Excluding the Nonfull Four-Cover Branch in Height-Five Frankl Counterexamples

This research preprint gives an exclusion conditional on the structural results in Tian’s preprint of the nonfull four-cover branch (branch C) in Chenxiao Tian’s August 2026 arXiv preprint classification of hypothetical 5-minimal empty-set-free counterexamples to Frankl’s union-closed sets conjecture. The proof uses critical-coordinate frequency identities, height-two deficit disjointness, a collision-free lower-to-upper doubleton transport map, and a maximal-avoider frequency argument. It rules out branch C under Tian’s Lemmas 2.1–2.3 and Theorems 4.1 and 7.8, subject to independent specialist verification. Tian’s branches A and B are not addressed by this result; the complete height-five case and the unrestricted Frankl conjecture are not claimed. Source, finite-check code, and SHA-256 checksums accompany the manuscript. This is a research preprint, not a peer-reviewed publication.