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

Let L be an infinite prefix-closed, right-prolongable regular language over a finite ordered alphabet of nonnegative integer digits. We ask when the lexicographic ranks of its fixed-width cross-sections, or more generally unit-spaced graded ranks, admit a normalized scalar positional representation

V_U(a₁ ··· a_n) = ∑_j=1ⁿ a_jU_n-j,    U₀ = 1.

Positionality at one root propagates to every live residual state. There are residual translations λ_q(n) such that

R_q⁽ⁿ⁾(w) = V_U(w) – λ_q(n),

and this residual affine form is equivalent to local tiling equations. If a_q is the least legal digit, p(q) = δ(q,a_q), and adjacent digits have targets r,s, then

λ_q(n+1) = a_qU_n + λ_p(q)(n),    U_n = N_r(n) + λ_r(n) – λ_s(n).

With at least one branching residual state, any genuine adjacency determines the only possible normalized basis; the basis and all residual translations are then unique and integral whenever compatibility holds.

The translation subsystem has a canonical finite quotient. Two states are equivalent when their infinite least-digit threshold itineraries agree. If the minimal live residual DFA has s states and this lower-itinerary quotient has b classes, census dynamics require s coordinates while translations require only b. The resulting (s + b)-dimensional realization has one common-translation gauge mode invisible to positional value. Hence

r_U ≤ s+b-1,    r_β^(q) ≤ s+b,

where r denotes Hankel rank and β^(q) = λ_q is the residual grading boundary. Both recurrence-complexity bounds are sharp for every 1 ≤ b ≤ s.

R_U(s,b) = s+b-1,    R_β(s,b) = s+b.

The same finite realization gives a compatibility-recognition bound of s + b − 2: if compatibility fails, some genuine adjacency detects the failure by that continuation depth.

This recognition bound is not universally sharp for genuine integer digit systems. At s = b = 4, the exact integer recognition radius is 5 rather than the universal bound 6. In the corresponding rational-threshold affine relaxation the radius is 6. Thus the one-step improvement is caused by an arithmetic integrality obstruction rather than by a smaller ambient realization dimension.

Equivalently, finite recognition complexity has two distinct sources: linear realization complexity and the integer lattice imposed by the digit thresholds. This motivates separate integer and rational recognition-radius problems beyond the universal s + b − 2 bound.

Thus maximal positional recurrence complexity splits into census complexity plus lower-itinerary complexity, with one dimension removed by the gauge, while finite recognition can be strictly shorter because of additional arithmetic constraints.

Fixed-width and genealogical positionality are root-specific specializations of the same intrinsic structure. Fixed-width rank occurs exactly when the chosen root boundary is zero. Genealogical rank occurs exactly when it equals the cumulative census

B_n = ∑_m<n |L ∩ Σ^m|,

in which case the least root digit is 1 and

U(z) = C(z) / [(1-z)g_*(z)].

The paper also develops a twisted-coboundary formulation of residual transfer, giving a root-change law and a path-independent description of affine boundary transport across the automaton.

For positive bases, both canonical root conventions have Perron-scale growth, while general intrinsic boundaries may support strictly faster positive growth.