Residual Transport in Batch-Normalization Adaptation: Mechanism and Parameter-Free Correction

Replacing BatchNorm (BN) statistics with moments estimated from an unlabeled target domain is a useful test-time adaptation baseline, yet partial replacement can help, do nothing, or harm. We isolate one mechanism behind this variability. If a BN layer keeps its source mean but adopts a target variance, the scale change transports the BN output mean by a signed, channel-specific amount $q_c={\gamma }_c({\mu }_{t,c}-{\mu }_{s,c})({\sigma }_{t,c}^{-1}-{\sigma }_{s,c}^{-1})$. We estimate this quantity on a calibration split and test it on disjoint evaluation images. Across CIFAR-10-C and a bounded external ImageNet-C blur-family replication with pretrained ResNets, the calibration-only prediction closely tracks the held-out observation. On fully trained models, injecting the predicted $+q$ into an otherwise frozen network degrades downstream performance, and exact-vector channel permutation substantially attenuates the effect, confirming that channel alignment matters. Relative to variance-only replacement, aligned residual compensation shows positive aggregate gains at both training durations and on both external ImageNet ResNet architectures, while permuted compensation yields a smaller gain. The mechanism yields a direct algorithm: Residual-Balanced BN (RB-BN) keeps the target scale and installs a mean that preserves the frozen target expected BN output, exactly realizing $-q$ through BN statistics alone. In an all-layer study on CIFAR-10-C and CIFAR-100-C with two independently trained 100-epoch checkpoints per dataset, all 15 corruptions, two calibration budgets, and disjoint evaluation images, sequential RB-BN improves accuracy over frozen inference by $13.77\mathrm{pp}$ and $14.48\mathrm{pp}$ on CIFAR-10-C and CIFAR-100-C, respectively; sequential AdaBN improves by $13.61\mathrm{pp}$ and $14.66\mathrm{pp}$. RB-BN uses no mixing weight. The evidence supports residual transport as a measurable, channel-aligned causal component of BN adaptation and provides a parameter-free correction derived from that mechanism.