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July 8, 2026Open Access

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

Key Points

This work aims to complete the framework behind the modular-period theory of pseudorandomly weighted Lucas sums.
Developed a general theory of observable linear state systems with companion matrices.
Formulated additive cocycles over product state orbits to analyze weighted sums.
Introduced representation-theoretic sequence families to explore irreducible representations and their implications.
Established new period theorems using matrix language for recurrence sequences and state systems.
Demonstrated that the weighted-sum period equals the least increment period times the order of the block drift.
Derived explicit finite-dimensional period bounds for polynomial composites and provided formulas for coset-cycle periods.

Abstract

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

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