All 23 cases at the 5/9 level are solved

On 21 July 2026, parametric solutions were completed for all 23 D₄ orbits at the 5/9 level.

For each orbit, the selected square-valued entries, the system of quadrics, the parametrization of E, x, and y, and the precise coverage status are recorded.

Completeness of the parametrization is proved for 15 orbits. For each of the remaining eight, the widest subset with proved coverage is stated together with the exceptional locus.

This completes the orbit-by-orbit construction for the 5/9 problem, but it does not solve the open existence problem at the 9/9 level.