A Unified Taxonomy of Modeling Paradigms: Linear, Nonlinear, Context-Variable, and Self-Directed Causal Frameworks with Theoretical Foundations and Empirical Validation

This manuscript establishes a comprehensive, unified taxonomy for modeling paradigms in complex systems, transcending traditional linear and nonlinear dichotomies by integrating context-variable (regime-switching) and self-directed causal models. Anchored in structural causal models (SCMs), stochastic processes, and advanced identifiability theory, the framework resolves longstanding challenges in heterogeneity, regime shifts, feedback loops, and causal inference under uncertainty. We provide precise mathematical formalizations, assumptions, identifiability conditions, and extensively expanded theoretical proofs, with a detailed emphasis on Markov-Switching Structural Vector Autoregression (MS-SVAR) including derivations from foundational literature, novel hybrid constructs like Causal Markov-Switching SCM (CMS-SCM), and robustness analyses. Rigorous comparisons with Structural VAR (SVAR), Time-Varying Parameter VAR (TVP-VAR), Bayesian VAR (BVAR), LSTM, Transformer-based models (including Informer), and contemporary extensions underscore the superiority of regime-switching frameworks in capturing structural breaks and dynamic dependencies, bolstered by graphical, quantitative, and sensitivity analyses. Empirical validation encompasses reproducible Python simulations on synthetic datasets and real-world applications, including FRED macroeconomic series, S&P 500 volatility regimes, Something-Something video datasets, neuroscience applications in EEG and fMRI brain connectivity, and extensive AI-driven time series forecasting scenarios. Quantitative metrics demonstrate substantial improvements (e.g., 25-50% MSE reduction over neural baselines in regime-volatile data, enhanced causal graph recovery F1-scores). Integrating recent advancements from high-impact literature (2005-2026), with verified DOIs and hyperlinks, this taxonomy furnishes a parsimonious, verifiable foundation for interdisciplinary advancements, pioneering identifiability proofs, empirical benchmarks, and policy implications in econometrics, causal inference, neuroscience, and artificial intelligence.