An actual serial half-diamond certificate

The limiting profile is random. The theorem does not assert convergence of the complete profile to a deterministic constant.

If $0\le s\le B$, Lemma lem:finitepath gives a finite-size failure probability for any chosen bound on the right side of (eq:serialcomparison). For general offspring-bounded $s$, the finite-variance theorem supplies the stated asymptotic conclusion; no numerical rate is invented.