Fig. 3.

Pattern dynamics for three kinds of random sequences in which the intervals between consecutive points are distributed 1) exponentially with the rate ν = 2; 2) uniformly with constant density ρ = 1; or 3) with Poisson rate μ = 5. Sample intervals are selected proportionally to the distribution scales (L_u = 25ρ, L_e = 25ν, and L_p = 25μ, so that each sample sequence contains about n = 25 elements) and are shifted by a single data point at a time. (A) The Kolmogorov parameter of the exponential sequence (red trace, λ_e), uniform sequence (blue trace, λ_u), and Poisson sequence (orange trace, λ_p) remains mostly within the “pink zone” of stochastic typicality (pink stripe is the same as on Fig. 2B, but stretched horizontally—note the illustration in the right corner). λ_u is the most volatile and often escapes the expected range, whereas λ_p is more compliant, lingering below the expected mean λ_p ≲ λ^* ≈ 0.87 (black dashed line). (B) The corresponding Arnold stochasticity parameters show similar behavior: β_u = 1.93 ± 0.2 fluctuates around the expected mean β^*(25) = 1.92 (black dotted line). The exponential sequence has smaller β-variations and a slightly higher mean, β_e = 2 ± 0.04. The Poisson sequence is the least stochastic (nearly periodic), with β_p = 1.22 ± 0.004, due to statistical suppression of small and large gaps. (C) The mean stochasticity scores, ⟨λ⟩ and ⟨β⟩ computed for about 10⁴ random patterns of each type. For sample patterns, (SI Appendix, Fig. 1A).