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 = Vₒbs − Vbar across 171 galaxies (3375 data points), modeling its dependence on baryonic acceleration proxy (log gbar) and normalized radius (Rₙorm). 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 gbar, Rₙorm) - 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) Paper B → unified nonlinear response model (this work) AUTHOR: Krzysztof TylecIndependent Researcher