Nonnegative Positionality in Regular Abstract Numeration Systems: Balance-Loop Normal Forms and Viability Filtrations

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 with a fixed initial state and nonzero source columns. We prove an effective equivalence between nonnegative positionality for infinite regular languages and nontermination of these balance loops. The forward reduction passes from nonnegative positional realizations to finitely generated transition monoids using observable normalization, exponential bounds, Hilbert-basis generation and a budget construction enforcing properness. The reverse reduction constructs an infinite regular language realizing any proper homogeneous balance loop by repeated lexicographic equality gadgets. We then develop a structural theory of balance-loop viability. Finite-horizon viable states form an effectively computable descending filtration of affine semigroups. Infinite viability is equivalent to viability at every finite horizon, so non-positionality has finite certificates. Consequently, regular abstract numeration system positionality lies in the arithmetical class Pi-0-1, while non-positionality is recursively enumerable. In one physical dimension, we derive an autonomous recurrence for the gcd of the viability semigroups. Proportional systems are governed by explicit divisibility growth, while nonproportional systems have an eventually stable viability lattice. In the nonproportional terminating regime, characterized exactly by 0 < b_i < a_i for every generator, both the least positive viable state and the conductor grow on the same exponential scale. For two generators, the late viability filtration admits a stronger classification. After normalization by the stabilized gcd, it becomes an exact finite union of arithmetic tails whose thresholds evolve through a finite phase map with bounded periodic affine corrections. This purification phenomenon fails already for three generators. The explicit example A = (2, 6, 8), B = (1, 1, 1) has a persistent boundary hole at every horizon, showing that higher factorization dimension introduces genuinely new arithmetic obstructions.