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.
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).
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:
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.
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.
