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

Let L ⊆ {0, 1, …, c}^* be an infinite regular language, ordered genealogically, and let G_L(w) denote the genealogical rank of w ∈ L. We ask whether there exists a sequence of nonnegative integer weights U = (U_n)_n≥0 such that

G_L(a₁⋯a_m) = ∑_i=1^m a_iU_m−i   (a₁⋯a_m ∈ L).

The companion paper [5, Theorem 6.1] proves that, after an effectively computable support preperiod τ and period p, the periodic tail can be stacked into a macrostate X_k ∈ ℤ^D satisfying

𝒜X_k + ℬX_k+1 = 0.

It also shows that the nonnegative problem already contains the Positivity Problem for linear recurrence sequences. Here we identify an exact normal form for that remaining nonnegative problem.

We define a proper homogeneous balance loop to be a system

Bλ_n = Aλ_n+1,    λ_n ∈ ℕ^r,

where A and B are nonnegative integer matrices and every column a_i of A is nonzero, together with a fixed initial vector λ₀. Properness gives ‖a_i‖₁ ≥ 1, which in turn makes the successor relation effectively finitely branching.

Our main theorem proves an effective equivalence between nonnegative positionality for infinite regular languages and nontermination of proper homogeneous balance loops. The forward reduction passes from the periodic descriptor relation to its nonnegative integer transition monoid, uses a Hilbert basis, and adds a budget coordinate to remove zero-source generators. The reverse reduction is constructive: a regular language built from radix fillers and finitely many repeated lexicographic equality gadgets realizes any balance loop.

The normal form has immediate consequences. Proper balance loops are effectively finitely branching, every trajectory has an effective exponential envelope, and nontermination is equivalent to survival at every finite horizon. Consequently, non-positionality of a regular abstract numeration system is recursively enumerable.

The result isolates the positivity barrier as an infinite nonnegative integer viability problem within a restricted homogeneous class of linear-constraint loops.