= Solution
Take $b>0$ for a stable quartic <Landau free energy>. The <cubic-term removal in a conserved quartic Landau free energy> uses $\phi=\psi-c/(3b)$ and gives, apart from a constant,
$$
f=\frac{a'}2\psi^2+\frac b4\psi^4+d'\psi,
\qquad a'=a-\frac{c^2}{3b},
\qquad d'=d-\frac{ac}{3b}+\frac{2c^3}{27b^2}.
$$
For a closed <binary fluid mixture>, the <compositional order parameter> has fixed integral. Adding $d\int\phi$ therefore adds the same constant to every allowed state. Equivalently, in the <common-tangent construction for phase coexistence>, a linear term changes the tangent slope, hence the <chemical potential>, by $d$, but leaves the coexistence compositions unchanged. This is why <linear composition bias leaves coexistence compositions unchanged>; it would not mean an unchanged composition at a fixed externally imposed <chemical potential>.
In the shifted variable, the <binodal> compositions are $\psi_\pm=\pm\sqrt{-a'/b}$ for $a'<0$. They merge at
$$
\boxed{a_c=\frac{c^2}{3b},\qquad \phi_c=-\frac{c}{3b}}.
$$
The <continuous phase transition> therefore has the same scalar quartic <Landau theory> as the symmetric <binary fluid mixture>, with a shifted critical composition and control parameter. In the <mean-field approximation>, the composition difference vanishes with <order-parameter critical exponent> $1/2$. This argument presumes the critical composition lies in the physically accessible composition range; conservation affects the dynamics, not this static <phase coexistence> construction.
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