Theorem: Finite half-diamond path-average certificate

Under (eq:offspring) and the marking contract,

Consequently, for every fixed $a>0$,

For $x>0$, $1\le L\le N-1$, the probability that the maximum in (eq:channelsum), before division by $\sqrt{N}$, exceeds $x$ is at most

For $1\le \ell \le L$, the probability that

is at most

Both bounds may be capped at one.

Proof. The spine mean of $a_j(\Lambda )$ is $a_j(\overline{\Lambda })$. If the half-diamond norm of a centered path sum exceeds $x$, Lemma lem:choicoord implies a coordinate deviation greater than $2x/d^4$. Apply Lemma lem:finitepath with $B=1$ to each coordinate, and use a union bound over $d^4$ coordinates. This proves (eq:channelabsolute).

For a path of depth $h$, a path-average error exceeding $\epsilon$ gives a centered sum error exceeding $h\epsilon$. The same argument proves (eq:channelrelative). The asymptotic statements follow from Corollary cor:boundedpath, the finite coordinate set, and division by the minimum depth.