= Solution
A <renormalization-group fixed point> satisfies $R_b(u_*)=u_*$. A critical fixed point is scale invariant and has divergent <correlation length>. The <critical surface> consists of couplings flowing toward it after relevant variables have been tuned; noncritical flows ultimately leave its neighborhood toward finite-correlation-length phases.
Linearize the <renormalization-group transformation> near the fixed point and choose scaling coordinates. If the thermal and magnetic coordinates transform as $t'=b^{y_t}t$ and $h'=b^{y_h}h$, the singular <free-energy density> obeys
$$
F_s(t,h)=b^{-D}F_s(b^{y_t}t,b^{y_h}h).
$$
Choose $b=|t|^{-1/y_t}$ to stop at an order-one thermal coordinate. This gives the <scaling hypothesis for critical phenomena>
$$
F_s(t,h)=|t|^{D/y_t}\Phi_\pm(h/|t|^{y_h/y_t}),\qquad
\xi(t,0)\sim|t|^{-1/y_t}.
$$
Consequently $\nu=1/y_t$ and $\Delta=y_h/y_t$, with $2-\alpha=D/y_t$ when hyperscaling applies. The fixed point determines universal powers, while microscopic couplings set metric factors and amplitudes. Dangerously irrelevant variables and marginal logarithms can qualify the simple homogeneous form.
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