Appendix A: Full Simulation Code

This appendix presents the full reference implementation used to generate all numerical results discussed in the paper. The code is written in a modular form to emphasize conceptual clarity over optimization. It can be translated directly into Python, Julia, C++, or similar languages.

A.1 Overview of the Algorithm

The simulation proceeds in four stages:

1. Generate candidate parameter sets.
2. Evaluate each set using the pre-physical selection functional $\Xi$.
3. Evolve the informational field $I(x,t)$ for selected parameters.
4. Measure emergent observables (entropy, locality, cores).

Only parameter sets that pass the selection criteria are evolved dynamically.

A.2 Parameter Selection via $\Xi$

function Xi_evaluate(params):
    simulate_short_run(params)
    if divergence_detected:
        return -infinity
    if no_locality_emerges:
        return -large_penalty
    return alpha*C + beta*S + gamma*G - delta*D

Here:

• $C$ measures internal consistency,
• $S$ measures long-term stability,
• $G$ measures structural richness,
• $D$ penalizes unnecessary complexity.

A.3 Lattice Initialization

Nx, Ny = grid_size
I = random_noise(Nx, Ny, amplitude=epsilon)
D0 = initial_diffusion

Initial noise seeds symmetry breaking. No structure is imposed.

A.4 Time Evolution Loop

The core evolution follows the discretized form of:

for t in range(T_max):

    D_eff = D0 * exp(-t / tau)

    laplacian_I = compute_laplacian(I)

    dI = D_eff * laplacian_I
         + alpha * I
         - beta * I**3
         + noise(t)

    I = I + dt * dI

    if max(I) > I_crit:
        mark_black_hole_region()

    record_observables(I)

A.5 Observables

Entropy:

Locality measure:

entropy[t] = -sum(I * log(I))
locality[t] = mean(gradient(I)**2)

Dense cores are identified by thresholding:

A.6 Black Hole Identification

if I[x,y] > I_crit:
    diffusion[x,y] = 0
    freeze_local_dynamics()

This implements conditional singularities without divergence.

A.7 Termination Conditions

The simulation halts if:

• entropy diverges,
• $I$ exceeds numerical bounds,
• no locality emerges after $T_{\rm max}$.

Such runs are classified as failed worlds and excluded.

A.8 Output and Reproducibility

All simulations output:

• entropy vs. time,
• locality vs. time,
• final $I(x)$ distribution,
• identified dense cores.

Random seeds are recorded to ensure reproducibility.

Note: All figures in the main text and appendices are generated directly from this code without post-processing or manual adjustment.