Gradient-Direction Sensitivity Reveals Linear-Centroid Coupling Hidden by Optimizer Trajectories

We show that replacing the rolling SVD of AdamW updates with a rolling SVD of loss gradients changes the diagnostic by 1-2 orders of magnitude. Performing SVD on the loss gradient instead of the AdamW update increases the measured perturbative coupling between SED directions and Linear Centroid Hypothesis (LCH) features from ${\bar{R}}_k\approx 3$–$9\times$ to $100$–$330\times$ across four single-task modular arithmetic operations, eliminating the apparent operation dependence in the original measurement. On a multitask transformer with a shared encoder, update-based SED gives ${\bar{R}}_k\le 1$ – an apparent failure of the diagnostic – while per-operation gradient-based SED recovers ${\bar{R}}_k=20$–$45\times$ across all four operations. Gradient aggregation across competing tasks is the main obstruction; performing SVD on per-task gradients resolves it. A causal intervention shows that constraining attention updates to any rank-3 subspace (whether SED-derived or random) accelerates grokking by approximately $2.3\times$ across random seeds and operations, while removing the rank-3 component has negligible effect under proper gradient-projection methodology. The SED-LCH coupling is therefore a strong diagnostic of where feature formation concentrates in parameter space, but it is not a unique causal pathway: the natural full-rank AdamW attention update is highly rank-redundant under our hyperparameters.