Nonexistence of Universal Lower Bounds for Multiplicatively Weighted Lucas-Sum Periods Modulo Primes

We study lower bounds for the least period of multiplicatively weighted Lucas partial sums modulo an odd prime. In the companion paper, a general upper-bound theory was established for weighted sums

$Sm=∑n=1m(an mod p)Un(P,Q)in Fp,Sm=∑n=1m(anmodp)Un(P,Q)in Fp$,

where $Un(P,Q)Un(P,Q)$ is the Lucas sequence of the first kind and $an≡kn−1(modt)an≡kn−1(modt)$. That theory naturally raises the question whether the least period $\tau$ admits any nontrivial lower bound in terms of the Lucas period $$\pi (p)\$$ and the ambient weight period $λ=ord⁡t(k)λ=ordt(k)$. We prove that, in complete generality, the answer is negative. More precisely, we construct an infinite family of parameters for which

$τlcm⁡(π(p),λ)lcm(π(p),λ)τ$

can be made arbitrarily small. Equivalently, for every $N\ge 1$ there exist admissible parameters such that

$τ<1Nlcm⁡(π(p),λ).τ<N1lcm(π(p),λ)$.

Thus no nontrivial universal lower bound depending only on $$\pi (p)\$$ and $\lambda$ can hold. The mechanism is explicit: we force the reduced weight period $λpλp$ to collapse to $1$ while keeping the ambient period $\lambda$ arbitrarily large, and then exploit vanishing drift on the shorter increment block. This shows that the upper-bound theory from the companion paper is genuinely one-sided: the ambient period $\lambda$ can be almost invisible from the viewpoint of the modular least period.