This paper adds a probability-theoretic layer to the deterministic finite-state modular‑period theory for pseudorandomly weighted Lucas sums. The deterministic theory shows that, for a cyclic finite‑state source with reduced output period ₚ, the least prime‑level weighted‑sum period satisfies p\, lcm ( (p), ₚ). The present manuscript asks which reduced periods are typical when the output labels are random. Three randomization paradigms are isolated: (A) uniform random output labelling on a fixed cycle, (B) random initial phase on a deterministic cycle, and (C) small random reset perturbations of the deterministic transition. For Paradigm A, exact formulas for the distribution of the least cyclic period are proved by Möbius inversion, sharp union bounds for collapse events are established, and exponential rarity of any fixed proportional collapse is shown. In particular, for fixed p the probability of being primitive tends to 1 as, and for fixed it also tends to 1 as p. These results provide a rigorous probabilistic explanation of the deterministic picture: collapse families are essential obstructions to universal lower bounds, but they occupy an exponentially small part of the natural random labelling space. Paradigm B shows that random initial phase does not change the reduced period. For Paradigm C, an -random reset perturbation of a deterministic cycle is proved to be an irreducible aperiodic finite Markov chain with uniform stationary distribution; exact transition formulas and quantitative estimates showing convergence of finite‑window laws to the deterministic cycle as 0 are given. Spectral and central‑limit statements for random increments are formulated only as conditional theorem‑safe propositions or open problems; unsupported Gaussian or large‑deviation assertions are never used as inputs to any proof.
Jianming Wang (Thu,) studied this question.