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Revealing a Singularity in Collatz Sequences

Authors: HAMDOUS, Ammar;

Revealing a Singularity in Collatz Sequences

Abstract

Understanding chaotic dynamical systems often begins with an effort to impose order — not to oversimplify, but to illuminate the hidden structures that chaos may conceal. The Collatz sequence, long considered erratic and unpredictable, is no exception. In this work, we construct a more organized and representative tree structure of the Collatz sequences. This restructured tree is not merely a visual aid; it is a strategic reformulation of the problem that allows underlying patterns to emerge more clearly. The act of ordering brings forth what appears to be a singularity; a critical and recurring structure within the dynamics of the Collatz process that has gone unnoticed in prior formulations. This singularity transforms our understanding of how the sequence behaves. Rather than relying on brute-force computation or stochastic models, this approach highlights deterministic structures encoded within the sequence itself.

Keywords

Mathematical singularity, Recursive sequences, Hidden Order method, Collatz Singularity, Discrete dynamics, Hailstone sequence, Structural patterns, Collatz tree, Number theory, Binary representation, Collatz conjecture, 3n+1 problem, Inverse Collatz, Dynamical systems, Mathematical logic, Syngularity, Conjecture resolution, Syracuse conjecture

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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
Green