To Create a Stone with Six Birds: Emergent Geometric and Thermodynamic Regimes from a Minimal Stochastic Substrate

Ioannis Tsiokos

Abstract:

We study emergence of macroscopic lawfulness in a minimal stochastic substrate defined on a finite state space Z with a time-dependent micro kernel P. A closure interaction algebra composed of six primitives is specified by an enable matrix E ∈ {0,1}^(6×6), yielding up to 36 directed actor←informant interactions that (i) rewrite and gate the micro kernel, (ii) adapt timescale τ, (iii) select coarse-graining lenses f:Z→X and packagings, and (iv) impose accounting constraints. For a given (τ, f) we construct an induced macro kernel P̂ = P^X_(τ,f) and the associated empirical endomap E_(τ,f), and we audit multiscale behavior along a resolution ladder k = |X|.

Across a manifest-defined campaign union spanning four micro sizes n ∈ {32, 64, 128, 256} (2307 runs total), combining an exhaustive ablation suite for n ≤ 128 with a computationally motivated scaling suite at n = 256, we find robust, resolution-dependent signatures reminiscent of geometry and stochastic thermodynamics, including least-action dominance, diffusion-consistent embeddings, and non-equilibrium time-asymmetry.

Beyond these substrate-specific results, the Primitive Interaction Closure Algebra (PICA) provides a reusable, implementation-independent object for specifying and comparing closure mechanisms in other emergent systems.