= Solution
In the high-tension, narrow-gap regime, the almost spherical membrane has area $4\pi(R+\delta)^2$. Its spherical bending energy $8\pi k_c$ is independent of radius, while <Helfrich repulsion> over area $4\pi R^2$ costs
$$
4\pi R^2\frac{(k_BT)^2}{64k_c\delta^2}.
$$
Up to terms independent of $\delta$, a suitable <free energy> is consequently
$$
F(\delta)\simeq
8\pi k_c+4\pi\sigma(R+\delta)^2
+4\pi R^2\frac{(k_BT)^2}{64k_c\delta^2}.
$$
Its <stationary point> satisfies
$$
8\pi\sigma(R+\delta)
-8\pi R^2\frac{(k_BT)^2}{64k_c\delta^3}=0.
$$
Using $\delta/R\ll1$ gives the <fluctuation-supported membrane--particle gap>
$$
\boxed{\delta=\left(
\frac{(k_BT)^2R}{64k_c\sigma}
\right)^{1/3}}.
$$
The inverse-square entropic repulsion prevents contact, while membrane tension limits the area gained by opening the gap.
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