Classical Minor Obstructions in a Nine-Region Census of Two-Layer Cube-Ideal Gluing

I study how two cube-ideal faces can be joined along a new Boolean coordinate while their parent fails to be cube-ideal. Here cube-ideality means integrality of the canonical 0-1 relaxation. The paper treats nine fixed regions of the five-dimensional Boolean cube. An exhaustive exact census examines 4,029,132 ordered seams of total size at most six. It finds 17,430 distinct nonideal parents whose two faces are cube-ideal. Their least nonideal minor dimensions are three, four and five, with counts 16,444, 888 and 98. Every parent has a certificate arising from a classical minimally nonideal example. A second census examines the 74,484 seams below the first legal sizes of the nine regions. Exactly 16 have two ideal faces and a nonideal parent. In each of the two exceptional regions, the eight ordered seams are the orientations of the four edges of a K₂,₂ disjointness graph. All conclusions are confined to the nine listed regions. The supporting data, exact checkers and certificates are preserved in the existing Finite Atlas, version 1.1.0: https://doi.org/10.5281/zenodo.21362847. This deposit contains the paper and links to that dataset.