Syracuse Random Variables and the Periodic Points of Collatz-type maps

The paper’s Correspondence Theorem states that an odd integer ω is a periodic point of Hp(x)=x/2 (even), (px+1)/2 (odd) if and only if there exists t with ω = χp(B(t)) = χp(t) / (1 − p#1(t)/2λ(t)) = (∑k 2λ(t)−βk(t)−1 pk−1) / (2λ(t) − p#1(t)) . The paper derives the key affine relation hβ−1(t)(x) = (p#1(t)/2λ(t))x + χp(t) (eq. (24)), which immediately yields the displayed fraction when one sets x=ω and imposes periodicity hβ−1(t)(ω)=ω . For the converse, the paper shows that if χp(B(t)) is an integer, then hβ−1(t)(ω)=ω for ω = χp(B(t)), and proves a claim that such a fixed point of a composition implies ω is periodic for Hp . The candidate solution proves the same equivalence by (i) writing an explicit n-step affine expansion 2n xn = pm x0 + ∑k 2n−βk−1 pk−1 and solving (i⇒ii), and (ii) constructing an inverse-step chain z_{i+1}=R_{b_i}(z_i) and giving a divisibility-induction to ensure all zi are integers and form a cycle (ii⇒i). Both approaches hinge on the same parity-vector/affine identity, but the model provides a more explicit integrality/divisibility argument where the paper appeals to a structural claim. The oddness of ω in the paper is justified 2-adically (numerator and denominator are odd) , which is consistent with the model’s modular observation.