Intrinsic Affine Positionality in Ordered Regular Languages: Fixed-Width and Genealogical Rank

Abstract

This paper develops an intrinsic affine theory of positional rank in prefix-closed, right-prolongable ordered regular languages. It characterizes when fixed-width lexicographic ranks and more general unit-spaced graded ranks admit a normalized scalar positional representation. Positionality at one root is shown to propagate to every live residual state through state-dependent affine translations. These translations satisfy exact local tiling equations determined by least-digit dynamics and adjacent lexicographic cylinders. When the minimal residual automaton contains a branching state, the normalized positional basis is uniquely determined. A canonical lower-itinerary quotient compresses the translation dynamics. If the minimal live residual DFA has s states and the quotient has b classes, the positional basis has Hankel rank at most s + b - 1, while residual grading boundaries have rank at most s + b. Both bounds are sharp for every 1 <= b <= s. The paper also develops a finite compatibility test, a twisted-coboundary formulation of residual transfer, and exact criteria for fixed-width and genealogical positionality as two root-boundary specializations of the same intrinsic affine structure. For positive bases, the canonical fixed-width and genealogical conventions have Perron-scale growth, while more general intrinsic boundaries may support strictly faster positive growth.