Polynomially Weighted Lucas Sums, Seed–Filter Finite-State Models, and Modular Periodicity

This paper gives a theorem-safe polynomial extension of the modular-period theory for pseudorandomly weighted Lucas sums developed in the multiplicative and affine settings. We study weights of the form aₙ ≡ f(kⁿ⁻¹) t, where f(x) ∈ Z[x], t ≥ 2, and (k, t) = 1. The basic observation is that the finite-state mechanism is controlled not by the weight values themselves but by the multiplicative seed orbit sₙ ≡ kⁿ⁻¹ t. The polynomial weight sequence is then obtained by applying the filter f to this cyclic seed orbit. This seed–filter viewpoint keeps the state space on the cyclic subgroup generated by k modulo t, so the prime-level dynamics reduce to the same one-dimensional seed coordinate that appears in the multiplicative theory. We prove a bijective state-space model modulo odd primes, pure periodicity of the weighted sums modulo p, and the upper bound τ p lcm(π(p), λp,poly), where π(p) is the Lucas period modulo p and λp,poly is the reduced polynomial-weight period modulo p. We then construct explicit collapse families for which λp,poly = 1 while the ambient seed period ordₜ(k) is arbitrarily large, showing that the lower-bound nonexistence phenomenon persists for polynomial filters. For the monomial case f(x) = xᵐ, we prove the exact formula λpʳ,poly = ordpʳ(kᵐ) = ordpʳ(k) / (m, ordpʳ(k)) for every r ≥ 1. Finally, we isolate the general prime-power divisor relations that remain valid for arbitrary polynomial filters and explain precisely where exact lifting laws depend on the filter. The paper is written so that every theorem-level claim is defined before use, proved in full, and does not rely on circular terminology.