HγC N1: Self-Selected Near-Critical Bands in the HγC Framework: A Minimal Numerical Route to RAR-Like Scatter

We propose a minimal numerical model of coarse-graining self-selection in the HγC framework. The coarse-graining parameter γ is treated not as an external smoothing choice but as a dynamically realized regime descriptor controlling how strongly accumulated relational structure remains effective after coarse-graining. A competing-loss dynamics is introduced in which excessive smoothing suppresses accumulated differentiation, while insufficient coarse-graining exposes microscopic heterogeneity. Their balance produces a preferred regime γc as an attractor of the effective flow. Allowing the microscopic bridge variables to vary across an ensemble promotes this sharp attractor into a finite near-critical band. We…