Entropy and BPL Structures of Profinite Eventually Periodic Sequences

Within the framework of profinitely eventually periodic sequences (RFEP) established in the first paper, we study the prime-power towers and the two kinds of layer increments of the characteristic spectrum, and introduce the exponent matrix together with the preperiod entropy and the period entropy. We characterize the class of bounded prime-power lifting (BPL) sequences by the criterion that the period entropy equals 1 and the non-principal-direction exponents vanish in the tail, and prove that this is equivalent to the condition that the tail period quotient of each tower equals only 1 or p, with at least one tower attaining p infinitely often. Finally we give the layer-by-layer formulas and BPL criteria for integer-valued polynomials, linear recurrences, and geometric sequences. The converse realizability of spectrum data is outside the scope of this paper.