FACET: Geometry and Guarantees at the Conformal Prediction-Box Interface

Multivariate conformal methods are calibrated and compared as prediction regions, while many downstream interfaces expose only coordinatewise lower and upper bounds: the axis-aligned box of that region. These are different prediction tasks. We characterize the resulting interface. For any nested box family, exact box membership depends only on its coordinate-envelope path, so two arbitrarily different native regions with the same envelopes become the identical box predictor after exact recalibration; for homothetic families this reduces the geometry to its axial support vector, and for Mahalanobis ellipsoids to $\sqrt{\text{diag}\Sigma }$. Containing conversions preserve coverage but can be asymptotically vacuous: the box of a calibrated Gaussian ellipsoid has coverage tending to one and is $\Omega (\sqrt{N/\log N})$ wider than a valid centered box. Finite non-containing conversions admit no distribution-free transfer and can instead lose nearly all coverage. We test these predictions in a locked, external-only benchmark on high-dimensional forecasting panels. At the native threshold, containing conversions add $9.8$-$11.6$ percentage points of coverage, whereas a rank-truncated non-containing conversion loses $86.3$ points in median. The box-task winner differs from the native-region winner in $69.3\%$ of eligible cells. Exact online box calibration reduces median absolute coverage error from $2.47$ to $0.1$ percentage points at a median width ratio of $0.980$, and equal envelope paths yield equal boxes to $4.25\times 10^{-10}$. The strongest external geometry remains $28\%$ wider in median than a validation-selected coordinatewise-support box. On four untouched panels, time-varying coordinate supports reduce width by $30.2\%$ over static supports, while pooled support models win $1$ of $140$ comparisons. The result is a task-faithful evaluation principle and a simple prescription: predict coordinate supports, then calibrate exact membership in the box that is actually delivered.