Multi-Fidelity Learning under Update Budgets: Cross-Fidelity Spectral Filtering

Updating a low-fidelity predictor with scarce high-fidelity labels can alter behavior that should be retained. We study multi-fidelity residual learning under an explicit budget on changes to the stored predictor. For fixed kernel models, the weighted norm of the cross-fidelity residual-to-anchor map gives the exact worst-case residual-normalized update, termed the scale-separation index (SSI). We construct a spectral filter that attenuates high-gain residual directions, decreases design-anchor SSI monotonically, and minimizes it among permutations of a fixed shrinkage spectrum. Validation selects the filter and blend step subject to an independently audited update cap. For a finite candidate family, we derive a selection-uniform population bound under independent and identically distributed audit sampling and bounded updates. Across 20 seeds at a normalized cap of 0.20, filtering reduces root mean squared error relative to matched quadratic regularization by 0.590% on QM7b and 0.241% on MF-PCBA, and outperforms equal-spectrum scrambled controls. The filter uses more of the available update budget than matched controls. Stronger external predictors remain more accurate in several comparisons, often with larger changes to the stored predictor. A separate quadratic-control study satisfies all 160 anchor certificates but meets the same cap on only 133 test blocks. Together, these results establish a mechanism for directional update control and delineate the role of anchor coverage in transferring local guarantees.