COL

We formalize a phase-refinement model for admissible inverse odd-to-odd trajectories generated by dyadic lifts and triadic reduction. The central object is a deterministic word router that assigns to each starting odd integer a finite lift-word of length j, together with a triadic phase label at each depth. The router is assumed periodic at depth j modulo M_j = 2 · 3^j+1. We introduce a refinement obstruction: to maintain a fixed lift-word step-by-step, the phase alignment is generically 1/3 out of phase after each step and must be repaired by passing to the next refined modulus. This forces an infinite, non-stabilizing refinement chain for any hypothetical cycle-word. Within this axiomatic system, we prove that no cycle-word is realizable by any finite starting integer.