Empirical Nonnegative Finite-Resolution MAR-PID for Continuous Variables

We develop a finite-resolution empirical framework for applying nonnegative Mages-Anastasiadi-Rohner partial information decomposition (MAR-PID) to continuous and non-binary discrete variables. The variables are represented by recursive quantile binarization. This provides a balanced binary-tree representation at each finite depth. MAR-PID is then applied to binary target components, and the resulting atoms are aggregated back to the original target and source variables. The construction gives nonnegative target-relative information summaries for observed variables up to the chosen empirical resolution. The pipeline consists of conditional channel estimation, bit-level MAR-PID computation, projection of source atoms to the original variables, and aggregation of the resulting information contributions. The obtained quantities are empirical estimates of finite-resolution population quantities. XOR and mixed redundancy-synergy examples show how the representation separates informational mechanisms, which signed interaction-information summaries can conflate. We focus here on the finite-resolution MAR-PID construction and its information-level quantities. Downstream summaries, such as resolution-normalized PID-dimension descriptors and thresholded support-based degrees of freedom, are indicated but left for separate work.