Prime-Power Periodicity and Lifting for Multiplicatively Weighted Lucas Sums

Abstract We study modular periods of partial sums obtained by multiplying a Lucas sequence by an independent multiplicative weight source, Sₘ=₍=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 =ₜ (k) of the weight source may collapse after reduction modulo p. The relevant increment cycle is therefore governed by the reduced period ₚ, not only by. For odd primes p Q, writing (p) for the first-kind Lucas state period, we build an explicit finite-state model and prove the block-discrepancy bound p\, ( (p), ₚ), with the sharper divisor ( (p), ₚ) when the common-cycle discrepancy vanishes. In the coprime non-collapsing zero-discrepancy regime, we characterize the extremal equality = (p) 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 ₑ+₁=ᵣ or ₑ+₁=pᵣ. 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**