Inverse-Coupling and Overconstrained Periodicity in the Collatz Dynamics

This paper is a direct continuation of the trace-compressed normal-form framework introduced in Collatz Normal Form: Time as Degree-of-Freedom Elimination and the Trace-Compressed Engine. The goal of the present work is not to establish convergence of Collatz trajectories, but to clarify the distinction between inverse-graph closure and genuine forward periodicity. Using the trace-compressed normal form, the paper derives an exact cocycle compatibility condition that every periodic orbit of the accelerated Collatz map must satisfy: 2K = 3E C, where (E) is the number of odd steps, (K) is the accumulated dyadic valuation, and (C) is a multiplicative correction cocycle generated by the additive (+1) term. A central contribution is the observation that inverse-coupling constructions, although capable of producing admissible closures in the inverse preimage graph, do not in general imply periodicity of the forward Collatz dynamics. The paper therefore separates two notions that are often conflated: 1. combinatorial closure in the inverse graph;2. arithmetic closure of the forward dynamics. Within this framework, periodicity appears as a highly overconstrained compatibility problem involving valuation sums, odd-step counts, and cocycle corrections. The work does not claim a proof of the Collatz conjecture and does not establish nonexistence of nontrivial cycles. Instead, it identifies an exact arithmetic bottleneck that any hypothetical cycle must satisfy, thereby providing a structural explanation for why inverse-construction approaches do not freely generate periodic orbits. This upload is intended as a citable structural reference within the broader trace-compressed normal-form program. This work extends the trace-compressed normal-form framework introduced in "Collatz Normal Form: Time as Degree-of-Freedom Elimination and the Trace-Compressed Engine" (DOI: 10. 5281/zenodo. 18233316).