to loops piercing only one object surface (hence, “belonging” to that object) is taken as zero.

Consider $n$ objects with potentials $V_1,\dots, V_n$. Then the total potential is $V=V_1+V_2+\dots +V_n$ and we use the freedom to shift the absolute energy level to define the “interaction Casimir energy” $E$ as in [@Gies:2003] as the energy difference relative to a configuration in which every body is at an effectively infinite distance from every other body: $$\label{eq:subtraction}
  E = \left(\Gamma[V] - \Gamma[V_1] - \Gamma[V_2] - \dots - \Gamma[V_n]\right)/\Delta \tau$$

Using this, we get $$E =  -\frac{1}{(4\pi)^2}\int_{1/\Lambda}^\infty \frac{ds}{s^5} \langle\Theta(sy,x)\rangle_y
\label{eq:loopenergy.final}$$ with $\Theta$ given by $$\Theta(sy,x) =
  \begin{cases}
    0 & \text{Re-scaled loop does not pierce any surface} \\
    1-n & \text{Re-scaled loop pierces the surfaces of $n\ge 1$ objects}
  \end{cases}$$

If $n$ objects come close to one another, every loop that pierces all of them can be regarded as the image of $n$ loops, each to be considered as being attached to (and moving with) that body. Hence, when objects are in proximity, we count a loop *once* that would have been counted $n$ times instead for separated objects. (Note that the counting weight of both a loop that pierces no surface, and a loop 