_{\ast }\le k\le K$,

A sufficient dimensional threshold is

In the qubit model, a sufficient cut is

provided $k_{\ast }\le n-q$.

Proof. Markov's inequality and (eq:K) imply

For a later cut, the complete complement is obtained by tracing the additional released cells:

The same trace maps ${{\sigma }_{B_{k_{\ast }}}}^U$ to ${{\sigma }_{B_k}}^U$. CPTP contraction proves (eq:leakall) on the single event; no independence or repeated Haar sampling is used. Lemma lem:approx gives (eq:recall). Equation (eq:thresholdsimple) follows from the final bound in (eq:K).