Cheap Per-Component Testing for PLS, Stable Under Rotation

Partial Least Squares (PLS) regression extracts a few outcome-aligned directions in a high-dimensional $\mathbf{X}$ and is widely used across applied science, but inference on the resulting fit is either expensive (CV-permutation-$Q^{2}$), biased and discouraged (jackknife $t$), or absent sklearn ships no test; the corresponding statsmodels feature request has been open since 2019). We close this gap by reducing inference to held-out OLS refits of the supervised subspace, a primitive shared by PLS, supervised PCA, linear probes, and sparse-autoencoder concept directions. We supply two tests on it: a Nadeau-Bengio corrected asymptotic $t$-test (NB-asymptotic) as the default, and a permutation-referenced variant (NB-permutation) for near-singular designs where the Fisher-$z$ asymptotic drifts. A rotation-invariance result shows the held-out predictions, and hence the test statistic, are unchanged under any orthogonal rebasing of the supervised span, so the same test applies to varimax-rotated word-readable axes. We validate on PLS (synthetic geometries, Tecator NIR chemometrics, cross-lingual GloVe valence prediction in EN/PL/ES at $K=3$, $r^{2}\approx0.63/0.69/0.55$, all $p<.001$); transfer to the other OLS-refit pipelines is argued but not tested here. NB needs $\sim20\%$ fewer samples than CV-permutation-$Q^{2}$ for matched power and is $50-100 \times$ cheaper at common, already compromised, defaults. We release a Rust library with Python, R, and Julia bindings, plus a Python text pipeline.