Exact Subspace Certificates and Adaptive Innovation Bounds

An exact minimum-energy criterion characterizes when a given linear feature space contains a valid spectral rank-$k$ approximation. It includes singular tail features and unequal leading singular values. For Gaussian block-Krylov features on interpolation nodes, the criterion becomes a $k\times k$ random matrix inequality. An exact reciprocal-Gram first moment replaces an earlier product of operator-norm bounds and gives an explicit success-probability bound. A conditional-rotation simulation then applies this bound to fully adaptive, individually charged queries, provided that the number of queries introducing directions outside the previous query-and-response span is controlled. The resulting theorem is a probability tradeoff for unrestricted algorithms, not a new unconditional bound on their total query complexity. A complementary algebraic theorem shows that the enlarged full-prefix space contains the exact leading eigenspace once its Gaussian feature dimension saturates the ambient dimension. Consequently, that enlargement cannot simply be used to eliminate the innovation qualifier.