The Self-Replication Phase Diagram: Mapping Where Life Becomes Possible in Cellular Automata Rule Space

TL;DR

This study exhaustively maps cellular automata rules to identify conditions supporting self-replication, revealing key features like background stability and mass conservation that define the self-replication phase boundary.

Contribution

It provides the first comprehensive phase diagram of self-replication in cellular automata rule space, highlighting the roles of background stability and mass conservation.

Findings

7.69% of rules support pattern proliferation.

Self-replicating rules are more mass-conserving.

Self-replication rate increases with neighborhood size.

Abstract

What substrate features allow life? We exhaustively classify all 262,144 outer-totalistic binary cellular automata rules with Moore neighbourhood for self-replication and produce phase diagrams in the $(\lambda, F)$ plane, where $\lambda$ is Langton's rule density and $F$ is a background-stability parameter. Of these rules, 20,152 (7.69%) support pattern proliferation, concentrated at low rule density ($\lambda \approx 0.15$--$0.25$) and low-to-moderate background stability ($F \approx 0.2$--$0.3$), in the weakly supercritical regime (Derrida coefficient $\mu = 1.81$ for replicators vs. $1.39$ for non-replicators). Self-replicating rules are more approximately mass-conserving (mass-balance 0.21 vs. 0.34), and this generalises to $k{=}3$ Moore rules. A three-tier detection hierarchy (pattern proliferation, extended-length confirmation, and causal perturbation) yields an estimated 1.56% causal self-replication rate. Self-replication rate increases monotonically with neighbourhood size under equalised detection: von Neumann 4.79%, Moore 7.69%, extended Moore 16.69%. These results identify background stability and approximate mass conservation as the primary axes of the self-replication phase boundary.