Initial Tukey structure below a stable ordered-union ultrafilter

Answering a question of Dobrinen and Todorcevic, we prove that below any stable ordered-union ultrafilter $\mathcal{U}$, there are exactly four nonprincipal Tukey classes: $[\mathcal{U}],[{\mathcal{U}}_{\min }],[{\mathcal{U}}_{\max }]$, and $[{\mathcal{U}}_{\mathrm{minmax}}]$. This parallels the classification of ultrafilters Rudin-Keisler below $\mathcal{U}$ by Blass. A key step in the proof involves modifying the proof of a canonization theorem of Klein and Spinas for Borel functions on ${\mathrm{F}\mathrm{I}\mathrm{N}}^{[\mathrm{\infty }]}$ to obtain a simplified canonization theorem for fronts on ${\mathrm{F}\mathrm{I}\mathrm{N}}^{[\mathrm{\infty }]}$, recovering Lefmann’s canonization for fronts of finite uniformity rank as a special case. We use this to classify the Rudin-Keisler classes of all ultrafilters Tukey below $\mathcal{U}$, which is then applied to achieve the main result.