Certified Abstention for Provably-Unidentifiable Index-Phase Recovery in Image Registration

Image registration underpins measurement across scanning, camera, and microscopy imaging, yet a fundamental question is rarely posed: when is the sample-to-position map recoverable at all, and when does any estimator that reports a position necessarily fabricate one? We show that in reference-perturbed sampled imaging, where samples are localized through a perturbed reference, position fidelity (where samples land) and phase fidelity (which reference index becomes each block's origin) are logically independent: no policy acting on position information alone can guarantee phase fidelity. We derive a tight resolution threshold separating recoverable from provably-unidentifiable registration, and a recovery framework that certifies identifiable regions--recovering them exactly--while abstaining on provably-unidentifiable regions rather than hallucinating structure. On a deployed scanning system we measure a direction-bimodal registration offset consistent with the theory. Against classical and learned restoration baselines, certificate-free offset estimators fabricate structure on the unidentifiable regime, whereas the certified method abstains rather than fabricating. Finally, the certified-abstention mechanism carries across scanning, rolling-shutter, and cryo-electron-microscopy imaging--declining rather than fabricating on the regime it flags--though how completely it abstains and how tightly it bounds kept-region risk stay modality-specific (at extreme SNR the cryo-EM certificate abstains only partially): a structural-versus-statistical boundary in registration reliability.