Resolving the Collatz Problem via a Finite-State Phase Machine Structural Absorption, Core Phases, and Exhaustive Bounded Delay

This paper presents a complete structural resolution of the odd-only Collatz problem via an explicit finite-state reformulation. The Collatz dynamics are reconstructed as a deterministic finite-state phase machine by compressing odd integers into phase states modulo a fixed power of two. Within this framework, global behavior is reduced from an infinite iterative process to a finite reachability problem on a fully enumerable transition graph. In contrast to the previous version (v0.5, DOI: https://doi.org/10.5281/zenodo.18446986), the present release provides an exhaustive certification of global reachability and bounded absorption. All states of the finite-state machine are shown to reach a small absorbing core consisting of the phases {1, 5, 17}, with a uniformly bounded delay. The maximal delay is explicitly determined, and its uniqueness is certified by reverse BFS over the entire state space. Moreover, the absorption structure is proven to be stable under resolution lifting: increasing the modulus refines the state space without introducing new cycles or escape paths, and instead collapses the maximal delay. A formally definable exceptional region is shown to be dynamically isolated and unreachable from all admissible initial conditions. This version consolidates all structural arguments, removes provisional statements, and separates observational verification from formal theorems. The Collatz problem is resolved at the structural level, not through probabilistic or asymptotic reasoning, but by explicit finite-state construction and exhaustive reachability analysis. 2026/02/15 v1.1 uploaded 2026/02/16 v1.5 uploaded https://doi.org/10.5281/zenodo.18661332