Data Consistency Reaches the Null Space

Sparse-view reconstruction is hard because most of the error lives where the measurements cannot see it, the null space of the forward operator. A measurement-consistency loss minimised over the image cannot move that component. We ask whether the same loss reaches the null space when it is minimised through a network instead, and what limits it there. Routed data consistency (RDC) frees one internal activation of a released reconstructor and fits the measurements through the frozen decoder behind it, with every weight fixed. How much it removes from the null space is bounded by one alignment, the cosine cos θ between its null-space update and the null-space error. For a fixed operator that alignment is set by the pretrained network and its target, not by the scan, and it caps the removable fraction at 1 − sin θ. With the relative length r of the update it forms an identity that ranks candidate variables from their decompositions alone, and a label-free test on withheld views defers to the weights where the frozen mapping is wrong. On eight-view cone-beam CT, forty steps finish 1.72 dB above the range oracle, the bound that no fit of the image can exceed, and surpass a budget-matched supervised finetune given forty-eight labelled volumes. On 18 clinical scans RDC leads weight-space adaptation in 16. Across six networks on four operators the identity predicts the better variable in 110 of 115 comparisons.