When Unpaired Sets Support Shared-Corruption Calibration: Moment Geometry and Two-Sample Precision

A mean shift between two data sources can be easy to detect but hard to remove without substantially changing their representations. We cast its removal as a statistical decision problem: from noisy differences between paired calibration measurements in $\mathbb{R}^d$, learn one linear map, applied to both sources under a hard distortion budget, that leaves as little of the shift as possible on fresh data. We derive the exact finite-sample minimax risk over all such maps, $(d-k) \mathbb{E}[1/(d+2J)]$ with $J\sim\mathrm{Pois}(κ/2)$, where the budget allows deleting $k$ directions and $κ$ is the calibration signal-to-noise ratio. Projecting out the mean calibration difference attains it without knowing $κ$ or the noise scale. This exposes a detection-repair gap: detecting the shift needs only $κ\gg\sqrt d$, whereas removing a fixed fraction of it at constant distortion needs $κ\asymp d$, as for estimating its direction. Standard linear concept erasers (MP, SAL, LEACE) remove the same calibration difference, so the formula gives, before fitting, exactly how much shift they leave on fresh data and how much calibration a target requires. The limit is robust: pairing keeps it exact for non-Gaussian shared content, the projection keeps its guarantee under anisotropic noise, and selective abstention cannot close the gap. On paired clinical and wearable sleep EEG, where differences between participants act as calibration noise, the formula predicts the device shift left in new participants, and more recordings per person soon stop helping. Together, these results tell whether a correction that falls short needs a better method, more recordings, or more participants.

A repair favored under an isolated input fault can be inferior when deployed modules share the faulty information. We introduce an information-interface audit for world models, distinguishing fidelity gaps, where exact inputs become estimates, from availability gaps, where inputs are missing. Fixed-weight interventions measure prediction error, input dependence, and paired closed-loop benefit, including dependencies introduced by reconstruction. In simulated quadrotor model predictive control, coupled, opposite-sign 10% mass/thrust calibration errors reduce a physics-anchored model's success from 69% to 8%; uncertainty training restores 65%. Wind reconstruction recovers control benefit but inherits calibration dependence. For a positive calibration offset, reconstruction-only corruption favors uncertainty-trained reconstruction, whereas shared corruption favors the baseline. Acceleration diagnostics reveal compensation between reconstruction bias and nominal-model error, also observed with a disturbance observer. Repair selection therefore depends on the information paths used in deployment.

Source labels tell us where a sample came from, but not how that source oriented its coordinates. In a sparse Gaussian multi-source model with source-specific sign gauges, that distinction changes the learning problem. Without calibration, pooling across sources erases the mean signal, and the source-labelled experiment reduces to sparse PCA. With task-independent relative-gauge metadata, synchronization aligns sources and the same observations become a sparse mean problem. We formalize the value of metadata through AlignedMass, which quantifies how much first-order signal a metadata-induced alignment restores. Source labels create only limited AlignedMass, whereas valid calibration can create AlignedMass proportional to the full sample size. This yields a statistical separation in micro-source regimes and, under standard sparse-PCA hardness assumptions, a computational separation as well. Synthetic experiments check the predicted scaling; controlled real-feature and physical sensor studies examine sign cancellation, separately acquired calibration, and reuse across tasks.