= Solution
For a no-slip spherical tracer of radius $a$ in an isotropic incompressible elastic continuum, a small translation produces restoring <force> $6\pi aG\,\mathbf x$. Thus
$$
k=6\pi aG.
$$
Combining this with the equilibrium three-dimensional positional variance gives
$$
\boxed{G=\frac{k_BT}{2\pi a\langle|\mathbf x-\langle\mathbf x\rangle|^2\rangle}}.
$$
Equivalently, if $P_\infty$ is the stationary three-dimensional displacement-MSD plateau, \b[$G=k_BT/(\pi aP_\infty)$]. For a single measured coordinate, $\langle x_i^2\rangle=k_BT/k$ and $G=k_BT/(6\pi a\langle x_i^2\rangle)$.
This is <passive microrheology>. Calibrate bead radius, <position> sensitivity, <temperature> and localization noise; subtract drift and test that a plateau is resolved. The continuum must be meaningful on the bead scale, the bead must couple without slip, and <thermal equilibrium> must hold. Boundaries, compressibility, active motion, probe attachment and unresolved relaxation can change the inferred response. An imposed optical trap adds stiffness and must be removed or independently calibrated rather than mistaken for material <elasticity>.
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