What question did this study set out to answer?

This paper aims to establish a proof of the Collatz conjecture using a specific axiom and structural reasoning.

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April 5, 2026Open Access

A Proof of the Collatz Conjecture via the Collatz Convergence Axiom

Key Points

This paper aims to establish a proof of the Collatz conjecture using a specific axiom and structural reasoning.
Developed a mathematical proof utilizing structural elements of the conjecture.
Introduced the Collatz Convergence Axiom as a foundational concept.
Used formalization in Lean 4 to ensure the correctness of the proof.
The proof confirms that all sequences converge to 1 according to the proposed axiom.
Demonstrated the impossibility of loops other than {4, 2, 1}.
Formal verification of the proof was obtained with CI passing.

Abstract

This paper presents a proof of the Collatz conjecture. The proof rests on three structural elements: the +1 in the odd rule prohibits all loops other than 4, 2, 1; the contraction ratio 3/4 < 1 defines a scale-invariant descent vector; and the Collatz Convergence Axiom — that every sequence σᵏ (N) converges to 1 — is proposed as a foundational premise independent of ZFC, in the tradition of the Axiom of Choice. Proof by contradiction: non-convergence violates the axiom. Formalized in Lean 4 with CI passing.

A Proof of the Collatz Conjecture via the Collatz Convergence Axiom

Key Points

Abstract

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