Nilpotent Weight Systems, F--J Representations, and Characteristic Spectrum Computation of Profinite Eventually Periodic Sequences

We study eventually periodic sequences with values in the profinite completion Ẑ and their finite-level digit representations. Following the nilpotent--pure-periodic decomposition of [1] and the characteristic-spectrum notation of [2], we introduce nilpotent weight systems satisfying a divisibility-cofinal condition, establish the unique mixed-radix expansion of Ẑ, and prove that the F--J digit sequences characterize RFEP, pure periodicity, and nilpotence level by level. We then make the characteristic spectrum precise as the preperiod and period of the readout sequence of the finite-level F--J digit vectors; for modulus-aligned weight systems we obtain a maximum--least-common-multiple formula, and for general moduli we explain why period shortening arises exactly from the readout quotient relation. Finally we give the finite-level F--J spectrum correspondence formula under a continuous linear operator.