Theorem: Fixed-weight nesting and fixed-mask commutation

For fixed weights, consistent tie breaking, and $K_f\le K_e$,

The statement holds layer by layer. Therefore early pruning followed by late pruning, with no intervening weight update, agrees with one-shot pruning to the same final integer counts.

For fixed masks $M,N$, their parameter projections satisfy

Proof. The top $K_f$ positions in a fixed ranking are a subset of the top $K_e$ positions. Ranking the eligible survivors therefore selects exactly the original top $K_f$. If some selected values are zero, the resulting weight vector still satisfies (eq:nesting). Equation (eq:fixedmaskcommutation) follows coordinatewise from multiplication of zeros and ones.