19 Simulation Results

We now present the results of numerical simulations of the informational dynamics. All simulations were performed using parameter sets selected via the pre-physical functional $\Xi$, as described in the previous section.

The results demonstrate that the framework produces stable, structured behavior without fine-tuning and without encountering singular or divergent regimes.

19.1 Entropy Growth and Saturation

A key diagnostic of global behavior is the evolution of entropy. We define an effective entropy functional:

Across all viable simulations, entropy exhibits a characteristic pattern: an initial rapid increase followed by saturation at a finite value.

The initial growth reflects symmetry breaking and structure formation seeded by primordial noise. Saturation indicates that the system reaches a stable regime in which coherence is redistributed but not destroyed.

Crucially, entropy never diverges. This confirms that the dynamics remain bounded and that information preservation is maintained throughout the evolution.

19.2 Emergence and Freezing of Locality

Locality is quantified by measuring the average informational gradient between neighboring sites. At early times, gradients are large and poorly correlated, indicating the absence of a well-defined notion of space.

As evolution proceeds, gradients decrease and stabilize. A clear separation of scales emerges, with strong correlations at short distances and suppressed correlations at long distances.

This behavior signals the dynamical emergence of locality. Once established, locality remains effectively frozen, providing a stable spatial substrate for subsequent dynamics.

19.3 Formation of Dense Cores

In the structured phase, localized regions of elevated informational density naturally form. These dense cores correspond to stable concentrations of coherence.

Depending on parameters, such cores may:

• remain as long-lived structures,
• merge hierarchically,
• or approach the critical density $I_{\rm crit}$.

Cores approaching the critical threshold exhibit horizon-like behavior, with suppressed information propagation across their boundaries. These configurations are identified as black-hole analogues within the framework.

19.4 Absence of Runaway Collapse

A central concern in any nonlinear dynamical system is the possibility of runaway collapse or unbounded growth. In all parameter regimes selected by $\Xi$, no such behavior is observed.

The nonlinear saturation term $-\beta I^3$ effectively regulates amplification, while diffusion suppresses excessive localization. As a result, the system avoids both global collapse and uncontrolled divergence.

This confirms that the absence of absolute singularities is not imposed externally, but arises dynamically from the structure of the equation and the selection principle.

Having established the robustness of the numerical behavior, we now examine the failure modes associated with parameter choices not selected by $\Xi$.