Proposition: The actual source count with ties

For an eligible layer with $A>0$ active magnitudes $w_1,\dots ,w_A$, the source requests

and deletes every magnitude at most the $k$-th order statistic. Its actual deletion count is

It equals $k$ only when no excluded position ties the cutoff. Even a small nominal fraction can delete all active positions when all magnitudes tie.

Thus the source recurrence is $A^{\prime }=A-K$, not generally $A^{\prime }=A-k$.

Proof. At least the first $k$ ordered values are at most the cutoff. Every additional tied value is also deleted. If all values agree, every position satisfies the threshold.