Local Dynamics and Periodic Lifts of Carry Depths in Canonical PLRS Numeration

For a canonical positive linear recurrence numeration system with threshold-reset legality automaton and maximal boundary block

P = p₁···p_L,

we give an exact local grammar for the reset-depth sequence produced by lexicographic succession. Every positive reset depth determines its predecessor’s branch-point state and depth exactly (Theorem A); every carry-depth excursion collapses to depth 0 in at most L deterministic steps (Theorem B); and every realized excursion skeleton admits every periodic lift of its initial depth, with width increased accordingly, and with no gaps (Theorem C).

Together these give an exact decomposition of the carry-depth sequence into a finite deterministic skeleton and an unbounded periodic lift parameter. We further show that carry-depth structure depends critically on which of two natural residual-state conventions is used: the source-word residual state is exactly depth-determined,

σ(w) = q_{T(w) mod L},

while the branch-point state is maximally mixed,

R^*_t = B   for every   t ≥ 0.

We apply this grammar to state-dependent carry-cost decoders, obtaining an exact single-edge observability criterion, exact backward resolving radii throughout the deterministic predecessor chain, including its first forced depth-zero endpoint, and a sharp characterization of the multivalued obstruction that governs that endpoint’s own predecessor.