Proposition: Finite hard pruning is not generally continuous

With frozen weights and exact retention counts, the hard-pruning map is a step function of the requested density. It can have nonzero jumps in logits and accuracy. Consequently a general Lipschitz theorem for $(d_e,f)\mapsto (A,S)$ is false without further assumptions.

Proof. The count $\lceil dN\rceil$ is locally constant between its integer thresholds and changes by one at a threshold. Retaining or deleting a nonzero weight therefore gives a parameter jump.

For an accuracy jump, consider a constant ReLU feature equal to one and two output logits

Retaining the positive output weight gives class $0$; deleting it gives class $1$. Extra inactive units and output classes embed the example in the stated architecture. For a class-$0$ test law, accuracy jumps from one to zero. Thus continuity, and hence a global Lipschitz bound, fails.