Renormalizing Causality: A Framework for Multi-Scale Information Flow

Causal emergence—the phenomenon where macro-scale dynamics possess greater informational efficacy than their micro-scale constituents—challenges reductionist assumptions in complex systems. While measures like Effective Information (EI) quantify this efficacy at specific scales, a unified framework for tracking causal flux across continuous length scales remains underdeveloped.

We introduce a Causal Renormalization Group (CRG) framework that treats Effective Information as a scale-dependent running coupling. We define the Causal Beta Function, β_C(λ), as the derivative of causal power with respect to the coarse-graining parameter λ. This formalism classifies systems not by their static complexity, but by their causal flow topology: reductionist systems exhibit monotonic decay (β_C < 0), while emergent systems exhibit regimes of positive flow (β_C > 0).

We apply this framework to the "Middle-Stack" problem in cosmology, demonstrating that the apparent complexity ridge at biological scales corresponds to a local zero-crossing of β_C, and predicting a resurgence of causal power (β_C > 0) at astrophysical scales driven by gravitational network topology.