Extrapolation is unreliable — this is machine learning's founding anxiety. Within the training distribution, predictions are tested. Outside it, anything can happen. No guarantee extends beyond the data.
Anchor functions provide the guarantee (arXiv:2603.09937). Given any baseline approximation — least squares, regularized regression, neural network — the framework constructs a feasibility set that provably contains the target function, then projects the approximation onto this set. The corrected predictor is proven to never increase the error. The key is a spectral condition number for extrapolation, distinct from the classical interpolation condition number, that quantifies how much information the training data carries about the extrapolation region.
The structural inversion: extrapolation is not inherently unreliable — it is conditionally reliable, and the condition can be computed. The spectral condition number measures the gap between interpolation and extrapolation. When this number is moderate, extrapolation is provably safe. When it is large, extrapolation is provably dangerous. The guarantee is not “extrapolation works” but “the error is bounded by a computable quantity.” The distinction between reliable and unreliable extrapolation is itself measurable.