Simulation
Convergence Dynamics
How 512 realms evolve toward a stable Gaussian distribution
Rapid Convergence
From any initial distribution, the occupancy histogram approaches a Gaussian bell curve centered near planes 4-5. The convergence metric drops below 0.05 within ~1,000 ticks.
Markov Chain Analysis
The dynamics form a finite-state Markov chain on F29. XOR transformations plus random noise ensure irreducibility and aperiodicity. By the Perron-Frobenius theorem, this guarantees convergence to a unique stationary distribution.
Initial Condition Independence
Four starting configurations — clustered at 0, uniformly random, centered near 4-5, clustered at 9 — all converge to the identical Gaussian. The stationary distribution is unique and universal.