Feature-measurable nearest-neighbour tie-breaking ($I(x)=\min {\mathrm{argmin}}_i\Vert X_i-x\Vert$)

For a fixed query, order training features by their distances and break equal distances by a rule depending only on features, such as original index. If auxiliary uniforms are independent of the features, the uniform selected by the resulting index remains uniform conditional on all features: sum over the possible selected indices. Label-dependent tie-breaking can invalidate Bernoulli coupling by a shared uniform random variable and the usual one-nearest-neighbour asymptotic risk.