Unified Nonlinear Response in Galaxy Rotation Residuals: Continuous Structure, Phase-Like Organization, and the Failure of Piecewise Laws

This work presents a unified nonlinear analysis of galaxy rotation curve residuals, extending previous results from Paper A (global model validation) and Paper A2 (residual structure analysis). We investigate the residual velocity ΔV = V_obs − V_bar across 171 galaxies (3375 data points), modeling its dependence on baryonic acceleration proxy (log g_bar) and normalized radius (R_norm). The goal is to determine whether galaxy dynamics are better described by discrete regime-based laws or by a single continuous nonlinear relation. Using strict galaxy-level Monte Carlo cross-validation (60 splits), we compare a hierarchy of models, including linear, nonlinear, and explicit regime-based formulations (hard splits and soft transitions). Key findings: - A global quadratic model provides the best overall performance, improving galaxy-level RMSE from 24.56 km/s (linear plane) to 22.89 km/s- Nonlinear terms are stable across all cross-validation splits, indicating a robust and reproducible signal- The data exhibit strong phase-like organization, with a simple classifier achieving ~95.6% accuracy in separating regions of (log g_bar, R_norm)- However, enforcing this separation at the model level does not improve predictive performance- Regime-based models (hard and soft transitions) are systematically outperformed by a single continuous nonlinear model These results demonstrate that the apparent phase structure is geometric rather than dynamical. Galaxy rotation residuals are best described as a unified nonlinear response surface, not as transitions between distinct physical regimes. This finding places strong empirical constraints on theoretical models: any successful description of galaxy dynamics must reproduce a smooth, continuous, and nonlinear mapping between baryonic acceleration, spatial structure, and residual velocity. CONTENTS:- Full manuscript (PDF)- Figures used in the analysis RESEARCH CONTEXT: This work is part of a structured research sequence:Paper A → global model validation (DOI: 10.5281/zenodo.19386393)Paper A2 → residual structure analysis (DOI: 10.5281/zenodo.19442108)