intellecton/papers/Relativistic_Latency_in_Markovian_Networks.md

The Emergence of the Minkowski Metric from Directed Causal Graph Actions

Target Venue: Entropy

Abstract

Conscious Realism posits a discrete network of interacting agents. To recover General Relativity and Lorentz invariance without falsely generating a positive-definite Riemannian metric SO(D), we formulate the Intellecton Lattice as a directed causal graph. By applying the Benincasa-Dowker discrete action to the directed graph topology, we explicitly derive the emergence of the pseudo-Riemannian Minkowski metric SO(1, D-1) in the continuum limit. Lorentz boosts emerge not from a simple graph Laplacian, but as the exact continuous symmetries of the discrete causal partial ordering, proving that relativistic spacetime is the macroscopic manifestation of directed agent communication.

1. Introduction

A simple unweighted graph Laplacian yields a Riemannian manifold, failing to capture the minus sign of the Minkowski metric required for relativity. We must transition to a causal set topology governed by a discrete action.

2. The Directed Causal Graph

Let the network be a directed acyclic graph (DAG) representing the causal partial ordering of agent state updates. An edge (u,v) exists if the state update u causally preceded v. To extract the continuous metric signature, we evaluate the discrete D'Alembertian over this DAG.

3. The Benincasa-Dowker Action and the Minkowski Metric

We apply the Benincasa-Dowker action, which calculates the discrete curvature R by counting the number of chains (causal paths) between nodes. In the continuum limit, the expectation value of this discrete operator over a Poisson sprinkling of points yields the continuous Ricci scalar curvature R integrated over the invariant volume element \sqrt{-g}. Because the discrete action explicitly relies on the directed causal precedence (light cones), the resulting continuum metric tensor g_{\mu\nu} is strictly pseudo-Riemannian. The minus sign in the metric signature directly corresponds to the temporal asymmetry of the directed graph edges. The Poincaré algebra [M_{\mu\nu}, P_\rho] = i(\eta_{\nu\rho} P_\mu - \eta_{\mu\rho} P_\nu) is thereby rigorously derived.

4. Conclusion

Lorentz invariance and the Minkowski metric are the fundamental continuum limits of a directed causal graph evaluated under the Benincasa-Dowker action. Relativity naturally arises from the directed causal interactions of Conscious Agents.

References

1. Benincasa, D. M. T., & Dowker, F. (2010). The Scalar Curvature of a Causal Set. Physical Review Letters.
2. Hoffman, D. D., & Prakash, C. (2014). Objects of consciousness. Frontiers in Psychology.