What question did this study set out to answer?

This research aims to explore the dynamics of the Collatz Conjecture using a structural representation of natural numbers.

March 25, 2026Open Access

Structural Proof of the Collatz Conjecture

Key Points

This research aims to explore the dynamics of the Collatz Conjecture using a structural representation of natural numbers.
Developed a unique structural representation of natural numbers
Established the Collatz transformation framework
Created an algebraic-cyclic formulation for closed orbits
Utilized global closing equations and structural rigidity results
Proved the nonexistence of nontrivial cycles in the Collatz Conjecture structure
Demonstrated foundational aspects of the Collatz dynamics through structural analysis

Abstract

This work is part of a two-part research program on the dynamics of the Collatz Conjecture, based on a unique structural representation of the natural numbers of the form \ (X = 2^n (2r + 1) - 1\). In this Part I, the structural framework of the Collatz transformation is established and an algebraic-cyclic formulation of closed orbits is developed. From global closing equations and structural rigidity results, the nonexistence of nontrivial cycles is proved. Part II will address the complementary problem of global convergence to the trivial cycle \ (\1, 4, 2\\).

Structural Proof of the Collatz Conjecture

Key Points

Abstract

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