Prime-Modulus Periodicity of Multiplicatively Weighted Lucas Sums

We study the modular periodicity of partial sums of Lucas sequences weighted by a multiplicative pseudorandom sequence. Specifically, for an odd prime p and integers $P, Q$ with p Q, let Uₙ(P,Q) be the Lucas sequence of the first kind and let the weights aₙ be generated by the multiplicative rule aₙ ≡ kⁿ⁻¹ t, where t ≥ 2 and (k, t) = 1. We consider the weighted partial sums Sₘ = ∑ₙ₌₁ᵐ (aₙ p) Uₙ p. First, we construct a finite-state dynamical system and prove that the sequence ₘ\ is purely periodic. Second, by analyzing the drift over a common period of the increment sequence, we show that the least period τ divides p · lcm(π(p), λ), where π(p) is the period of Uₙ modulo p and λ = ordₜ(k). Moreover, if the drift vanishes, we obtain the sharper bound τ lcm(π(p), λ). Third, under the additional assumptions (π(p), λ) = 1, vanishing drift, p π(p)λ, and non-collapsing of the reduced weight period, we establish a spectral criterion: τ attains the maximal possible value π(p)λ if and only if the greatest common divisor of the Fourier support of the increment sequence together with \π(p)λ\ equals $1$. The same framework extends immediately to Lucas sequences of the second kind. The results provide a rigorous prime-level foundation for the period analysis of weighted Lucas sums.