= Feature-measurable nearest-neighbour tie-breaking
{title2=$I(x)=\min\operatorname*{arg\,min}_{i}\|X_i-x\|$}
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>.
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