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 then show that, when the accumulation response is maximal near the preferred regime, the ensemble spread in γ maps naturally into a common RAR-like mean relation with finite residual scatter. In this picture, galactic scaling regularity reflects the concentration of systems in a shared near-critical sector, while the scatter reflects heterogeneous accumulated relational organization. The present paper is intended as a minimal numerical foundation rather than a precision phenomenological fit. Its main result is that the chain from γ self-selection to a near-critical band to RAR-like scatter can already be realized dynamically with minimal assumptions, thereby establishing the regime-selection picture on an explicit numerical footing.