December 4, 2025Open Access

The Unified Theory of Collatz Dynamics: Geometric Dissipation and Rigorous Arithmetic Constraints

Key Points

Global convergence of trajectories was demonstrated through geometric dissipation and arithmetic constraints.
The framework addresses non-existence of infinite k=1 sequences and shows positive density of collapse events.
Analysis applies rigorous principles including Δₖ Automaton and Structural Decay Principle for deeper insight.
Significant implications for understanding the Collatz Conjecture may arise from this closed formal proof.

Abstract

This work presents a unified structural proof framework for the Collatz Conjecture,combining geometric intuition (Vacuum Funnel) with rigorous arithmetic constraints (Δₖ Automaton, Residue Mixing Lemma, Structural Decay Principle). The paper establishes:• non-existence of infinite k=1 sequences• positive density of collapse events (k ≥ 2)• strictly negative drift in logarithmic potential (εₖ > 0) These combined mechanisms force global convergence for all trajectories. This is the closed formal version used for the public release of“The Unified Theory of Collatz Dynamics (Closed Proof Attempt) — Part 5” posted on Reddit r/Collatz on December 4, 2025.

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Moon， Kyung-Up (Thu,) studied this question.

synapsesocial.com/papers/694025742d562116f28fddc8 — DOI: https://doi.org/10.5281/zenodo.17810874

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The Unified Theory of Collatz Dynamics: Geometric Dissipation and Rigorous Arithmetic Constraints

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