A Pre-Registered Invariance Audit of Post-Hoc Out-of-Distribution Detectors under Exact Network Symmetries

We present, to our knowledge, the first pre-registered invariance test for post hoc OOD detectors: is detector behaviour a property of the network function or of its coordinates? Two exact symmetries rescale or permute the penultimate representation without changing predictions. Before training, we registered predictions for 28 constants across 21 detectors; across six backbones, 13 of 16 testable rows matched, and the three misses are reported as falsifications. Whether rescaling reaches the penultimate feature depends on the classifier head, not the backbone. Rebuilt references gave ΔAUROC ≈ 10⁻⁸; frozen ones shifted AUROC by up to 0.64, and frozen unit-indexed masks by up to 0.63 under permutation. Algebra fixes whether a detector moves and how far its score shifts, with closed forms matched for two detectors, and approximated for a third, across backbones and a second seed; AUROC loss and its direction depend on checkpoint and OOD set. A post hoc ImageNet-1K replication, on two architectures and five OOD sets, reproduces the refit–frozen split. ViM moved in float32 but was ranking-invariant in float64. As practical guidance, we recommend logit-only scores, or references rebuilt on the deployed checkpoint, and avoiding constants fixed in absolute units, as in NCI and GMM, which move regardless. Predictions, manifests, results and a claim-checking script are released; registrations are verifiable from git ancestry.