What question did this study set out to answer?

The aim is to explore how macroscopic laws emerge from a minimal stochastic substrate using defined interactions.

March 4, 2026Open Access

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

Key Points

The aim is to explore how macroscopic laws emerge from a minimal stochastic substrate using defined interactions.
Analyzed a stochastic substrate defined on a finite state space with a time-dependent micro kernel.
Specified an enable matrix for six primitives to create directed interactions.
Constructed a macro kernel and empirical endomap for varying timescales and coarse-graining lenses.
Performed extensive simulations across various micro sizes with different ablation and scaling suites.
Found resolution-dependent signatures consistent with geometry and thermodynamics.
Identified behaviors like least-action dominance and diffusion-consistent embeddings.
Observed non-equilibrium time-asymmetry across the simulations.

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 = PX_ (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. Keywords: Emergence; Coarse-graining; Markov; Thermodynamics; Irreversibility; Spectral; Diffusion; Six Birds Theory; Emergence Calculus

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

Key Points

Abstract

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