What Explains Quadratic Feature Gains in Semi-Supervised Classification?

Adding quadratic interactions can improve classification with sparse annotations. Does that gain require additional feature directions, or merely a different regularizer? We study this question for spectral graph features using exact controls, a classification theorem, and paired experiments. Projected products are exactly a low-dimensional linear component with an anisotropic penalty. More surprisingly, we construct connected graphs on which full and projected models make identical decisions, although fitting the complementary block separately achieves perfect classification. No positive quadratic reweighting of the fixed linear-feature space recovers that result; the error separation persists as the labeled fraction tends to zero. On three image graphs, a 52-budget study establishes a reference gain. A separate matched decomposition with a common free intercept shows that jointly fitting a separate complementary coefficient block helps on two graphs, while recoupling offsets part of that benefit. A theory-prescribed scalar control leaves further gains in some annotation regimes, while stronger model selection remains competitive. Finally, signed class-margin bounds identify decisions invariant along the entire coupling path. Their numerical coverage rises from 33.02% for a norm bound to 93.17% with signs and 98.20% after refinement. The resulting lesson is operational: separate reweighting, complementary directions, and their coupling before interpreting an enrichment gain or a near-zero ablation gap.