Prime-Power Periodicity and Lifting for Multiplicatively Weighted Lucas Sums

We study modular periods of partial sums obtained by multiplying a Lucas sequence by an independent multiplicative weight source,Sₘ=∑ₙ₌₁ᵐ aₙUₙ(P,Q), aₙ≡ k\,n-1 t. Classical work describes periods of Lucas and Fibonacci recurrences themselves, and many identities are known for weighted recurrence sums; the period of the accumulated weighted sum is a separate dynamical question. One recurring obstruction is that the ambient order λ=ₜ(k) of the weight source may collapse after reduction modulo p. The relevant increment cycle is therefore governed by the reduced period λₚ, not only by λ. For odd primes p Q, writing π(p) for the first-kind Lucas state period, we build an explicit finite-state model and prove the block-discrepancy boundτ p\,(π(p),λₚ),with the sharper divisor (π(p),λₚ) when the common-cycle discrepancy vanishes. In the coprime non-collapsing zero-discrepancy regime, we characterize the extremal equality τ=π(p)λ by finite Fourier support, equivalently by triviality of a cyclic annihilator, and relate this support to the split and nonsplit discriminant cases. We also give a second-kind analogue and prime-power lifting criteria: without synchronization there is a uniform one-step upper bound, while under a verifiable synchronization hypothesis a single block defect decides whether τᵣ₊₁=τᵣ or τᵣ₊₁=pτᵣ. Computations illustrate period collapse, nonzero discrepancy, extremal and non-extremal behavior, synchronized lifting, and synchronization failure. --- Keywords **Lucas sequence; weighted partial sums; modular periodicity; finite-state dynamics; discrete Fourier criterion; prime-power lifting framework** --- MSC2020 Classification **11B39, 11B50, 11T06, 37B15**