9 Emergence of Space

Having identified the structured dynamical phase as the only physically admissible regime, we now address the emergence of space itself. In this framework, space is not a pre-existing container in which fields evolve. It is an effective concept arising from the behavior of the informational field $I(x,t)$.

This section shows how locality, spatial organization, and dimensionality emerge dynamically from the reaction–diffusion process.

9.1 Locality as a Dynamical Outcome

Locality is often treated as a fundamental axiom: interactions occur only between nearby points in space. Here, locality is instead a dynamical outcome.

The diffusion term in the evolution equation,

suppresses long-range correlations while favoring the formation of stable local neighborhoods. As a result, interactions become effectively short-ranged, even though no fundamental notion of distance was assumed.

Locality emerges when the timescale of diffusion is shorter than that of global amplification. Under these conditions, information propagates preferentially through nearby relations, giving rise to effective spatial adjacency.

Thus, what is perceived as spatial proximity is an emergent property of coherent information exchange, not a primitive metric structure.

9.2 Graph-Based and Continuum Limits

At early stages of the physical phase, the system is most naturally described in terms of a graph or network. Nodes represent informational degrees of freedom, and edges represent channels of coherent propagation.

The diffusion operator can be implemented as a graph Laplacian:

where $i,j$ label relational nodes.

As coherence grows and fluctuations are smoothed out, the graph admits a continuum approximation. In this limit, the discrete Laplacian converges to a differential operator, and the system can be described by a continuous field $I(x,t)$.

The continuum spacetime manifold is therefore an effective description, valid only once sufficient coherence and scale separation have emerged. At fundamental scales, the graph-based description remains primary.

9.3 Dimensionality as an Emergent Quantity

In conventional physics, the dimensionality of space is fixed by assumption. In the present framework, dimensionality is emergent and dynamical.

The effective dimension is determined by how information propagates across the network. One operational definition is provided by the spectral dimension $d_s$, inferred from diffusion behavior:

where $P(t)$ is the return probability of a diffusion process.

Numerical simulations show that $d_s$ converges to a stable value in the structured phase, corresponding to the observed macroscopic dimensionality. Different generative worlds may yield different effective dimensions, but only those supporting stable, low-dimensional propagation are selected by $\Xi$.

Dimensionality is thus neither arbitrary nor imposed. It is selected indirectly as the dimensionality most compatible with sustained structure and informational coherence.

With the emergence of space established, we now turn to the emergence of time. In the next section we show how temporal ordering and the arrow of time arise from the same underlying dynamics.