Remark: Exact order is not generally stable

The weight bound does not preserve exact classicality order. Arbitrarily small perturbations can create nonzero singleton weights where exact cancellation previously gave zero. The FR perturbation in Proposition prop:frrobustness is an explicit example.

For thresholds ${\tau }_J>0$, membership in

is preserved whenever the ideal distance from each threshold exceeds $2^{|J|-1}ϵ$. Exact order can instead be preserved under additional restrictions that enforce the lower-order zeros exactly.