A Classical Circular Method for the Accelerated Collatz Map

We study the accelerated odd Collatz map by keeping together three elementary structures: finite products along odd orbits, circular phases given by fractional logarithms, and the residue obstruction modulo 3. The note formulates a classical Eulerian reduction of non-principal obstructions to the existence of a self-sustaining branch preserving a circular mode while remaining compatible with the many-to-one structure of the accelerated map.