2 Setup

2.1 Configuration Space and Entropy Evolution

The system is defined by a binary configuration string S_t of fixed total length L. The initial state S₀ corresponds to the all-zero bitstring, establishing an initial boundary condition where Shannon entropy is minimised:

Consistent with Paper I’s rejection of any externally imposed clock or agent [meskanen2001], the evolution is not driven by an external ”bit-flipper.” At each raw tick, one of the L positions is selected uniformly at random — a choice determined entirely by internal, countable facts about the current configuration (how many positions currently hold each value), not by any external rule — and that position’s value is toggled. Tracking k(t_raw), the number of 1-bits in S_t after t_raw raw ticks, this process is exactly the Ehrenfest urn model [ehrenfest1907]:

Writing p(t_raw) = k(t_raw)∕L and the rescaled time τ = t_raw∕L, the mean trajectory obeys dp∕dτ = 1 − 2p, with exact solution

a closed-form, L-independent relaxation curve from p = 0 toward the equilibrium value p = 1∕2. Note that τ is a rescaled variable, not a raw tick count: τ = O(1) corresponds to O(L) actual bit-flip events having occurred. This distinction matters throughout, and especially in Section 5.3.

Two distinct timescales govern this process and should not be conflated: the relaxation time L∕2 (in raw ticks), governing how fast the mean of k approaches equilibrium, and the sharp mixing (cutoff) time (L∕4)lnL, governing when the distribution of k becomes statistically indistinguishable from its equilibrium shape [diaconis1990hypercube]. We use the exact process above, not the mean-field curve alone, wherever the distinction matters.

2.2 Recursive Spatial Fabric and Structural Filtering

Level-0: The Spacetime Fabric

We define a minimal local target pattern string P of length w₀, with a ones and b = w₀ − a zeros. The global string S_t is parsed into non-overlapping blocks of size w₀, yielding a raw background capacity of N_max = L∕w₀ elements. A Level-0 “space fabric” token exists at index m if the block matches the binary target integer value V _P :

The raw count of detected space fabric elements is N_G = ∑ σ_G[m].

Level-k: Hierarchical Matter Sub-Scales

Higher-order structural tiers (representing Hadrons, Atoms, and Compounds sequentially) are extracted by recursively filtering the underlying layer’s output array. For a given structural level k ∈{1,2,3}, the input array is scanned using a non-overlapping window of width w_k subject to a density activation threshold τ_k:

The raw abundance of entities at level k is given by N_k = ∑ σ_k[m]. As discussed in Section 3, the qualitative shape of N_k as a function of entropy is fixed by the effective composition (a,b) implied by each level’s threshold τ_k, and should be checked explicitly rather than assumed.