Direction, Not Distance: Task-Conditioned Tangent Alignment Predicts Cross-Subject EEG Transfer

Cross-subject EEG transfer remains highly variable even after Riemannian Alignment (RA) removes subject-wise covariance-location differences. We investigate which component of the remaining task-conditioned geometry predicts this variability. Prior geometric source-selection approaches use class-mean distances, dispersion and accuracy features, domain-transferability scores, or rotation-invariant scalar summaries such as the pole ratio. Our contribution is narrower and mechanistic: we isolate the signed relationship between full class-contrast tangent matrices and test whether transfer is governed by directional alignment, tangent distance, or manifold curvature. Using BNCI2014_001, we represent trials as covariance matrices on the affine-invariant Riemannian manifold of symmetric positive definite matrices. For each subject, we construct a class-contrast tangent matrix from the difference between the logarithms of the two recentered class means. Across all 72 ordered pairs from nine subjects, we compare three quantities: signed Frobenius alignment, tangent distance, and the normalized sectional curvature of the plane spanned by the subjects’ discriminative directions. To prevent shared target-label noise, the target direction is estimated on a stratified half of the target trials, while transfer degradation and the target-trained ceiling are evaluated on the disjoint half. Dependence among subject pairs is handled using node-label permutation inference. Predictive generalization is evaluated using subject-blocked cross-validation with either one subject or both subjects in each pair held out, with models refitted inside every fold and $R^2$ calculated from pooled out-of-fold predictions. The proposed curvature mechanism is not supported. Curvature’s marginal association reverses the predicted direction and is not independently significant after controlling for alignment ($r=-0.230$, $p=0.104$). Signed alignment is the strongest individual and independently significant predictor ($r=-0.442$, node-label permutation $p=0.0001$), generalizing out of sample with $R^2=0.125$ when one subject is unseen and $R^2=0.058$ when both are unseen. It remains associated after controlling for source self-accuracy estimated by within-subject cross-validation ($r=-0.297$, $p=0.039$), indicating information beyond general source quality. Tangent distance is not predictive and performs worse than the mean baseline. The distinction is geometrically meaningful. Tangent distance conflates directional mismatch with the magnitude of each subject’s discriminative structure, whereas signed alignment isolates orientation. Anti-aligned pairs are present, and signed cosine is more informative than its absolute value, showing that information discarded by the unoriented curvature plane matters for transfer. Curvature is only modestly correlated with alignment ($r=0.28$). Perturbation analysis shows that curvature can become unstable near degenerate planes, although the empirical subject pairs are not near degeneracy. A controlled synthetic intervention that varies directional discrepancy and curvature separately leaves curvature uninformative when direction is fixed, while directional discrepancy remains strongly predictive when curvature is fixed. Curvature may nevertheless carry complementary predictive information. Adding it to alignment improves prediction when both subjects are unseen by $\Delta R^2=0.098$. This improvement is supported by a conditional node-permutation test that keeps alignment and the outcome intact while permuting curvature and refitting the full cross-validation pipeline ($p=0.013$), although a paired Wilcoxon sensitivity analysis is borderline ($p=0.056$). We therefore distinguish curvature’s lack of an independently significant partial association from its possible incremental predictive contribution. As a source-selection diagnostic, alignment improves target balanced accuracy from 0.673 under random selection to 0.731, compared with an oracle value of 0.748. It does not outperform a label-free source self-accuracy heuristic when selecting one source, and its advantage for longer ranked candidate lists is exploratory. Alignment is therefore best interpreted as a label-dependent transfer diagnostic rather than a zero-shot predictor. Finally, all three tested predictors are symmetric in subject order, whereas transfer degradation is directed. Only 44% of total pairwise degradation variance lies in the symmetric component, imposing a structural ceiling: the tested predictor family cannot explain most observed variability, including source–target asymmetry. This identifies directed transfer geometry as a central target for future work.