Calibration or Location? Predictive Uncertainty under Gauge Confounding

Representation learning often aims to remove sensor or acquisition variation while preserving task information. These goals conflict when nuisance and task transformations induce the same observation: nuisance invariance then collapses the corresponding target orbit. For coherent arrays, we prove that an unknown sensor phase affine in position is observationally equivalent to a change in incident wavevector. Neither raw I/Q nor conjugate-phase features can then identify absolute direction from one snapshot. Lower bounds for point risk and prediction-set width require valid predictions to represent this ambiguity unless additional information breaks the gauge. We test these implications through controlled interventions on public 77 GHz raw ADC data. A prespecified label-only pipeline retains 281 automotive frames. Across 843 injected phase ramps, every retained beam peak translates as predicted: mean circular spatial-frequency error rises from 0.042 to 0.144 and returns to 0.042 with oracle correction. An independent audit confirms complete-spectrum translation and noncoherent range-Doppler invariance. In learned benchmarks, three five-model ensembles achieve 88.6–90.2% coverage at the training drift bound but only 6.1–7.1% when the test bound is sixteen times larger, while widths remain nearly constant. Under gauge-paired worlds, byte-identical inputs leave every prediction unchanged and shifted coverage falls to zero. Given a specified nuisance law, an analytic posterior and a learned orbit head recover about 90% coverage by widening along the unidentified orbit; a noisy known-direction anchor restores 90.4%. The theorem covers arbitrary geometries under narrowband far-field propagation; the learned study uses a single-source ULA without multipath.