Affine Weights, Local Lifting, and the Matrix–Sequence Theory of Pseudorandomly Weighted Lucas Sums

Abstract This paper completes the matrix–sequence framework behind the modular-period theory of pseudorandomly weighted Lucas sums developed in the companion works. Its first layer develops a general theory of observable linear state systems \ ( (V, A, x₀;f) \), with \ (sₙ=f (Aⁿ x₀) \), and companion matrices as the canonical systems attached to constant-coefficient recurrences. The second-order Lucas bridge becomes a special case of a general cyclic-subspace theorem: over a finite field, the state period of \ (x₀\) is the order of the transition operator restricted to the cyclic subspace generated by \ (x₀\). This yields the usual identity \[₏, ₐ (p) = (pmatrixP companion matrix; affine weights; modular periodicity; weighted sums; reduced period; prime-power lifting; synchronization; matrix–sequence theory; representation theory; period collapse --- **MSC2020: ** 11B39, 11B50, 11T06, 37B15