Classify character gradings under the full automorphism group

Classify the orbits of the character-induced gradings of the complex Exceptional Periodicity algebras under the full automorphism group, and determine which gradings admit nonuniform refinements or compatibility with an integral order and the transported real structure.

Background

The paper proves the existence of balanced tetracode character gradings at every level, but it does not classify all character gradings up to the full automorphism group. The computational section likewise notes that classification under the full automorphism group and classification of arbitrary Z_32 gradings remain separate issues.

The conclusion explicitly asks how the gradings decompose into automorphism-group orbits and which ones support further refinements or compatibility with the integral and positive-real structures.

References

How do the character gradings split into orbits under the full automorphism group? Which admit nonuniform refinements or are compatible with an integral order and the transported real structure?

— Okubo algebras and Exceptional Periodicity: tetracode gradings, root-lattice obstructions and Clifford cones  (2610.18643 - Marrani et al., 13 Oct 2026) in Section “Conclusions and further questions”