Let P = e1 ⋯ eL be the primitive maximal boundary block of a canonical positive linear recurrence sequence (PLRS) numeration system. Earlier local carry theory determines individual successor depths but not which such depths can occur consecutively. We solve this global realization problem.
For each maximal-orbit phase i and width m, the state carry words satisfy
Wi(m) = (W0(m−1)(m − 1))ei+1 Wi+1(m−1).
If u is a finite depth word with M = max u, then
u is globally realizable ⇔ u ∈ Fact(W0(M+L)).
Thus realization has an exact bounded-width criterion, although the corresponding root carry word may be exponential in that width.
The same recursion yields a generalized ruler word with exact renormalization and a countable-alphabet substitution structure. At matched scale, phase-containment recursions connect global realization to the anti-Lyndon criterion: P is anti-Lyndon exactly when the root phase already contains every matched-scale carry factor at every depth.
For non-anti-Lyndon boundaries we identify the exact first failure depth and distinguish immediately visible local defects from globally masked witnesses.
Leave a Reply