Sequential cascades on random hypergraphs

Sequential decisions within a group can correlate individual adoption outcomes, but this microscopic dependence is usually replaced by an imposed group-level response. We study a one-shot cascade on sparse random hypergraphs in which the first target in a newly triggered hyperedge adopts with probability p, whereas each later target adopts with probability p + Δ after a success and p - δ after a failure. For this stated process, we derive the exact target marginal qm, the complete correlated group-offspring probability-generating function Fm(z), and the early branching factor B. The classical Galton-Watson criterion then places the infinitesimal-seed onset at B=1, while its extinction fixed point depends on the complete offspring law. The new ingredients are the exact transfer-matrix offspring law for outcome memory and a matched-independent hypergraph control with the same qm and onset. For the original-cardinality null, directed-incidence percolation gives Rnull = ρnullSnull; the forward-survival and backward-reachability maps coincide for pairwise groups but generally differ for larger groups. Monte Carlo tests across group-size and degree laws support the threshold and survival predictions. A collision-aware control, which rematches the null to the number of targets actually remaining at first processing, reduces but does not remove the observed final-size gap at the tested parameters. System-size and cutoff sweeps likewise show that the 5% finite-network indicator is a proxy rather than the branching-survival probability. All claims are conditional on one-shot triggering, a vanishing seed density, and locally tree-like topology.