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A Proof of the Collatz Conjecture via Finite Fractal Structure and the Extremum Principle | Synapse

March 16, 2026Open Access

A Proof of the Collatz Conjecture via Finite Fractal Structure and the Extremum Principle

Key Points

The aim is to provide a proof of the Collatz conjecture using a specific mathematical framework.
Utilized F-Theory axiomatic framework
Recognized Collatz sequence as a finite fractal
Applied Extremum Principle (A1) to the framework
Verified two component theorems using Lean 4 with Mathlib
Demonstrated that assuming non-convergence contradicts the Extremum Principle
Confirmed the proof of the Collatz conjecture is complete

Abstract

The Collatz conjecture (1937) is proven within the F-Theory axiomatic framework. The Collatz sequence is recognized as a finite fractal structure: the same rule σ applied repeatedly within the space of positive integers. By the Extremum Principle (A1), the extremum of this space is 1. Assuming non-convergence leads to contradiction with A1. Two component theorems are formally verified in Lean 4 with Mathlib (CI: green). The proof is complete.

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Cite This Study

Takeo Yamamoto (Thu,) studied this question.

synapsesocial.com/papers/69b79e398166e15b153ab474 https://doi.org/https://doi.org/10.5281/zenodo.19015856

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