Score Transport Cannot Recover Power: The Geometry of Correcting a Shifted Detector

A detector built on a frozen encoder and thresholded on one set of domains sits at the wrong operating point when it meets a new domain. Two corrections are available, and they are usually compared as if they differed only in strength. One transports the distribution of scores, which in one dimension is the increasing quantile map. The other transports the distribution of features, by a translation, a diagonal rescaling, a second-moment alignment, or a rescaling confined to an estimated low-dimensional subspace. We show the two are separated exactly rather than by a trade-off. A monotone map on scores only relabels thresholds, so it leaves the ROC curve pointwise unchanged: it can move an operating point and can never recover discriminative power. A feature map recovers power in proportion to how much it reorders examples, and on 24 held-out-domain cells both statements hold to four decimal places. Feature maps, though, need the second moment of the encoder's response to the nuisance at the deployment domain. We prove that reusing a source estimate of that operator's leading subspace costs a spectrally weighted squared Grassmann distance, then ask which of these quantities predicts the cost. Subspace rotation does, almost deterministically; the operator distance our worst-case bound is written in barely does, and that bound is vacuous on our data. An exact non-identifiability result and an affine-hull certificate say why no source-only estimate suffices.