Vertex-link activation for weak tetrahedron saturation in random three-uniform hypergraphs

Let G = G⁽³⁾(n,p) be the binomial three-uniform hypergraph on [n], write b_n = binom(n−1,2), and let A_n be the event wsat(G,K₄⁽³⁾) = b_n. For every sequence p = p(n) in [0,1] with np⁶ ≥ 100 log n for all sufficiently large n, we prove that A_n holds with high probability. For each fixed 0 < ε < 8/3, its probability tends to zero when p ≤ ((8/3−ε) log n/n)1/3 for all sufficiently large n. A fixed-vertex seed attains the sufficient range through connected graphs of admissible pair cores and six-edge witnesses. The failure range follows from uncovered host edges, a boundary-rank estimate resilient to one edge deletion, and edge-count anti-concentration. At five vertices, the equality event A₅ is neither increasing nor decreasing.