Vanishing of first path homology above the random-digraph exponent −2/3

Let $G(n,p)$ be the loop-free random digraph in which every ordered edge is present independently with probability $p$, with reciprocal edges permitted. We prove that its first non-regular path homology over $\Q$ vanishes with probability tending to one if $n^2p^3/\log^4(2n)\to\infty$. Consequently, vanishing holds for $p=n^\alpha$ for every fixed $-2/3<\alpha\leq0$, as conjectured by Chaplin. The proof uses only boundaries of directed diamonds and transitive triangles. A uniform estimate for the diamond quadratic form shows that its annihilator consists of vertex gradients and the constant edge cochain. A transitive triangle removes the latter direction. The estimates retain the independent ordered-edge model and the non-regular boundary convention throughout.