Growth and Bounded Rigidity of Profinite Eventually Periodic Sequences

We study the ordinary absolute-value growth of integral profinitely eventually periodic (RFEP) sequences. We use the nilpotent–pure-periodic decomposition and the closure under the operations of the first paper, and the characteristic-spectrum notation and the filtration framework of the second paper; we do not discuss the realizability of spectrum data, nor do we repeat the spectrum realization and finite-rigidity theory. We introduce the growth envelope and its eventual domination order, and prove that the integral RFEP class has no maximal growth element; using least-common-multiple support sets we construct arbitrarily slow yet unbounded nilpotent sequences, and prove that the unbounded RFEP class has no minimal growth order. We further prove in the finite-coordinate topology that, given any tail growth lower bound, the nilpotent sequences and the pure-periodic sequences satisfying that lower bound are each dense. Finally we prove that for an integral RFEP sequence, boundedness, finiteness of the value set, and eventual periodicity in the ordinary sense are equivalent, and we obtain the stability of the characteristic spectrum at large moduli in the bounded case.