Rank-Positional Threshold Languages: ResetStructure, Census Rigidity, and Finite Reconstruction

Let L be a prefix-closed language over a finite ordered digit alphabet, and assume that at every reachable residual the legal next digits form an initial interval

{0, 1, …, d(q)}.

Write

C_n = |L ∩ Σⁿ|,   C₀ = 1,

and let R_n(w) be the zero-based lexicographic rank of w ∈ L ∩ Σⁿ. We call L rank-positional if

R_n(a₁···a_n) = ∑_j=1ⁿ a_jC_n−j

for every legal word. Thus the language’s own width census supplies its positional place values.

Viewed from abstract numeration theory, this is a restricted form of the broader positionality problem: when does a language whose numerical values are defined by enumeration admit a genuine positional realization? Here the basis is not arbitrary; it is determined intrinsically by the language itself through its width census. Under the threshold hypothesis, we give a complete characterization of this census-positional class.

We prove that the fixed-width rank identity is equivalent to a local reset condition: every strict-deficiency successor has the same continuation census as the root, and in the accessible minimal automaton every such transition returns literally to the root. Hence every regular rank-positional threshold language has a root-reset lollipop automaton whose live states form one maximal-transition orbit.

The maximal thresholds define an intrinsic boundary

e = e₁e₂···.

Its tails classify residual states, regularity is equivalent to eventual periodicity, and a boundary with minimal preperiod t and primitive period p yields exactly t + p live states. The same boundary is the length inventory of a canonical prefix-free reset code, giving

L = B(e)^*P(e)

and

C_n = 1 + ∑_i=1ⁿ e_iC_n−i.

This relation is triangular, so the census determines the boundary, the minimal automaton, and the complete language.

For regular systems we prove an optimal finite rigidity theorem: if two systems have s and s′ live states, equality of

C₀, …, C_s+s′−1

forces equality of the languages, and the bound is sharp for every pair (s, s′). We also determine the exact order of the minimal homogeneous linear recurrence satisfied by the census:

r = t + p − deg gcd(B, S_p),

where B is the periodic boundary polynomial and

S_p = 1 + z + ··· + z^p−1.

Thus the gap between automaton-state complexity and scalar recurrence complexity is an explicit cyclotomic defect.

The threshold hypothesis is sharp: without it, rank-positional languages can have distinct residuals with identical continuation-count sequences. Purely periodic and self-admissible boundaries recover, respectively, PLRS and beta/Bertrand specializations of the general boundary theory.

The paper therefore gives a complete intrinsic solution to the census-positional threshold case of the general positionality problem for regular-language numeration, while leaving the unrestricted positionality problem open.