A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding

Disproved. An explicit 32-leaf tree has least isometric $\ell_\infty$-dimension six, not the conjectured five, and keeps dimension six under every assignment of positive edge lengths, so the weighted sharp-threshold conjecture falls too. Every tree with at most 31 leaves does attain $\lceil \log_2 t \rceil$, so 32 is the first failure; Brigham et al. had checked through 21. The proof is finite and exact: six explicit orientation masks cover all leaf pairs, and a recurrence on rooted branches rules out any five-orientation cover.

Disproved. An explicit 32-leaf tree has least isometric $\ell_\infty$-dimension six, not the conjectured five, and keeps dimension six under every assignment of positive edge lengths, so the weighted sharp-threshold conjecture falls too. Every tree with at most 31 leaves does attain $\lceil \log_2 t \rceil$, so 32 is the first failure; Brigham et al. had checked through 21. The proof is finite and exact: six explicit orientation masks cover all leaf pairs, and a recurrence on rooted branches rules out any five-orientation cover.