= Solution
Let $\delta\ll a$ be the minimum gap as the <spheres> approach, and $w=-\dot\delta>0$ their normal relative approach speed. For smooth equal-radius <spheres> the gap has local form $\delta+O(s^2/a)$, so the thin-gap region has lateral scale $\ell\sim(a\delta)^{1/2}$. <Incompressibility> drives a radial escape speed of order $w\ell/\delta$. The <lubrication pressure> therefore scales as
$$
p\sim\mu\frac{w\ell^2}{\delta^3}\sim\frac{\mu aw}{\delta^2},\qquad
F_n\sim p\ell^2\sim\frac{\mu a^2w}{\delta}.
$$
The available gravitational forcing remains finite. Thus <lubrication resistance> gives $w=O(W\delta/(\mu a^2))$: the normal approach becomes proportionally slower as the gap closes. In particular, for a bounded coefficient $C$, $\dot\delta\geq-C\delta$, so $\delta(t)\geq\delta(t_1)e^{-C(t-t_1)}>0$ at every finite time. Equivalently the estimated time to reach zero contains $\int_0^{\delta_1}d\delta/\delta$, which diverges. This is how <lubrication prevents finite-time collision of smooth spheres>.
The small nonzero initial lateral offset breaks the exactly collinear configuration; the <spheres> can deflect sideways and pass with a positive gap rather than touch. The condition $R>2a$ leaves space for a side-by-side passage. Smooth surfaces, continuum hydrodynamics and the <no-slip boundary condition> are essential to the noncontact conclusion; the argument is not a model of surface roughness or molecular contact.
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