Theorem: Scaled, all-later-cut certificate

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

then, with Haar probability at least $1-p$, the same sampled device has

A sufficient condition is

For full-diamond target $0<\epsilon <1$, the choice $\zeta ={\epsilon }^2/8$ gives the sufficient condition

Proof. Markov's inequality bounds

Proposition prop:nested propagates this one event to every later retained cut. Apply Theorem thm:correction. The full-diamond error is at most $2\sqrt{2\zeta }$.