Affine Positionality in Abstract Numeration Systems: Phase Dynamics, Decidability, and a Positivity Barrier

We study positionality for abstract numeration systems built from arbitrary infinite regular languages over finite ordered digit alphabets. Unlike the synchronous prefix-closed setting, residual languages need not support every continuation width, so fixed-width rank data live on partial state-width supports. We show that these supports are eventually periodic and derive an exact phase-local…