= Solution
Advance the projected contact line by $dx$ per unit transverse length without wetting the gaps. A fraction $f$ converts solid–vapor interface into solid–liquid interface, contributing $f(\gamma_{SL}-\gamma_{SV})dx$. The remaining fraction creates liquid–vapor interface above trapped gas, costing $(1-f)\gamma_{LV}dx$. The outer droplet surface contributes $\gamma_{LV}\cos\theta^*dx$. Thus
$$
\frac{dG}{dx}=
f(\gamma_{SL}-\gamma_{SV})+(1-f)\gamma_{LV}
+\gamma_{LV}\cos\theta^*=0.
$$
Substituting <Young's equation> gives
$$
\boxed{\cos\theta^*=f\cos\theta-(1-f)=f(1+\cos\theta)-1}.
$$
This is the ideal <Cassie–Baxter state> relation. Its $f$ is projected solid-contact fraction, not the roughness factor of <Wenzel wetting>. A suspended interface over the gaps is essential; invasion of the texture changes the area bookkeeping.
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