Surjectivity of the Characteristic Spectrum and Finite Rigidity of the Period Spectrum for Profinite Eventually Periodic Sequences

We study the surjectivity of the characteristic spectrum map of profinitely eventually periodic (RFEP) sequences and its integral boundary, and treat the finite rigidity of the period spectrum as an independent spectral property. Keeping the notation of the first paper, write Λ_A(m) = (N(A)(m), T(A)(m)), where N_A and T_A are the preperiod spectrum and the period spectrum; we keep the spectrum order of the second paper, but do not repeat its decomposition and filtration theory. We first define the preperiod candidate spectra and the period candidate spectra, and prove that they are characterized respectively by max-type and lcm-type compatibility conditions on the divisibility lattice. For Ẑ-valued RFEP sequences we give an explicit prime-power-by-prime-power construction: any preperiod spectrum and period spectrum satisfying these compatibility conditions can be realized simultaneously. Hence the Ẑ-valued characteristic spectrum map is surjective. In the integral direction, the preperiod coordinate must take into account the fact that a nonzero integer has only finitely many divisors. We give a necessary and sufficient condition for an integral nilpotent preperiod spectrum, and construct a realizing sequence. For the period coordinate, we define finite rigidity by the requirement that the modulus preimage of each spectral value ℓ be finite, and prove: if some period-spectrum value of a pure-periodic integral RFEP sequence has an infinite preimage, then that integer is an ordinary period of the sequence; consequently, a sequence that is not periodic in the ordinary sense has a finitely rigid period spectrum. Finally we propose the surjectivity conjecture for finitely rigid period spectra.