intellecton/papers/Holographic_Entanglement_Entropy_in_Markovian_Networks.md

Holographic Trapped Surfaces via Directed Graph Edge-Cuts

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

Abstract

Mapping the Bekenstein-Hawking entropy bound to a discrete pre-geometric network requires replacing continuum metrics with rigorous graph theory. Previous attempts contained algebraic dimensional errors and failed to distinguish thermal graph bottlenecks from true gravitational event horizons. We rectify this by defining the holographic bound via the max-flow min-cut theorem: S \le |C_{min}| \log(d), where C_{min} is the minimum edge cut and d is the local Hilbert dimension. Furthermore, we introduce directed causal edges. A gravitational singularity is rigorously defined as a sub-graph where the directed edge cuts form a strict trapped causal surface (all directed paths point inward), completely isolating the internal network's entanglement entropy from the exterior topology.

1. Introduction

In a Markovian network, "space" is the relational connectivity between agents. We formulate black holes not as tears in a spatial manifold, but as trapped topological surfaces in a directed graph.

2. Correcting the Holographic Algebraic Bound

By the max-flow min-cut theorem of network information theory, the maximum entropy that can flow across a boundary \partial V separating an internal sub-graph V_{int} from the exterior V_{ext} is proportional to the number of edges, not the logarithm of the edges. The corrected discrete Bekenstein bound is:

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

where |C_{min}| is the number of edges in the minimal cut, exactly mirroring A / 4G.

3. Directed Edges and Trapped Causal Surfaces

A saturated edge cut alone only indicates a maximal thermal state, not a black hole. To form an event horizon, the graph must possess directed causal links. As the internal entanglement S(V_{int}) increases and the node density grows, the local gravitational coupling alters the graph's transition probabilities. When the transition probabilities across the cut C_{min} become strictly unidirectional (all external paths point inward, with zero probability of an outward path), the sub-graph forms a Trapped Causal Surface. The interior agents continue to compute, but their state updates cannot causally influence the exterior network.

4. Conclusion

Black holes in Conscious Realism are sub-graphs bounded by purely unidirectional directed edge-cuts. By correctly applying the max-flow min-cut theorem, we mathematically unify graph theory with holographic black hole thermodynamics.

References

1. Bekenstein, J. D. (1973). Black holes and entropy. Physical Review D.
2. Penrose, R. (1965). Gravitational collapse and space-time singularities. Physical Review Letters.