Emergent Spacetime Geometry and the Einstein Equations from Möbius-Decomposed Informational Synergies in the DAGI Framework

Context: This manuscript represents the gravitational track of the DAGI research program at Whytics, deriving effective continuum spacetime geometry from discrete information-theoretic constraints. Abstract: The Directed Acyclic Graph Interpretation (DAGI) models physical reality as a discrete causal network of informational events and records. Recent DAGI extensions use Möbius inversion on subset posets to decompose global information into irreducible k-body contributions. This note proposes a concrete (and computable) map from those Möbius terms to an effective continuum geometry. On spatial slices, adjacency lengths are defined from a bounded mutual-information overlap with a physically interpreted vacuum-correlation floor $epsilon$ so that strongly correlated regions lie closer and weakly correlated regions lie farther. The Lorentzian signature is supplied by the DAG's intrinsic arrow of time: the time component g₀₀ is fixed by a local record/branching rate (Generalized Record Rate; GRR), consistent with prior DAGI "informational time dilation" hardware observations. For dynamics, we follow Jacobson's thermodynamic route: assuming local equilibrium and an entropy-area relation for local causal horizons, the Clausius relation reproduces the Einstein equation. In the DAGI setting, the horizon entropy is interpreted as a coarse-grained graph-cut count of horizon-crossing synergistic records. Higher-order synergies $k ge 3$ then act as curvature via discrete embedding defects (Regge-like deficit angles) in the coarse-grained causal graph. Key Highlights: Computable Metric Dictionary: Maps pairwise mutual information density over shared causal intersections directly to spatial distances. Time from Irreversibility: Derives local time dilation $g₀₀$ directly from the intrinsic rate of irreversible record creation (GRR). Curvature from Synergy: Identifies the positive-tail Möbius atoms $f^+$ of interaction order k ge 3 as the rigid embedding defects that source spacetime curvature. Simulation Validation: The kinematic primitives proposed in this theoretical dictionary have been empirically parameterized on IBM Quantum hardware and successfully utilized in companion 10,000-node classical DAG simulations to generate localized time-dilation wells and demonstrate weak-field Einstein-Poisson closure.