Non-negative Positionality in Regular Abstract Numeration Systems: Balance-Loop Normal Forms and Infinite Integer Viability

Abstract

This paper develops an exact normal form for the nonnegative positionality problem in regular abstract numeration systems. A companion paper showed that, after an effectively computable finite preperiod and periodic phase lift, affine positionality is governed by a finite homogeneous integer descriptor relation, while the nonnegative problem already contains the classical Positivity Problem for linear recurrence sequences. Here we identify the structure of that remaining nonnegative problem. We introduce proper homogeneous balance loops, systems of nonnegative integer states satisfying a fixed homogeneous balance relation in which every source column is nonzero. We prove an effective equivalence between nonnegative positionality for infinite regular languages and nontermination of these balance loops. The forward reduction converts a nonnegative positional realization into a finitely generated nonnegative transition monoid, using observable normalization, uniform exponential bounds, Hilbert-basis generation and a budget construction that enforces properness. The reverse reduction constructs an infinite regular language that realizes any proper homogeneous balance loop by repeated lexicographic equality gadgets. As consequences, the resulting transition systems are effectively finitely branching and admit uniform exponential bounds. Infinite viability is therefore equivalent to viability at every finite horizon, giving finite certificates for non-positionality. In particular, regular abstract numeration system positionality lies in the arithmetical class Pi-0-1, while non-positionality is recursively enumerable. The paper also examines deterministic and branching subclasses and shows that even low-dimensional balance systems contain residue-affine dynamics closely related to generalized Collatz-type integer loops.