Exact Spectral Degeneracies in Kernel Discriminant Operators Under Small-Sample Designs

For a family of kernelized discriminant operators built from class-membership structure alone, this note proves two closed-form spectral theorems. Under an exactly balanced sampling design, the operator relevant to several fused objectives collapses to an exact scalar multiple of an orthogonal projector, so that every direction in its signal subspace receives identically the same eigenvalue. Under a general, possibly imbalanced, design the spectrum is given in closed form by class size alone, with tied eigenspaces pooling across classes of equal size. Neither result assumes that the underlying kernel matrix is invertible, a condition that never holds exactly for a doubly-centred kernel matrix, and both are verified numerically to machine precision. A companion empirical finding is reported but not resolved. Within the resulting tied eigenspace, ranking directions by ascending rather than descending variance produces a large, classifier-independent accuracy gap, the opposite of what the objective's literal formula would suggest, and consistent with the hypothesis that this is an extreme-value instability of small-sample empirical-covariance eigen-directions. Establishing that hypothesis rigorously is posed as an open problem, one that would connect this exact, combinatorial phenomenon to the distributional small-sample breakdown of sliced inverse regression and related estimators.