Exact Cascade Uncertainty Under Pairwise Compression

Abstract

Higher-order data are often compressed into pairwise co-occurrence counts, but this can hide how pairs are assembled into triples. We ask for the largest possible difference in deterministic cascade size between simple 3-uniform hypergraphs with the same labeled exact pair-codegree matrix. Starting from two active vertices, a triple activates its third vertex when its other two are active. We define M₂(n) as the maximum closure-size gap inside any projection fiber on n vertices. A direct proof gives M₂(n)=0 for 2≤n≤5 and M₂(n)=n−3 for n≥6. The upper bound follows because equal codegrees force both cascades either to stop at two or reach at least three vertices. A five-edge Pasch-trade-plus-context construction attains the bound and extends to every larger n. Fresh exhaustive enumeration of all 2²⁰ simple 3-uniform hypergraphs on six labeled vertices independently checks the first ambiguous order, a maximum gap of three, and edge thresholds: four edges for different final sets, five for different sizes, and seven when non-isomorphism is additionally required on six vertices. Two separately implemented censuses agree on 69 shared summary fields. The result is a precise warning about information lost under pairwise compression for one abstract deterministic closure rule; it is not an empirical claim about real epidemics or social networks.