15 Breakdown of the Spacetime Description

The emergence of spacetime as an effective description relies on the ability of the informational field to propagate coherence locally. In extreme regimes, this condition can fail. Black holes correspond precisely to such regimes: not as singular points of infinite curvature, but as transitions beyond which the spacetime description ceases to apply.

In this section, we identify the conditions under which spacetime breaks down and clarify the physical meaning of this breakdown.

15.1 Critical Density I_\mathrmcrit }

The informational field admits a maximal sustainable coherence density. We define a critical threshold:

When this threshold is exceeded, the nonlinear saturation term in the dynamical equation dominates, and further amplification becomes dynamically suppressed. This threshold is not universal in numerical value, but universal in its role: it marks the boundary beyond which the physical phase cannot be maintained.

The condition $I>I_{\mathrm{crit}}$ does not indicate divergence. Rather, it signals the failure of the assumptions underlying the emergent spacetime description.

15.2 Vanishing of Information Propagation

As $I$ approaches the critical threshold, the effective diffusion coefficient tends to zero:

When information propagation ceases, coherence can no longer be redistributed locally. This loss of propagation corresponds to the formation of an event horizon in the emergent spacetime description.

Beyond this point, no signal or excitation within the physical phase can escape. Importantly, this is not due to infinite curvature or energy density, but to the collapse of informational connectivity.

Thus, the horizon is an informational boundary, not a geometric singularity.

15.3 Failure of Locality and Time

Locality and time both rely on the propagation of information. When $D\to 0$, neither concept remains well-defined.

Locality fails because there is no longer a meaningful notion of neighborhood. Temporal ordering fails because the sequence of configurations cannot be extended beyond the horizon.

From the perspective of the physical phase, evolution halts. However, this does not imply that information is destroyed. It implies only that the spacetime-based description has reached the limits of its applicability.

The breakdown of spacetime is therefore a controlled and finite transition. It replaces the notion of a singularity with a phase boundary beyond which new descriptive tools are required.

In the next section, we formalize this distinction by introducing the concept of conditional singularities and explaining why absolute singularities are forbidden by the selection principle.