Collatz convergence is a Hydra game

TL;DR

This paper introduces a novel automaton-based approach to analyze Collatz sequences, proving their convergence by modeling the problem as a Hydra game embedded in a specific number set, and offers new insights into its mathematical structure.

Contribution

It establishes an automaton framework for Collatz sequences using five arithmetic rules, demonstrating their convergence and embedding in 24N-7, and provides a new proof of the conjecture within ZFC.

Findings

Proves Collatz sequences converge using an automaton model.

Shows Collatz graph is embedded in 24N-7, supporting the conjecture.

Introduces a Hydra game analogy to understand Collatz dynamics.

Abstract

The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers which inflation propensity remains so unpredictable it could be used to generate reliable proof of work algorithms for the cryptocurrency industry. Here we establish an ad hoc equivalent of modular arithmetic for Collatz sequences to automatically demonstrate the convergence of infinite quivers of numbers, based on five arithmetic rules we prove apply on the entire Collatz dynamic and which we further simulate to gain insight on their graph geometry and computational properties. We then formally demonstrate these rules define an automaton that is playing a Hydra game on the graph of undecided numbers we also prove is embedded in 24N-7, proving that in ZFC the Collatz conjecture is true, before giving a promising direction to also prove it in Peano arithmetic.