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Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
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II. Cylinder collision, bit non-reuse and effective non-degeneracy in the 2-adic Collatz dynamics

Authors: Miguel Cerdá Bennassar;

II. Cylinder collision, bit non-reuse and effective non-degeneracy in the 2-adic Collatz dynamics

Abstract

This paper proves the effective non-degeneracy conjecture formulated in the author’s previous study on the 2-adic structure of the compressed Collatz dynamics 10.5281/zenodo.18685581. It establishes that the restricted survival sets inside a fixed modular class decrease exactly by a factor of one half at each step. The proof relies on the precise cylindrical structure of the 2-adic valuation levels, the affine behavior of the dynamics within each branch, and a structural separation argument based on cumulative 2-adic depth. It is shown that distinct itineraries cannot collide at any finite resolution, ensuring injectivity of the correspondence between admissible trajectories and initial residues. As a consequence, the infinite survival set has Haar measure zero. The question of its emptiness, which is equivalent to the classical Collatz conjecture in the principal modular class, remains open. Version 1.1: Maintenance update including minor revisions and the resolution of issues identified in the previous version. 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

Keywords

survival sets, Collatz conjecture, valuation theory, 3n+1 problem, discrete dynamical systems, affine dynamics, modular structure, cylinder sets

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
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