Width Controls Wasserstein Robustness in Learned Spectral Projectors

A spectral projector may vary continuously even when every narrow continuous factor incurs order-one error somewhere. To the best of our knowledge, we establish the first sharp joint characterization of output-factor width, and Wasserstein robustness for spectral projector prediction on real Grassmann manifolds. For predictors $W{W}^{\top }$ on $\mathrm{Gr}(k,N)$, let $W$ have $m$ output columns, $\Vert W{\Vert }_F\le \sqrt{k}$, and ${\mathrm{Lip}}_F(W)\le L$. For $k\le m<N$, set $r=m-k+1$. At fixed dimensions, in the large-$L$, small-$\epsilon$ regime, the optimal in-distribution squared operator-norm risk under Haar sampling is $\Theta (L^{-r})$, while the optimal risk over a $W_p$ ball of radius $\epsilon$ is $\Theta (L^{-r}+{\epsilon }^{pr/(p+r)})$ for $1\le p<\infty$. Imposing a density-ratio cap $M\ge 1$ yields $\Theta (\min \{M{L}^{-r},L^{-r}+{\epsilon }^{pr/(p+r)}\})$. Matching lower bounds and explicit continuous constructions turn a topological existence obstruction into quantitative approximation and distribution-shift exponents. The lower bounds apply to every neural architecture satisfying the stated constraints; matching neural upper rates remain architecture-dependent. The exponents persist under input densities bounded above and away from zero, bi-Lipschitz coordinate changes, and uniformly gapped spectral deformations. Consequently, at fixed subthreshold output width, vanishing in-distribution error can coexist with an unavoidable robust-error floor at a fixed positive shift radius, while the density cap determines whether this floor persists. Frozen-backbone Fourier neural operator experiments on elliptic, Darcy, and plate problems support this distinction: widening only the output factor head substantially reduces targeted shifted error, whereas a sixteenfold increase in the size of a one-column plate model leaves its shifted-error floor essentially unchanged. Analytic rate tests and structural-mode experiments provide additional checks across ranks and discretizations.