Fit polynomial coefficients to minimize the sum of squared gaps on each dataset’s five targets. The mean function is known only for constructing and evaluating this toy; it does not enter the fitting procedure. For degree 4 five distinct X positions identify a unique interpolant. Curves plot 81 evaluated X positions joined by short straight segments, including both probe positions.

Each training target’s sign is independently equally likely plus or minus. Draw ID minus1 is its five-bit mask, from lowest bit at X = −2 to highest at X = 2. A fresh response at the probe has an independent plus/minus sign and the same amplitude. Its conditional mean is f(X) and noise variance is amplitude². Average over the full32-fit distribution, using variance divisor 32. Expanding squared prediction error under this independence gives squared bias + prediction variance + response noise. Expected error is also computed directly from all 64 combinations.

A single fit’s squared residual is neither squared bias nor repeated-training prediction variance. Training draw does not change the underlying distribution. Noise-free collapses all 32 weighted outcomes to identical targets; degree 2 or higher recovers this quadratic with zero error. At X = 0.5 degree 1 has more squared bias than degree 0, so neither a monotone bias curve nor a universal U-shaped total error is assumed. No sampling estimate, confidence interval, real model-selection procedure, causal conclusion or guarantee about other populations is made.