An exact verification of the fourteen-quartet optimum for seven taxa

The minimum number of four-taxon subsets that distinguish all unrooted binary trees on seven fixed labelled taxa is fourteen. This verification note connects an existing partition-crossing lower bound with the fourteen-quartet construction in Fischer and Pott, Example 3.23. It presents a pair-link proof of the lower bound, a compact certificate for the published construction, and reproducible checks of all 350 four-block partitions and all 945 unrooted binary seven-taxon trees. Bujtás and Tuza's account of the lower bound and its earlier origins is credited. The contribution is an attributed verification and synthesis of these existing results. The public edition also corrects text serialization for Windows reproducibility. Byungwoong Yoo selected the problem and directed the research and certification requirements. Substantive assistance from ChatGPT and Codex with mathematics, source comparison, programming, execution and writing is described in the note. The original note is CC BY 4.0; original code is separately licensed under MIT.

Sources: Fischer and Pott (2025), https://doi.org/10.1007/s11538-025-01457-7; Bujtás and Tuza (2018), https://doi.org/10.14232/actacyb.23.3.2018.6; Steel and Sanderson (2010), https://arxiv.org/abs/0906.4875.