Affine Weights, Sharp Local Lifting Criteria, and Synchronization Questions for Pseudorandomly Weighted Lucas Sums

This paper continues the modular-period theory of pseudorandomly weighted Lucas sums. We address three residual directions from that program. First, we extend the prime-modulus finite-state formalism from purely multiplicative weights an≡kn−1(modt)an≡kn−1(modt) to affine weights an≡αkn−1+β(modt)an≡αkn−1+β(modt). We prove a bijective state-space model, pure periodicity modulo odd primes, and the upper bound τ∣plcm⁡(π(p),λp,aff)τ∣plcm(π(p),λp,aff), where λp,affλp,aff is the reduced affine-weight period modulo pp. We also construct explicit affine collapse families for which the reduced affine period equals 11 while the true affine weight period is arbitrarily large, so the lower-bound nonexistence phenomenon persists beyond the purely multiplicative setting. Second, on the lifting side we isolate the exact one-step sharp-growth criterion for the local orders dr=ord⁡pr(k)dr=ordpr(k), namely ord⁡pr+1(k)=pord⁡pr(k)ordpr+1(k)=pordpr(k) if and only if kdr≢1(modpr+1)kdr≡1(modpr+1). We then show by explicit examples that no analogous universal sharp law can hold for the reduced weight periods λprλpr once one moves past the pp-primary depth νp(t)νp(t). Third, we reframe the multi-step synchronization problem by giving a clean factorwise criterion for increment synchronization and an automatic one-step propagation principle under separate divisibility hypotheses on the Lucas side and the weight side. We also explain why the full transitivity problem for synchronization remains genuinely open at the level of theorem-safe proofs. All theorem-level statements are proved in this manuscript, and each main result is accompanied by concrete numerical examples.