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

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 $(λ, F)$ plane, where $λ$ 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 ($λ\approx 0.15$--$0.25$) and low-to-moderate background stability ($F \approx 0.2$--$0.3$), in the weakly supercritical regime (Derrida coefficient $μ= 1.81$ 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.

Figures (12)

• Figure 1: Worked example of the three-tier detection hierarchy using HighLife (B36/S23). (a) Tier 1: component count $n_c$ increases from 1 to 10 over 112 steps, which triggers the pattern-proliferation criterion. (b) Tier 2: the same pattern tested at extended length (160 steps); proliferation continues ($n_c = 13$), which confirms sustained replication. (c) Tier 3: a seed pattern is isolated on an empty grid and self-replicates ($n_c = 1 \to 8$); a single-cell deletion from the seed prevents replication ($n_c = 1$), which establishes causal fragility.
• Figure 2: Phase diagram for all 262,144 Life-like (outer-totalistic $k=2$ Moore) rules. Columns correspond to the 19 exact $\lambda = n/18$ values; $F$ is binned into 22 intervals (the 147 distinct $F$ values, which are weighted subset sums of binary rule-table entries, are too dense to resolve individually). White cells indicate structurally empty regions where no rule exists. (a) Rule density (log scale): the binomial concentration at $\lambda \approx 0.5$ reflects $\binom{18}{9} = 48{,}620$ rules. (b) Self-replication rate: the "island of life" at low $\lambda$ and low-to-moderate $F$ concentrates in the weakly supercritical regime identified by the Derrida analysis (Section \ref{['sec:derrida']}).
• Figure 3: Smoothed 3D surface of the self-replication rate over the $(\lambda, F)$ plane (Gaussian filter $\sigma=0.8$, cubic spline interpolation; colour encodes log rule density). The sharp peak at low $\lambda$ and low $F$ shows that life-supporting rules occupy a narrow island in parameter space. The Derrida analysis (Section \ref{['sec:derrida']}) places this peak in the weakly supercritical regime ($\mu \approx 1.4$–$1.8$), just above the edge of chaos. Note: both $\lambda$ and $F$ are discrete; the smooth surface is a visual aid, not a claim of continuity.
• Figure 4: Self-replication rate versus $\lambda$, overall (black) and conditioned on $F$ terciles (coloured). The peak shifts with $F$ level, which indicates that background stability adds discriminatory power beyond $\lambda$ alone.
• Figure 5: Self-replication rate versus $F$, overall (black) and conditioned on $\lambda$ terciles (coloured). Self-replication peaks at low-to-moderate $F$ ($\approx 0.2$--$0.4$) and drops at both extremes, consistent with the localised island in the phase diagram.