Nonexistence of nontrivial algebraic periodicity in normal-play partizan games

Determine whether normal-play partizan octal games admit no nontrivial algebraic periodicity, and, if so, identify a structural feature of the partizan normal-play setting that distinguishes it from misère play.

Background

Nontrivial algebraic periodicity is observed in several misère octal games, but the authors report finding no such examples among normal-play partizan octal games. They ask whether this absence is universal and whether it reflects a structural distinction between normal and misère play analogous to the role of the Sprague–Grundy theorem for impartial normal-play games.

References

Do normal-play partizan octal games admit no non-trivial algebraic periodicity? If so, does this suggest some specific structure of the partizan setting under normal play that distinguishes it from misère play (analogous to the Sprague--Grundy theorem in the impartial case)?

— A Finite Automaton Approach to Combinatorial Games  (2608.13273 - Liang, 13 Aug 2026) in Section 3, subsection “Algebraic Periodicity”