Nilpotent--Pure-Periodic Decomposition of Profinite Eventually Periodic Sequences

Let Z-hat be the profinite completion of the ring of integers. We study sequences valued in Z-hat that are profinitely eventually periodic. A sequence A: N -> Z-hat is called an RFEP sequence if, for every modulus m >= 1, the reduced sequence A mod m is eventually periodic. For each RFEP sequence A, write N(A)(m) and T(A)(m) for the least preperiod and least period modulo m, and call Lambda_A(m) = (N(A)(m), T(A)(m)) the RFEP spectrum of A. We first establish the basic arithmetic of the spectrum: minimality, divisibility monotonicity, the Chinese remainder law for coprime moduli, and the prime-power primary decomposition. We then prove the main result: every RFEP sequence decomposes uniquely as the sum of a pure-periodic sequence and a nilpotent sequence. In this decomposition the pure-periodic part carries exactly the period spectrum, and the nilpotent part carries exactly the preperiod spectrum. We further describe the algebraic structure induced by addition, pointwise multiplication, and convolution, proving that the nilpotent class is an ideal under pointwise multiplication but in general not under convolution. Finally, we compute the spectra of integer-valued polynomials, the Fibonacci sequence, the factorial sequence, geometric sequences, and the Fermat sequence, and use the Thue--Morse sequence to show that automaticity does not imply the RFEP property. The boundary of the integer-valued decomposition is determined by the integrality of the pure-periodic projection, and the sequence 2^n gives a concrete example where this boundary fails.