= Solution
<Hyperscaling> adds the physical assumption that the singular <free energy> associated with one <correlation volume> is of order $k_BT_c$:
$$
f_s\xi^d\sim k_BT_c,\qquad f_s\sim\xi^{-d}.
$$
At fixed scaled field $x=h/s^\Delta$, this predicts $f_s\sim s^{d\nu}$, whereas the given thermodynamic form has power $2-\alpha$. Hence
$$
\boxed{d\nu=2-\alpha.}
$$
The product $g_f(x)[g_\xi(x)]^d$ can be a finite dimensionless function of fixed $x$; it need not be constant. The homogeneity of $\xi$ alone is insufficient: the correlation-volume free-energy assumption supplies the connection. Above an <upper critical dimension>, a <dangerously irrelevant coupling> can invalidate this <argument>; at a marginal dimension logarithms may qualify pure <powers>.
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