A Möbius-Inversion Enhanced Formalism for Multiscale Information Dynamics in the DAGI Framework

Context: This manuscript provides the rigorous mathematical and information-theoretic toolkit for the DAGI research program at Whytics, bridging abstract mereology with effective field theory and macroscopic observables. Abstract: We extend the Directed Acyclic Graph Informational (DAGI) framework by integrating a rigorous mereological decomposition based on Möbius inversion on partially ordered sets. The central output is a multiscale expansion that decomposes a macroscopic informational observable $g(X)$ on node-sets X into unique, irreducible contributions $f(Y)$ across all sub-configurations Y ⊆ X. To bridge this abstract mereology with physical kinematics, we apply the formalism to total correlation (multi-information). Its Möbius inversion isolates order-resolved k-body dependency atoms that can be positive (synergy) or negative (redundancy). We introduce the positive-tail truncation f⁺(X) = max(X),0\ to rigorously isolate active, irreducible synergistic constraints; this "f⁺ tail" yields the exact geometric source terms employed in the companion Emergent Spacetime Geometry (ESG) framework. We also show how the same decomposition can be coupled back into dynamics via a coarse-grained effective field theory (EFT) Lagrangian. Finally, we summarize computational pathways and modern experimental status, including superconducting-qubit demonstrations that extract irreducible 3-body and thresholded high-order synergies from IBM quantum processors using Möbius/HOIC diagnostics. Key Highlights: Order-Resolved Information: Deploys Möbius inversion on subset lattices to cleanly separate multi-partite redundancy from irreducible synergy. Effective Dynamics: Proposes a method to couple higher-order informational constraints back into an effective Lagrangian, providing a concrete mathematical basis for context-dependent dynamics ("downward causation") while avoiding double-counting via connected-cluster normalizations. Hardware Extraction: Details the analytical foundation used to isolate multiscale High-Order Information Content (HOIC) in real quantum circuits (e.g., IBM Torino benchmarks). Companion Toolkit: Serves as the mathematical prerequisite for defining the discrete curvature sources used in the DAGI Emergent Spacetime Geometry (ESG) mapping.