Necessity of angle admissibility for the c+a and c+b periods

Prove that, whenever a primitive non-additive three-move subtraction game with a>=2 and gcd(a,b,c)=1 is purely periodic with least period p, the angle rho=c mod (a+b) belongs to the admissible set for that period: rho∈Theta_{c+a} when p=c+a and rho∈Theta_{c+b} when p=c+b.

Background

The admissible sets Theta_{c+a} and Theta_{c+b} are defined through finite avoidance conditions involving rotations of the two-move base-game P-position pattern. The paper proves the forward implication that membership in either admissible set yields pure periodicity with the corresponding candidate period.

The unresolved direction is whether every purely periodic game whose least period is c+a or c+b must satisfy the corresponding rotation-avoidance condition. Together with the period-restriction conjecture, this would imply the full necessity conjecture for c>=2(a+b), while the inert branch is already settled in that range.

References

Let $S$ be non-additive with $a\ge2$ and $\gcd(a,b,c)=1$, and suppose that $G_S$ is purely periodic with least period $p$. If $p=c+a$ then $\rho\in\Theta_{c+a}$, and if $p=c+b$ then $\rho\in\Theta_{c+b}$.

— Purely Periodic Three-move Subtraction Games  (2609.05358 - Manabe, 4 Sep 2026) in Conjecture~\ref{conj:residue}, Section 6, subsection “The periods c+a and c+b”