= Solution
Let $E=\langle\nabla u\rangle$ and retain only the independent response of each well-separated small <sphere>. In the leading <dilute spherical-inclusion conductivity> approximation,
$$
P(x)\simeq\beta\sum_{j=1}^N\chi_{B_j}(x)E,\qquad
\beta=\frac{3a_0(a_1-a_0)}{a_1+2a_0}.
$$
Averaging gives $\langle P\rangle\simeq p\beta E$, where $p=\sum_j|B_j|/|\Omega|$. Since $a\nabla u=a_0\nabla u+P$,
$$
a^*E=a_0E+\langle P\rangle.
$$
The spherical response holds for every $E$, so
$$
\boxed{a^*\simeq a_0I+\frac{3pa_0(a_1-a_0)}{a_1+2a_0}I.}
$$
This retains the order-$p$ term and neglects interactions. A precise dilute regime can be obtained by shrinking the <spheres> while holding separated centres fixed. <Volume fraction> alone is not enough if the inclusions instead remain in close clusters whose collective polarization persists at leading order.
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