3 Results

Starting from the pristine H = 0 boundary state, the following macroscopic phenomena emerge consistently as entropy increases:

Zero entropy yields a structureless state. The all-zero configuration contains no detectable structural matches (N_G = 0), mapping naturally under any relational decoding scheme to a single, unextended spatial point. This is the initial state – the singularity.

Spatial expansion. As entropy grows, the density of 1-bits climbs toward 0.5 following the exact relaxation curve p(τ) = (1 − e^−2τ) derived in Section 2, precipitating a freeze-out of background L₀ fabrics. The decoded spatial extent expands, following this closed-form approach to equilibrium.

Three regimes of structural abundance. For a fixed pattern of length k = a + b (with a ones and b zeros) counted against a bitstring whose global 1-density p(τ) sweeps monotonically from 0 to 1∕2, the expected count is proportional to p^a(1 − p)^b. This function’s unconstrained maximiser is p^∗ = a∕(a + b). Since p(τ) only ever reaches the range [0,1∕2], the observed shape depends entirely on where p^∗ falls relative to that range:

• a > b (so p^∗ > 1∕2): the abundance curve is monotonically increasing over the entire reachable range, with no interior peak.
• a = b (so p^∗ = 1∕2): the abundance curve is monotonically increasing and flattens smoothly as it approaches its maximum exactly at equilibrium.
• a < b (so p^∗ < 1∕2): the abundance curve exhibits a genuine interior maximum at p = p^∗, i.e. a true rise-and-fall hump, reachable within [0,1∕2], closely approximated by a lognormal shape.

3.1 Hump-Class Dominance Across the Relaxation Trajectory

The classification above answers which shape a given composition (a,b) produces. A distinct question is which class is typical: if a window of fixed length k = a + b is drawn at time τ, which regime does its observed composition tend to fall into? Answering this requires weighting each composition by the number of arrangements that share it, , rather than examining a single representative pattern in isolation.

The number of 1-bits in a length-k window is distributed as Binomialk, p(τ). Since p(τ) ∈ [0,) for every finite τ, the mean and mode of this distribution lie strictly inside the hump region a < b throughout the relaxation, approaching the a = b boundary only as τ →∞. Formally, define the composition-weighted probability mass in each regime:

with M_hump + M_non-hump = 1 by construction.

Direct evaluation, with the crossover τ_c(k) located by root-finding on M_hump(τ) − M_non-hump(τ) = 0, shows that M_hump(τ) holds the overwhelming majority of the mass from just above τ = 0 up to τ_c(k), and that τ_c(k) grows with k, with p(τ_c) → as k →∞:

The hump regime is therefore not merely one of three equally weighted possibilities: it is the typical outcome of the relaxation process at any window length, dominating essentially the entire trajectory. The non-hump regimes only attain majority mass in the immediate approach to final equilibrium — a fraction of the trajectory that shrinks as k grows, vanishing entirely in the k →∞ limit except at τ →∞ itself.

This is a fully predictive, closed-form classification: the qualitative shape of any given filter’s abundance curve can be determined in advance from (a,b) alone, without running the simulation.