= Solution
With equal <Substrate>[substrate] abundance and association rates, let $a=k_-^S<b=k_-^R$ and $c=l_-^{S*}<d=l_-^{R*}$. Then
$$
F(m,V_A)=\frac{a+m}{b+m}\frac{c+V_A}{d+V_A}.
$$
Both factors increase with their forward rate: for example $\partial_m[(a+m)/(b+m)]=(b-a)/(b+m)^2>0$. Thus maximum discrimination requires enough waiting at each stage for the less stable incorrect complex to be rejected. In the slow-forward limit,
$$
\boxed{m\ll k_-^S,k_-^R,\quad V_A\ll l_-^{S*},l_-^{R*},\qquad
F\longrightarrow\frac{k_-^S}{k_-^R}\frac{l_-^{S*}}{l_-^{R*}}.}
$$
The two rejection stages multiply their discrimination. Fast modification or fast product formation bypasses the corresponding <Dissociation rate constant>[dissociation] test and drives that factor towards one.
There is a <kinetic proofreading fidelity-throughput tradeoff>: taking either forward rate literally to zero gives zero <Reaction rate>[product flux], not a useful finite-output optimum. If <Dissociation rate constant>[dissociation] rates are well separated, a practical high-yield discrimination window is $k_-^S\ll m\ll k_-^R$ and $l_-^{S*}\ll V_A\ll l_-^{R*}$. Correct complexes then usually pass both stages while incorrect complexes usually dissociate; its fidelity is not the mathematical infimum obtained with vanishing forward rates. The irreversible modification represents a driven, energy-consuming step, permitting successive nonequilibrium checks rather than merely repeating one equilibrium binding comparison. No finite positive optimum can be selected from minimization of $F$ alone without a speed or yield requirement.
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