Beyond the Straight Line: What Polynomial Regression Actually Learns

Polynomial regression is often presented as the natural extension of a straight line: add powers of the predictor until the curve appears to fit. That recipe is incomplete. A polynomial model remains linear in its coefficients, yet its geometry, numerical conditioning, variance, boundary behavior, and susceptibility to selection bias change sharply with degree. This paper develops the Polynomial Complexity Validity Map, a five-level framework connecting mathematical fit, statistical validity, predictive validity, transport validity, and decision validity. The framework is executed on 392 vehicles from the UCI Auto MPG dataset, using horsepower to predict fuel economy. Model years 1970–1979, comprising 307 vehicles, form the development period, while model years 1980–1982, comprising 85 vehicles, remain a temporally separated final test. Polynomial degrees 1–10 are evaluated through leave-one-year-out validation. A prespecified practical-tolerance rule selects the lowest degree within 0.10 mpg of the best validation MAE. Although degree 7 obtains the lowest internal MAE, degree 2 is selected because its additional validation error is practically negligible relative to its substantially lower complexity. The selected quadratic model achieves a final temporal-test MAE of 6.55 mpg, RMSE of 7.57 mpg, and R² of −0.588. It outperforms the development-mean baseline, whose test MAE is 11.04 mpg, but retains substantial systematic underprediction across the later technological period. A post-hoc restricted cubic spline also retains negative test R², showing that additional flexibility does not resolve the underlying distribution shift. Stability analyses strengthen the selection result: degree 2 is selected in 215 of 300 bootstrap repetitions and in all ten delete-one-year analyses. Sensitivity analysis also demonstrates that the selected degree depends on the declared practical tolerance, reinforcing that model selection is partly a decision procedure rather than a purely mechanical consequence of fit. The results support the paper’s central claim: polynomial degree is not a measure of knowledge. Additional curvature is scientifically useful only when it survives honest validation, remains numerically stable, stays within the support of the data, and supports the intended claim. The record includes a reproducibility package containing source code, data, vehicle-level predictions, fold-level metrics, figures, manifests, and structured results.