= Solution
Subtract the ambient <velocity> and <stress>, writing $\mathbf u'=\mathbf u-\mathbf u_\infty$. This perturbation decays at infinity and on the <sphere> has <velocity> $\mathbf U+\boldsymbol\Omega\times\mathbf x-\mathbf u_\infty$. The ambient flow is regular through the ball and has no <body force>. Its net <torque> on the imaginary spherical surface is zero: the <divergence theorem> converts the <torque> to the <volume integral> of the <stress> <divergence> plus its antisymmetric part, both zero. Thus the perturbation has zero <torque> too.
Reciprocity with the auxiliary <rotlet> gives
$$
\boldsymbol\Omega\cdot\mathbf G=\int_{r=a}\mathbf u_\infty\cdot\boldsymbol\sigma_G\mathbf n_D\,dS.
$$
Using $\boldsymbol\sigma_G\mathbf n_D=3(\mathbf G\times\mathbf x)/(8\pi a^4)$ yields
$$
\boldsymbol\Omega=\frac3{8\pi a^4}\int_{r=a}\mathbf x\times\mathbf u_\infty\,dS.
$$
On this <sphere> $\mathbf x=a\mathbf n$, so the <vector> <divergence theorem> gives $\int\mathbf x\times\mathbf u_\infty dS=a\int_{r<a}\nabla\times\mathbf u_\infty dV$. Taking the <curl> of the homogeneous <Stokes equation> makes $\boldsymbol\omega_\infty$ <harmonic>. Its ball average is its central value: the derivative of each spherical mean is proportional to the <volume integral> of its <Laplacian>, hence zero, and its small-radius limit is its value at the center. Therefore
$$
\boxed{\boldsymbol\Omega=\frac3{8\pi a^3}\frac{4\pi a^3}{3}\boldsymbol\omega_\infty(0)=\frac12\boldsymbol\omega_\infty(0).}
$$
This proves <Faxén rotation law> directly from the reciprocal theorem, including why no finite-radius <Laplacian> correction is needed for a body-force-free ambient flow. Regularity of the ambient field throughout the <sphere>'s prospective interior is part of this argument.
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