A JOIN-TREE SUFFICIENT CONDITION FOR THE NON-CANCELLING INTERSECTIONS CONJECTURE

The Non-Cancelling Intersections (NCI) Conjecture of Amarilli, Monet, and Suciu asks whether the union of a finite inclusion-incomparable family can always be constructed from exactly those intersections whose collected inclusion–exclusion coefficients are nonzero, using only disjoint union and subset complement. We prove a structural sufficient condition formulated in terms of an acyclic hypergraph representation. Let N be the family of non-cancelling intersections and let B be its inclusion-maximal members. If B admits a join tree satisfying the running-intersection property and every nonempty edge separator of that tree belongs to N, then ⋃︁F has an explicit left-linear construction over N. Rooting the join tree, one subtracts the parent separator from each child bag and then adjoins the resulting residue by disjoint union. Running intersection guarantees that each residue is disjoint from everything assembled earlier. This yields a linear-size structural certificate and a direct polynomial-time verifier once the join tree is supplied. We compare the theorem with the known partial cases and strengthenings in the original NCI paper and with the recent failure of unrestricted left-linear witnesses. As supporting evidence only, a public implementation verified 50,000 independently generated theorem-eligible instances, including examples with eleven collected cancelling intersections. The theorem and verifier are independent of any private search acceleration.