Canonical Chain–Node Encoding and Finite-Width Return Renormalization for the Accelerated Collatz Map

Abstract

We select a canonical orientation of an alternating signed-binary representation of positive odd integers and refine it into a chain--node encoding, called the canonical CPE. The signed representation itself is classical in spirit and closely matches the alternating binary system of Hajnal; the present contribution begins with its Collatz-adapted canonical selection and CPE dynamics. The lowest chain length is exactly the $2$-adic valuation removed by the accelerated Collatz map. A zero-node statistic $H_K$ gives an exact finite-descent reformulation of the positive Collatz conjecture and reduces any hypothetical counterexample to a barrier orbit. Barrier orbits split into an infinite-width-growth branch and a fixed-width neutral-cycle branch. For the latter we derive a sequence of exact affine return systems, correct the valuation-cylinder conditions at every intermediate step, and identify a signed normalization current with the boundary of the ordinary binary carry sequence. At sufficiently large fixed width, an induced six-candidate return family clips to two active affine branches. After normalization to $[0,1)$, the induced dynamics closes in three coordinates $(a,b,\theta)$. A balanced manifold has a common branch fixed point and exact coincidence of interval and slope thresholds. The finite-width endpoint defect has affine inverse transport; this yields exact width thresholds for successive return depths. We prove that these thresholds increase and diverge, so no finite width supports the same terminal-two continued-fraction real-domain return skeleton at every induction depth. This is a finite-depth structural obstruction, not a proof of the Collatz conjecture.