Invariance Audits for Quantum Kernels and Variational Rewinding: A Real-to-Hermitian Taxonomy of Projector, Flag, Anchor, and Density Geometry

Abstract

Machine-learning models often replace vectors by normalized directions, projectors, covariances, subspaces, ordered flags, quantum states, or density operators before any classifier is fitted. This replacement is an invariance decision: it determines which distinctions are kept and which are quotiented out. We develop a self-contained real-to-Hermitian taxonomy for auditing such representations in quantum machine learning. On the real side, we formalize Grassmann and flag projector kernels, prove positive semidefiniteness and block-gauge invariance of a weighted flag kernel, and give a same-span block-swap witness showing when whole-span Grassmann geometry must fail while ordered flags succeed. On the quantum side, we prove that a noiseless fidelity kernel is exactly the Hilbert--Schmidt inner product between the associated rank-one Hermitian projectors, and that a QVR-style return probability is exactly an overlap score between the input projector and a learned anchor operator. Rank-constrained returns correspond to complex Grassmann anchors, while mixed or multimodal class models are naturally represented by density or positive-semidefinite anchors. Controlled vector, subspace, statevector, anomaly, finite-shot, and quotient-witness experiments support the same conclusion: quantum and geometric lifts are useful when their invariances match the task, and fail correctly when discarded information is label-bearing. The paper makes no hardware-speedup or quantum-advantage claim.