[Fig. \[fig:fig3\]]{} to the *right* (shrinking the attractive region) as $h$ decreases. Correspondingly, if we choose a radius $R=57$ nm just to the right of isolated-sphere bifurcation, then as $h$ decreases the Si–PS–Au combination should push the bifurcation past $R=57$ nm leading to the creation of a stable/unstable pair for small $h$, and precisely this behavior is observed in the inset of [Fig. \[fig:fig2\]]{}. Conversely, for the Si–PS–Tef case of a teflon plate with Si–PS spheres, the sphere–plate interaction is primarily attractive, and the opposite behavior occurs: choosing a radius $R=25$ nm to the left of the isolated-sphere bifurcation, decreasing $h$ increases the attraction and moves the bifurcation to the *left* in [Fig. \[fig:fig3\]]{}, eventually causing the disappearance of the stable/unstable equilibrium at $R=25$ nm. Correspondingly, for the Si–PS–Tef curve in [Fig. \[fig:fig2\]]{}, we see the disappearance of a stable/unstable pair for sufficiently small $h$.

The additive regime {#sec:additive}
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In general, three-body effects can expected to disappear in various regimes where key parameters of the interaction become small. First, for large radii, where $h$ (the sphere–plate separation) and $d$ (the sphere–sphere separation) become small compared to $R$, eventually the Casimir interaction is dominated by nearest-surface interactions, or the proximity-force 