The Collatz Singularity as a Global Binary Structural Attractor

In our fifth work [3] entitled A Structural Approach to the Collatz Conjecture via the Binary Singularity, we introduced the notion of binary singularity of Collatz sequence as a structural invariant and proposed a structural reformulation of the Collatz conjecture: every positive odd integer either is, or eventually reaches, a singularity configuration under a finite number of iterations. We further hypothesized that this property could be characterized through an appropriate metric definedon binary structures.In the present work, we develop this idea by introducing a global binary structural attractor governing the evolution of odd Syracuse iterates. This attractor is based on the canonical binary decomposition of the binary structure of odd integers.We prove that every odd Syracuse iterate that is not a singularity admits a unique decomposition into a finite sequence of nested concatenation of suffixes of generators of the suffix of singularity followed by a single suffix of singularity, On = G(p,n)∥G(p−1,n)∥ · · · ∥G(1,n)∥Sn, and define a structural complexity function P(On) equal to the number of suffix generators contained in this decomposition.The evolution of P is entirely determined by the complementarity of the overflow generated by the most significant suffix generator G(p,n), leading to a complete structural classification of increasing, stationary, and decreasing transitions. This decomposition provides a symbolic description of the odd Syracuse dynamics and establishes the structural foundation for an eventual Lyapunov function whose minimal state is precisely the binary singularity P = 0. Rather than studying the arithmetic growth of Collatz trajectories, the proposed approach describes the dynamics as the progressive elim1 ination of nested binary suffixes of generators toward a unique global structural attractor.