Endpoint Decomposition and an Anti-Lyndon Aperiodicity Criterion for PLRS Numeration

Canonical positive linear recurrence sequence numeration systems admit a natural deterministic threshold-reset automaton for their legality languages. Paper IV showed that, after minimization, every presentation of a fixed system yields the automaton of the canonical reduction. The present paper studies the internal transition structure of that canonical automaton. We introduce anchored deficient-block languages and prove a Generalized Endpoint Decomposition describing every live computation from an arbitrary starting state as either a forced threshold march or an anchored first reset followed by a universal restart grammar. We then identify an exact aperiodicity criterion. If

$P = c1c2 · · · cL−1(cL − 1)$

is the primitive maximal boundary block of the canonical reduction, then the canonical threshold-
reset automaton is aperiodic if and only if $P$ is lexicographically maximal among its nontrivial cyclic rotations, equivalently, if and only if $P$ is anti-Lyndon. The forward implication follows from a total
order on live states induced by the rotations of $P ∞$, under which every digit acts monotonically. For the converse, the least rotation exceeding $P ∞$ yields an explicit word whose induced transformation has a nontrivial periodic orbit. Since the canonical automaton is minimal over its finite natural digit alphabet, this is also equivalent to star-freeness of the legality language. Exhaustive computations for all canonical reductions of order at most six with coefficients at most four agree exactly with the
theorem and recover the previously observed cycle families as concrete manifestations of the same restart structure.