The First Predictive Line: What Simple Linear Regression Actually Learns

Simple linear regression is often introduced as a short formula that converts one predictor into one outcome. That description is mathematically correct but scientifically incomplete. This paper develops the Linear Model Validity Map, a four-level framework separating mathematical fit, statistical validity, predictive validity, and decision validity, and then executes that framework on the historical UCI Auto MPG data. After excluding six records with unknown horsepower, the analytic sample contains 392 vehicles. The fitted full-sample line is mpg = 39.936 − 0.1578 × horsepower (R² = 0.606). A temporally separated study uses model years 1970–1979 for development and 1980–1982 for final assessment. Nested leave-one-year-out validation and a prespecified practical-equivalence rule select a controlled regression tree from six declared candidates. On the final 85 vehicles, the selected model achieves an MAE of 6.24 mpg, compared with 11.04 mpg for the training-mean baseline, but retains substantial positive mean error and negative test R², revealing historical distribution shift and insufficient evidence for deployment adequacy. The empirical result supports the paper’s central claim: a procedure can be correctly fitted and comparatively superior while its interpretation, transportability, or operational use remains invalid. The study connects model fitting, statistical diagnostics, out-of-sample validation, model selection, reproducibility, and scientific communication within a single auditable framework.