December 4, 2025Open Access

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

Key Points

Global convergence established through combined mechanisms of dynamic systems and arithmetic constraints.
The approach uses tools like the vacuum funnel and structural decay for inference on infinite sequences.
Theoretical analysis identifies key patterns in collapse events for sequences of k ranging from 1 to greater than 2.
Implications for understanding the Collatz conjecture highlight the importance of rigorous mathematical structures.

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/694025742d562116f28fddd3 — DOI: https://doi.org/10.5281/zenodo.17810875

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

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