From a conjecture of Collatz to Thompson's group F, via a conjunction of Girard

TL;DR

This paper explores a novel connection between the Original Collatz Conjecture, Girard's Geometry of Interaction, and Thompson's group F, revealing new algebraic and categorical insights into their relationships.

Contribution

It introduces a new framework linking the OCC operator with Thompson's group F through Girard's conjunction, supported by category theory analysis.

Findings

Realization of Thompson's group F as congruential functions

Categorical interpretation of the core operator as a coherence isomorphism

New algebraic perspective on the OCC operator

Abstract

The famous 3x + 1 problem of L. Collatz needs no introduction; however, this paper concerns a lesser-known, but similarly unresolved, precursor problem : the Original Collatz Conjecture, or OCC. We demonstrate that the core arithmetic operator from the OCC, when combined with a conjunction of J.-Y. Girard from his Geometry of Interaction system, leads to a realisation of R. Thompson's group F as congruential functions, in the sense of J. Conway. We also give the underlying category theory that accounts for this, and describe the core operator from the OCC as a canonical coherence isomorphism.