A Finite-State Symbolic Automaton Model for the Collatz Map and Its Convergence Properties

TL;DR

This paper introduces a finite-state automaton model for the Collatz map that captures its dynamics through symbolic digit transitions, demonstrating finite convergence to a unique cycle and providing a new structural perspective.

Contribution

The paper develops a novel finite-state automaton that models Collatz dynamics symbolically, revealing convergence properties and a canonical normal form of the map.

Findings

All trajectories converge to a unique cycle in finitely many steps.

The automaton's states encode digit and carry information for precise modeling.

A symbolic drift function and ranking potential explain the convergence process.

Abstract

We present a finite-state, deterministic automaton that emulates the Collatz function through digitwise transitions on base-10 representations. Each digit is represented as a symbolic triplet (r, p, c) encoding its value, the parity of the next digit, and an incoming carry propagated from the lower digit. This yields exactly 60 possible local states. The automaton applies local, parity-aware rules that collectively reconstruct the global arithmetic of the Collatz map. We show that all symbolic trajectories converge in finitely many steps to a unique terminal cycle (4, 0, 0) -> (2, 0, 0) -> (1, 0, 0), with all higher digit positions degenerating to the absorbing state (0, 0, 0). This collapse reveals a canonical symbolic normal form of Collatz dynamics. In parallel, a binary view explains the dynamics as alternating bit-length growth and contraction, aligning with known heuristics for Collatz convergence. This structural perspective is further reinforced by a symbolic drift function and a ranking potential that together explain and formalize the convergence process.