When Spectral Accuracy Controls Low-Rank Updates

Equivalent factor coordinates of a low-rank product lead to different Euclidean factor-gradient updates, and a spectral curvature objective can be precisely optimized over such coordinates while the resulting product update remains undetermined. We characterize when the spectral objective's accuracy controls the product response. A latent coverage matrix is formed by a spectral optimality certificate for convex monotone spectral criteria on a bounded factor-coordinate budget. Compared with the current gradient, this matrix provides an explicit upper bound on the response error that allows for repeated eigenvalues, optima on the budget boundary, and rank-deficient factors. Full-support criteria identify the response under gradient compatibility. For the largest eigenvalue, at interior optimal points with a simple top eigenvalue, the criterion is exact: the response is fixed if and only if the gradient has no leakage into the free subspace. When identification fails, we provide the extremal first-order response pairing for a certified spectral block in closed form. On reduced-rank digit classifiers, coordinates whose relative spectral excess is certified to be less than $7\times 10^{-6}$ still change the product response by as much as 4.9% under squared loss. Under softmax cross-entropy, a measured relative objective deviation of at most $7\times 10^{-6}$ from a solved reference allows changes of up to 54%. Each evaluated certificate contains the observed change; under squared loss, projecting the gradient onto the predicted zero-leakage subspace eliminates the effect to numerical precision, and the block extremum improves the response pairing over both uniform endpoints exactly when predicted.