Symmetry-Projected Neural Collapse: Anisotropic Contraction of Structured Within-Class Nuisance

Neural Collapse (NC) describes terminal classifier geometry in which within-class features concentrate around their class means. Explicit collapse penalties encourage this NC1 component by contracting every within-class residual direction at the same rate, without distinguishing structured nuisance from task-relevant variation. We introduce Symmetry-Projected Neural Collapse (SP-NC), an online alternating adversarial procedure that learns a positive-semidefinite Laplacian $\bar{\Delta }G=\frac{1}{K}{\sum }_k{L}_k^{\top }{L}_k$ from prototype-preserving skew-symmetric generators and contracts residuals along its non-kernel eigendirections. Our analysis characterizes key parts of the method: for a fixed operator, SP-NC contracts selected modes at a rate of at least $\lambda \lambda {\min }^{+}$, and the NC1 reduction achievable by a fixed rank-$r_G$ support is bounded by $r_{\perp }\cdot m_{r_G}\cdot \mathrm{Tr}({\Sigma }_W)$; at a fixed representation, the one-generator adversary has an exact top-eigenspace solution with controlled prototype leakage, and finite warm-started ascent has a conditional local tracking bound under spectral separation, a dominant inner iteration, bounded drift, and suitable initialization. Controlled synthetic settings test generator recovery and selective contraction; four image and sensor benchmarks test downstream utility under injected or natural nuisance: Colored MNIST, Fashion-MNIST, Waterbirds, and UCI-HAR. SP-NC improves shifted or shortcut-sensitive accuracy over CE in every seed on all four benchmarks; matched frozen PCA-SP and JTT controls do not reproduce the gains, while isotropic collapse is competitive or stronger on some datasets but does not uniformly reproduce SP-NC's shifted-accuracy and selectivity profile. A Rotated MNIST failure analysis shows that SP-NC cannot recover a group action the encoder does not expose and demonstrates that it is complementary to equivariant architectures.