A single Idea seed placed at the center of a 4-dimensional stochastic cellular automaton — two temporal layers coupled to a spatial grid — evolves through five semantic states: Idea, Concept, System, Flaw, and Dead. The update rule has three components: a cardinal-neighbor condition governing spatial propagation, stochastic decay with probability 0.3, and temporal coupling between layers via BitOr injection. No cross-shaped template is imposed. The cruciform pattern emerges from the interaction of cardinal propagation geometry with the lifecycle transition rules.
This post implements the automaton defined in Mathematical Belief I (Sprenger, 2025), translating the original Python code into Mathematica with array-based state masking via Unitize. The simulation runs on a 31×31×2×2 grid for 150 steps and produces: (1) the emergent cross in all four temporal layers, (2) chromatic composition evolution over time, (3) Blue component tracking across the full simulation, (4) attractor analysis of late-time state fractions, (5) a decay-rate sensitivity sweep demonstrating that cross shape persists while chromatic balance varies, (6) breach analysis, and (7) the lifecycle path rendered on the {0,1}³ cube.
The cross is the first of three emergent geometries. Post 7 extends the same lifecycle to 3D, producing cubic symmetry. Post 10 adds axial coupling, producing a tree. The operator is the same throughout. Only the dimensionality changes.
Open Questions:
The cruciform symmetry emerges from the clean-tip propagation rule: exactly 1 cardinal System neighbor, 0 other active cardinals. Is this the minimal local rule producing stable cross-symmetry on a square lattice?
The four temporal layers develop distinct phase relationships through BitOr coupling. Does the system exhibit computational irreducibility?
The lifecycle visits 5 of 8 vertices of {0,1}³. What role would Cyan (0,1,1) play if introduced as a seventh lifecycle state?
Under what propagation rules does breach become inevitable? Is there a critical decay probability above which the cross destabilizes?
Notebook Attached.
https://www.wolframcloud.com/obj/58ccf07e-69a8-4d84-8864-0de8dccd2000
Attachments:
