= Solution
Let $t,h$ be the thermal and magnetic scaling fields, with eigenvalues $b^{y_t}$ and $b^{y_h}$. After discarding irrelevant variables whose vanishing is nonsingular, the homogeneous singular <free-energy density> satisfies
$$
f_s(t,h)=b^{-D}f_s(b^{y_t}t,b^{y_h}h).
$$
Choose $b=|t|^{-1/y_t}$. Then
$$
f_s(t,h)=|t|^{D/y_t}\mathcal F_\pm(h|t|^{-y_h/y_t}),\qquad\xi(t,h)=|t|^{-1/y_t}\mathcal X_\pm(h|t|^{-y_h/y_t}).
$$
Two thermal derivatives give the <heat-capacity critical exponent>; one and two field derivatives give the <order-parameter critical exponent> and <magnetic-susceptibility critical exponent>. The <correlation-length critical exponent> follows directly from the length rescaling. Thus
$$
\boxed{\nu=\frac1{y_t},\qquad\alpha=2-\frac D{y_t},\qquad\beta_m=\frac{D-y_h}{y_t},\qquad\gamma=\frac{2y_h-D}{y_t}.}
$$
At $t=0$, choose $b=|h|^{-1/y_h}$. Differentiating $f_s(0,h)\propto |h|^{D/y_h}$ gives $M\propto\operatorname{sgn}(h)|h|^{(D-y_h)/y_h}$, so $\delta=y_h/(D-y_h)$. The <order parameter> has scaling dimension $x_M=D-y_h$; therefore its critical <connected correlation function> has power $r^{-2x_M}$. Comparing with $r^{-(D-2+\eta)}$ yields
$$
\boxed{\delta=\frac{y_h}{D-y_h},\qquad\eta=D+2-2y_h.}
$$
These formulas express the requested <critical exponents> in terms of the relevant scaling exponents. Their hyperscaling assumptions must be respected: above an <upper critical dimension>, a <dangerously irrelevant coupling> can invalidate the free-energy homogeneity used here. Applying the formulas to the <Gaussian thinning transformation> in the root solution gives the requested Gaussian indices and their physical qualifications.
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