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Three timed ternary changes steer binary lattice growth to denser states
The Binary Crisis Clock: Controlled by Sparse Ternary Interventions
Sparse interventions reset the crisis clock in automata, reducing fragmentation without altering core binary rules.
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We investigate modular Laplacian automata on triangular lattices with evolution governed by binary and ternary moduli. Extending previous studies on square lattices, we examine how lattice geometry influences long-term growth, density, fragmentation, and the emergence of self-similar structures. We further investigate whether sparse ternary interventions can stabilize predominantly binary dynamics.
The experiments reveal that mask geometry is the primary determinant of large-scale morphology. Full hexagonal masks generate recurrent density crises and fragmentation, whereas triangular masks support persistent growth and reveal a threshold phenomenon governed by growth-capable nuclei. Although seed symmetry influences transient behaviour, the asymptotic morphology is inherited mainly from the mask.
To control binary fragmentation, we investigate sparse developmental ternary perturbations in which a small number of carefully timed occurrences of modulus 3 are inserted into an otherwise binary sequence. A Monte Carlo optimization demonstrates that as few as three interventions are sufficient to redirect the subsequent binary evolution toward substantially denser carpet-like configurations. The effectiveness of this strategy depends primarily on the timing of the interventions rather than on their number.
Analysis of the post-intervention dynamics shows that ternary shaping does not replace binary evolution. Instead, it produces denser self-similar structures, substantially reduces crisis depth, and resets the phase of the binary crisis clock. The results suggest that geometry determines the family of admissible morphologies, whereas sparse developmental perturbations select favourable long-term trajectories within that family.
