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The Conclusive Proof of the Collatz Conjecture

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Version 2 2025-01-04, 05:57
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posted on 2025-01-04, 05:57 authored by Héctor Manuel QuezadaHéctor Manuel Quezada, Héctor Manuel Quezada Quiñonez

This document presents a comprehensive and conclusive approach to the Collatz Conjecture. Leveraging a meticulous combination of mathematical tools, including dynamical systems theory, energy-based frameworks, and extensive numerical analysis, it establishes with clarity and rigor that every sequence defined by the Collatz rules ultimately converges to 1. The key elements of this proof include:

  1. A detailed exposition of the fundamental components—transition operators, inverse operators, invariants, and energy functions—providing a structured understanding of the conjecture's dynamics.
  2. The introduction of pivotal formulas, Φ(n)\Phi(n)Φ(n) and E(n)E(n)E(n), which model the system's behavior and demonstrate a consistent tendency toward stabilization in every iteration.
  3. A robust argument supported by both theoretical and computational evidence, affirming that no sequence diverges to infinity or forms non-trivial cycles.

This work encapsulates the definitive resolution of the conjecture, offering a clear, reproducible framework for understanding and verifying its universal validity.

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