III. Arithmetic obstruction to indefinite survival in the 2-adic Collatz dynamics

This paper continues the investigation into the 2-adic structure of the Collatz dynamics, specifically focusing on the arithmetic possibility of indefinite survival for positive integers. Building on previous results that established the measure of survival sets to be zero, this work proves that no positive integer can stay indefinitely within the survival classes.

The author introduces the residual parameter as a discrete Lyapunov function and demonstrates its strict descent under the condition of survival. The argument shows that any such orbit must reach a base case in finite time, leading to eventual convergence to the number one. Additionally, the paper establishes the absence of non-trivial cycles within the specific modular class C1, effectively ruling out purely internal cycles in the compressed dynamics. The approach is strictly arithmetic, avoiding probabilistic models or circular reasoning regarding the bounding of exponents.

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