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.
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.
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.
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.
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.
