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ZENODO
Preprint . 2026
License: CC BY
Data sources: ZENODO
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
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Modular Closure and Internal 2-adic Structure in Collatz Dynamics

Authors: Miguel Cerdá Bennassar;

Modular Closure and Internal 2-adic Structure in Collatz Dynamics

Abstract

We present a structural analysis of Collatz dynamics based on an early modular closure and on the study of the 2-adic valuation of odd terms. We prove that, after the first odd term, trajectories permanently avoid multiples of 3 and become confined to a rigid modular cycle modulo 9. Within this framework, the dynamics reduces to the analysis of the factors 3n+1 and their 2-adic valuation. A systematic decomposition of trajectories into odd segments and even tails is introduced, showing that the global structural organization of the orbit —though not its exact length— is determined from the initial value. The balance between episodes of odd growth and divisions by 2 is interpreted as a structural constraint of the dynamics, without recourse to probabilistic hypotheses or heuristic arguments.

Keywords

Modular closure, Discrete dynamics, Collatz conjecture, Even tails, Modular arithmetic, 2-adic valuation, Iterative dynamics, Odd segments

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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
Average
Average
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