Lemma: A tree-law Bell-overlap interval

For $1\le \ell \le L\le N-1$ and $r>0$, set

With probability at least $1-{\delta }_{\mathrm{tree}}$, simultaneously for all $v$ with $H(v)\ge \ell$,

Proof. Apply (eq:finiterelative) to the three correlator labels with $B=1$, and use a union bound. Their path-average correlators are exactly those of ${{\Lambda }_v}^{\mathrm{av}}$, by linearity. The fidelity combination has sum of absolute coefficients $3/4$.