Energy-Based and Finite-State Analysis of Collatz Dynamics

This paper develops a finite-state framework for analyzing the accelerated Collatz map on odd integers using residue classes modulo 32, valuation labels, and a coarse transition graph. The approach emphasizes explicit finite combinatorial structure and energy-based drift bookkeeping. We prove that the coarse residue-level drift envelope does not enforce strictly negative drift on all return cycles by explicitly enumerating admissible return paths of bounded length and identifying a sharp obstruction. This converts a global dynamical question into a finite combinatorial problem. A Markov-type surrogate is introduced as a computational device for summarizing finite-state transition structure, without asserting probabilistic modeling of true Collatz orbits. All results are finite, explicit, and computationally verifiable. The work does not claim a resolution of the Collatz conjecture.