Readout Dependent Parameter Identifiability of Quantum Kernels

A circuit parameter can change measured features without changing their projected Gaussian kernel. We characterize this lost dependence and distinguish recovery by metric design from nonlinear feature expansion. With analytic feature coefficients, a fixed finite data basis affinely independent on the input domain, a fixed positive definite metric, and fixed bandwidth, the coefficient Gram matrix determines all-input kernel equivalence, global coordinate inactivity, and generic differential rank. An exact four-qubit family exhibits a sharp boundary: diagonal metrics give rank two, dense metrics reach but cannot exceed three, and no fixed affine expansion crosses this ceiling. One appended nonlinear scalar coordinate suffices for rank four and is minimal when the original features are retained. A single pairwise product achieves this with an explicit nondiagonal metric; a scalar sum of two products achieves it with a diagonal metric. Both use existing ideal local means. Exact certificates establish ranks in 54 selected circuit/readout settings. Matched finite-shot comparisons show that minimal representation size need not give the best sensitivity among the tested designs. Under a factorized measurement distribution, conditional product estimation cannot increase variance at a fixed design. Execution audits further separate kernel identifiability from compiled cost.