Linear Operators, Kernel Structures, and Riordan Operators of Profinite Eventually Periodic Sequences

We study linear operators, kernel operators, and Riordan-type operators acting on profinitely eventually periodic (RFEP) sequences valued in Zhat. Using the RFEP definition, the nilpotent--pure-periodic decomposition, and the characteristic-spectrum notation of the first five papers, we first establish the continuous kernel-matrix representation on H = Zhat^N0, and obtain a kernel criterion for RFEP linear operators in terms of the row-summability condition and the eventual periodicity of arithmetic-progression row sums. We then analyze in detail the shift, pointwise multiplication, convolution, prefix sum, difference, and binomial transform, giving their kernels, Riordan generating functions, and characteristic-spectrum estimates. The group law, identity, and inversion formula of the Riordan group are taken from the existing standard theory, on the basis of which we prove that mod-rational Riordan operators are RFEP linear operators. Finally we study the RFEP classification and the evolution of the characteristic spectrum under operators. For B = TA we define the operator-induced reduction map on each modulus level and give exact formulas for the preperiod spectrum and the period spectrum of B; we further distinguish RFEP linear operators from those with bidirectional preservation, and prove that in the invertible case an operator and its inverse jointly preserve RFEP if and only if both the RFEP class and the non-RFEP class are preserved. The binomial transform and mod-rational Riordan operators provide the typical examples.