IV. Arithmetic obstruction to mixed orbits in the 2-adic Collatz dynamics

This paper concludes a series on the 2-adic structure of the compressed Collatz dynamics. Previous works showed that no positive integer has an orbit remaining indefinitely in the class of odd integers with 2-adic valuation at least 2, and ruled out non-trivial cycles in both that class and the class with valuation 1. The open scenario was orbits visiting both classes arbitrarily, without cycling or converging to 1.

We rule out this scenario via a global 2-adic budget argument: excursions in both classes consume non-regenerating bits from the initial parameter. Each excursion consumes bits equal to its length or budget, with disjoint blocks ensured by the cylindrical structure and local affine relations. An infinite orbit would demand unbounded bits, contradicting the finiteness of the parameter. This proves all orbits starting in the valuation-2 class converge to 1. The analysis is purely arithmetic, building on prior results without new proofs.

Version 1.1-1.2: Minor corrections and editorial revisions.

This paper is part of a series of six works on the Collatz conjecture. In reading order:

I. 2-adic structure of tails and survival sets in Collatz dynamics https://doi.org/10.5281/zenodo.18831439

II. Cylinder collision, bit non-reusage, and effective non-degeneration in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831527

III. Arithmetic obstruction to indefinite survival in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831690

IV. Arithmetic obstruction to mixed orbits in 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18831791

V. The ϕ function and the extension of the 2-adic budget argument to arbitrary k0 in Collatz dynamics https://doi.org/10.5281/zenodo.18831874

VI. Structural reduction of the Collatz conjecture: stretches, portals, and 2-adic survival sets https://doi.org/10.5281/zenodo.18831607

VII. Structure of entries to C1 and the rigid regime https://doi.org/10.5281/zenodo.18879276

VIII. Return map, rigid regime, and invariance gap in the 2-adic Collatz dynamics https://doi.org/10.5281/zenodo.18879361