On the Infinitude of Synchronization Clusters in the Collatz Iteration

【Abstract.】 This paper establishes an algebraic framework based on the tillering function to rigorously prove that the length k of synchronization clusters (k-clusters) in the Collatz iteration is unbounded. The tillering function f(x) = 4x + 1 generates tillering sequences exhibiting strict modular periodicity (period 0 \to 1 \to 2 modulo 3) and the structural characteristic of congruence to 5 modulo 8. Building upon these structural properties, we present a formal proof that the probability of a 2-cluster is exactly 2/3, and derive therefrom that since the tillering function can be iterated infinitely for any seed odd number, the cluster length k has no upper bound. This theoretical prediction is in remarkable agreement with computational findings (e.g., the observation of an 1.8 \times 10^7-cluster near 10^{123}), demonstrating that the unbounded growth of synchronization clusters is an inevitable consequence of algebraic structure rather than a statistical artifact.