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

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 \((V,A,x_0;f)\), with \(s_n=f(A^n x_0)\), 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 \(x_0\) is the order of the transition operator restricted to the cyclic subspace generated by \(x_0\). This yields the usual identity \[πP,Q(p)=({pmatrix}P&-Q\\1&0{pmatrix} p)\] for first-kind Lucas sequences, and extends it to arbitrary order, arbitrary initial state, and arbitrary observable. The matrix language produces new period theorems. Products of recurrence sequences are realized on tensor-product state systems, with diagonalizable matrix order \[(A⊗ B)=i,j(α_iβ_j).\] Polynomial composites are realized on truncated symmetric algebras, giving explicit finite-dimensional period bounds in terms of the eigenvalue monomials of symmetric powers. Algebraic pullbacks become pullbacks of observables, matrix polynomials become finite linear combinations of shifts, and compatible subquotients become invariant-subspace or quotient systems. The second layer reformulates weighted sums as additive cocycles over product state orbits. If \(\) is the least increment period and \(B_\) is the block drift, then the weighted-sum period is exactly \[τ=\,_p,+(B_).\] This recovers the affine-weight upper bound \[τ p\,(π(p),λp,aff)\] and places affine collapse and local lifting inside one finite-state language. At prime-power level the synchronization hypothesis is kept explicit: without it there is a uniform one-step upper bound, while under it a single additive defect gives the exact dichotomy \(τᵣ₊₁∈\{τ_r,pτ_r\}\). The third layer develops representation-theoretic sequence families: irreducible representations give state-indecomposable sequence families, the least common multiple of full-state periods is the exponent of the reduced image, and induced representations have an explicit coset-cycle period formula. --- Keywords Lucas sequence; 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