Affine Networks of C-Finite Recurrence Signatures

Building on the affine framework for weighted binomial itineraries developed in the preceding work, we fix two elementary procedures - binomial translation and geometric scaling - and study the relational network that they generate on C-finite recurrence signatures and their Taylor fibers. Binomial translation shifts each characteristic root by h, while geometric scaling multiplies each root by k. Starting from a signature A, we define four additional nodes: B by translating A by 1, C by scaling A by k, D by scaling B by k, and E by translating C by 1. The five nodes form a finite chart inside the affine orbit generated by the affine group of the line. We obtain a universal formula for every directed transition, identify the subgroup generated by translation by 1 and scaling by k together with their inverses, and describe the induced orbit of an arbitrary-order signature. Binomial inversion is exact on sequences and formal signatures, while passage to a minimal signature may lose a zero-root factor and therefore obstruct reconstruction if the numerator is discarded. The network is then lifted from signatures to Taylor fibers. For a rational C-finite state expressed as the quotient of a numerator polynomial N(z) and a recurrence denominator D(z), the affine operator transports the numerator by a linear automorphism whenever the scaling parameter is nonzero. Consequently, two procedures may collide on the recurrence base while remaining distinct in the numerator fiber. Fibonacci, Pell, and the triple-root signature (3, -3, 1) provide explicit examples and OEIS anchors. Finally, we study the Hadamard product of nodes E and D. In arbitrary order, the exchanged-pair symmetry of a Hadamard square disappears generically when k is neither 0 nor 1, restoring formal order d squared; in order two, an exact discriminant factorization classifies the algebraic components on which the order decreases.