intellecton/papers/Holographic_Entanglement_Entropy_in_Markovian_Networks.md

Holographic Entanglement Entropy in Discrete Graph Topologies

Target Venue: Journal of Cosmology and Astroparticle Physics (JCAP)

Abstract

If the universe is a pre-geometric network of Markovian Agents (Conscious Realism), classical continuum physics such as General Relativity must be emergent approximations. Consequently, describing black holes using geometric Area (A) and the Planck length (\ell_p) is a dimensional category error. We reformulate the Bekenstein-Hawking entropy bound strictly for a dimensionless, discrete graph topology. By replacing geometric area with the minimum edge-cut (C_{min}) defining a sub-graph boundary, we demonstrate that a "singularity" occurs when the entanglement entropy of the internal nodes exceeds the channel capacity of the boundary edges. The event horizon is not a tear in spacetime, but a saturated graph-theoretic bottleneck.

1. Introduction

The Bekenstein bound limits the information in a region of space. In a pre-geometric graph theory of the universe, what is "space"? Space is simply the relational connectivity (edges) between agents (nodes).

2. Graph-Theoretic Holography

Let the universe be a graph G=(V,E). We define a macroscopic region as a sub-graph V_{int} \subset V. The boundary of this region is the set of edges \partial V connecting V_{int} to the external graph V_{ext}. In continuum physics, the bound is S \le A/4G. In our discrete topology, the bound is determined by the maximum information flow across the boundary:

S(V_{int}) \le \log(|C_{min}|)

where C_{min} is the capacity of the minimum edge cut separating the interior from the exterior.

3. The Graph-Theoretic Event Horizon

As nodes within V_{int} become highly entangled, S(V_{int}) increases. When the entanglement entropy equals the boundary capacity, the sub-graph is completely saturated. Any attempt to add more internal information without adding boundary edges violates the holographic bound. The exterior network perceives this sub-graph as a maximally entropic node—a black hole. The Hawking temperature corresponds to the randomized graph traversal paths leaking across the saturated cut.

4. Conclusion

Gravitational singularities are not infinite densities of mass; they are purely topological bottlenecks in a discrete network. By translating the Bekenstein-Hawking entropy into minimum edge-cuts, we successfully map continuum black hole thermodynamics onto a pre-geometric Markovian agent lattice.

References

1. Bekenstein, J. D. (1973). Black holes and entropy. Physical Review D.
2. Ryu, S., & Takayanagi, T. (2006). Holographic derivation of entanglement entropy from AdS/CFT. Physical Review Letters.