A central limit theorem for the first path Betti number of a random digraph

We prove a central limit theorem for the first non-regular path Betti number of a random digraph with independent ordered edges of probability $p_n$, in the full range $np_n\to\infty$ and $n^{2/3}p_n\to0$. This establishes Chaplin's Conjecture B.2. The variance is asymptotic to $n(n-1)p_n(1-p_n)$. The proof compares centered Betti numbers with the edge count: a monotone family of short two-chain generators controls the variance of the boundary rank, while a component estimate handles graphs that need not be connected.