= Solution
For $\mathcal A_2\ne0$, rescale the <order parameter> by $m=(|\mathcal A_2|/\mathcal A_6)^{1/4}\psi$. Write $\sigma=\operatorname{sgn}\mathcal A_2$ and introduce dimensionless scaling variables
$$
X=\frac{\mathcal A_4}{|\mathcal A_2|^{1/2}\mathcal A_6^{1/2}},\qquad Y=\frac{h\mathcal A_6^{1/4}}{|\mathcal A_2|^{5/4}}.
$$
Every term then has the common energy-density factor $|\mathcal A_2|^{3/2}/\mathcal A_6^{1/2}$:
$$
\mathcal A=\frac{|\mathcal A_2|^{3/2}}{\mathcal A_6^{1/2}}\left(\frac\sigma2\psi^2+\frac X4\psi^4+\frac16\psi^6-Y\psi\right).
$$
Minimizing over $\psi$ defines the <tricritical crossover scaling> functions
$$
\Phi_\sigma(X,Y)=\min_{\psi\in\mathbb R}\left(\frac\sigma2\psi^2+\frac X4\psi^4+\frac16\psi^6-Y\psi\right).
$$
For the minimized potential, following the paper's notation $\mathcal F$, this proves
$$
\boxed{(u,v,w,x,y,z)=\left(\frac32,\frac12,\frac12,\frac12,\frac14,\frac54\right).}
$$
The exponent names $x,y$ here are unrelated to the couplings in question 2. The expression concerns the specified order-parameter <polynomial>; a general material can also have a smooth, field-independent background <free-energy density>. Such a background should be separated before writing a homogeneous singular scaling form. A potential at prescribed <magnetization>, instead of prescribed field, is a different <Legendre transform> and is not obtained by this minimization.
There is a needed qualification to the printed single $\Phi$ and nonzero-value condition. At $(X,Y)=(0,0)$, the ordered branch has minimizers $\psi=\pm1$ and $\Phi_-(0,0)=-1/3$, whereas the disordered branch has minimizer zero and $\Phi_+(0,0)=0$. Thus
$$
\boxed{\Phi_-(0,0)=-\frac13,\qquad \Phi_+(0,0)=0.}
$$
For example, $\mathcal A_2>0$, $\mathcal A_4=h=0$ gives an identically zero minimized <polynomial>, contradicting a nonzero $\Phi(0,0)$ on that side. The valid scaling statement uses separate sign branches, or restricts the asserted nonzero value to approach from the ordered side. At $\mathcal A_2=0$ the displayed coordinates are singular; the critical isotherm is obtained as a limit or directly from the equation of state.
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