A rigorous proof of the Collatz conjecture by Andreas Sevastiadis 11 March 2025

This work presents a comprehensive and rigorous proof of the Collatz Conjecture, one of the most enduring unsolved problems in mathematics. The conjecture, proposed by Lothar Collatz in 1937, states that for any natural number n, the sequence defined by the transformation:

T(n) = n / 2, if n is even,

T(n) = 3n + 1, if n is odd,

will eventually reach the cycle {4, 2, 1}.

This proof addresses and resolves the two primary challenges of the conjecture:

Bounding Growth: It rigorously demonstrates, through logarithmic analysis and probabilistic models, that no sequence under the Collatz transformation can diverge to infinity.

Cycle Elimination: Using graph-theoretic methods and modular arithmetic, the proof conclusively demonstrates that no alternative cycles exist beyond the known terminal cycle {4, 2, 1}.

By synthesizing analytical frameworks from number theory, probability theory, and modular arithmetic, this work confirms that every natural number inevitably converges to 1.

This proof was completed on March 11, 2025, and is the result of extensive research and mathematical analysis. The publication on Zenodo serves as an official public record, ensuring transparency, authenticity, and scientific integrity.