Finite-State Weight Sources for Lucas Sums: Unified Periodicity, Closure, and Collapse Mechanisms

This paper gives a theorem-safe finite-state unification of the modular-period theory for weighted Lucas sums developed earlier for multiplicative, affine, and polynomial weights. The new framework replaces the special algebraic source kⁿ⁻¹ by an arbitrary autonomous finite-state Moore source. We show that the correct level of generality is not "every deterministic finite-state source" in isolation, but rather the periodic orbit generated by the chosen initial state. Once that orbit is finite and cyclic, the weighted Lucas sums modulo an odd prime remain purely periodic and satisfy the same universal upper bound τ p lcm(π(p), λₚ), where π(p) is the Lucas period modulo p and λₚ is the least positive period of the reduced weight sequence modulo p. The proof uses only a bijective finite-state model on the effective orbit and is independent of the special algebraic form of the source. We then identify the exact collapse criterion in terms of the cyclic reduced output word on the effective orbit, prove that injective reduction is sufficient for the absence of collapse but not logically necessary, and construct explicit cyclic Moore families with λₚ = 1 and arbitrarily large ambient orbit length. This implies that no universal lower bound depending only on π(p) and the ambient orbit size can hold in the full finite-state category. Finally, we prove closure under synchronized product constructions and formulate theorem-safe algorithms for computing the effective orbit, the reduced period, and the resulting period bound. Every object is defined before use, and the paper is written to avoid circular terminology, hidden synchronization assumptions, and unjustified identification of distinct period scales.