Rigorous proof of algebraic periodicity

Prove rigorously the algebraic periodicity observed for the listed misère impartial octal games, including the validity of their finite-parameter reduction schemes.

Background

The paper defines algebraic periodicity through two reduction rules involving the largest and second-largest heaps and reports computational evidence for numerous misère impartial octal games. The observed patterns were obtained from finite computations, and the authors explicitly state that no rigorous proof method has yet been found. Establishing such proofs is necessary for complete mathematical solutions of these games.

References

The above results all come from observations of computations on smaller positions; we have not yet found a method for rigorously proving such algebraic periodicity.

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

Can one define a notion of ``weak algebraic periodicity'' that encompasses the two exceptional cases \mathbf{0.145} and \mathbf{0.54}?

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