Transient Geodesic Alignment in Life-Like Cellular Automata on the {7,3} Heptagrid: Empirical Findings and a Structural Mechanism

We report an empirical investigation of two-state totalistic cellular automata on the {7,3}heptagrid — the regular tiling of the hyperbolic plane by heptagons meeting three to avertex — implemented in the Poincaré disk model with a 7-neighbor topology. While priorwork on hyperbolic cellular automata has focused predominantly on questions ofcomputational universality, the dynamical behavior of Life-like birth-survival rule families onthis tiling remains largely uncharacterized. We introduce a geodesic alignment metric defined as the mean angular distance between alive-cell positions and the nearest of sevenprimary geodesic directions through the disk center, with a random baseline of 12.857°derived analytically. Surveying eight Life-like rule families across random initial conditionson grids of depth 4 (190 cells) and depth 5 (519 cells), we find that under rule B2/S23, alltwenty tested random initial conditions at both grid sizes produce minimum alignmentscores below the random baseline: mean minimum 7.998° (37.8% below baseline) atdepth 4, and 9.516° (26.0% below baseline) at depth 5. The effect is transient, appearingduring the early growth phase and relaxing toward baseline as grid occupancy increases.We derive a structural mechanism from two properties of the {7,3} neighbor graph —concentration of geodesic adjacency along geodesic directions (7.5–11.6× ratio at depth 4)and differential shared-neighbor density between on- and off-geodesic cell pairs — bothrule-independent properties of the tiling that intensify at larger scales. We also retract aninitially observed period-4 oscillator under B34/S234, which does not persist at depth 5 andis a finite-grid artifact. The interactive computational explorer and all verification code arereleased as open-source artifacts.