Theorem: Exact finite-count depletion

Suppose each triggered update deletes exactly

and no repair occurs between updates. Then

and

After $T$ updates,

The difference from the continuous multiplicative approximation is nonnegative and less than $T$.

For constant $s\in (0,1)$, a positive active count cannot reach zero under this floor rule, and deletion stops once $sA<1$.

For the distinct exact-count rule

each triggered update removes at least one coordinate and active exhaustion occurs after at most $A_0$ triggered updates.

Proof. Take ${ϵ}_t=s_t{A}_t-\lfloor s_t{A}_t\rfloor$ and iterate the recurrence. Each summand in the rounding remainder is less than one. For $s<1$, $\lfloor sA\rfloor \le A-1$ at every positive integer $A$. The minimum-one rule instead decreases every positive count by at least one.