Equal-length moves can have different local costs (left: mathematical example). In E3 (right), Fisher predicts small KL changes well but loses accuracy at the original edit strength.
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Left: equal-length arrows cross different cost contours of the exact quadratic form ; this is not measured Qwen geometry. The rotating arrow keeps its Euclidean length while its quadratic cost changes. Right: E3 on 32 evaluation prefixes, with Fisher estimated separately for each prefix. Gold points show all measured prefixes at the highlighted dose. Highlights step between tested doses; lines join their medians. The full-dose statistic is the median of per-prefix prediction ratios, not the ratio of medians.
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