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 affine characterization of positionality in terms of active lexicographic cylinders. After lifting one support period, the full affine problem becomes a finite linear descriptor relation.
Over the rationals, affine positionality is decidable by finite-dimensional viability and every realization admits a C-finite representative. Over the integers, the periodic tail defines a viability subgroup for an integer linear relation. We identify this subgroup with the constant part of an explicit S-saturated polynomial module, where S consists of polynomials with unit constant term. Primary decomposition over ℤ[z] therefore gives an effective decision procedure and an effective C-finite replacement.
The nonnegative problem encounters a substantially stronger barrier. We construct, from any integer linear recurrence sequence (an), an infinite regular language whose genealogical abstract numeration system is positional with nonnegative integer weights if and only if an ≥ 0 for every n. Thus LRS Positivity reduces effectively to general regular-ANS positionality. The results separate the finite affine and integral layers from the remaining infinite sign constraint and show that any decision procedure for general regular-ANS positionality would in particular decide LRS Positivity.
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