would expect the Casimir force between two parallel flat plates to not disappear when connecting them with a thin wire (so that they become a single object), the regularization prescription that amounts to attributing forbidden loops to specific objects cannot be compatible with (unconstrained) shape changes.

Even if we limit ourselves to shifts and rotations of bodies, retaining eq. \[eq:subtraction\], it makes sense to describe geometry changes in a more general way. We hence consider the potential $V$ to be a function of multiple geometry parameters (positions and angles), e.g. $V=V(r^a_1,r^a_2,r^a_3,r^b_1,r^b_2,r^b_3,\alpha^a_1,\beta^a_2,\gamma^a_3;\ldots)$.

If we perform integration over the scaling parameter $s$ analytically for every loop and consider the intervals over which the subset of objects pierced by that loop does not change, then the interval endpoints, will become analytic functions of the geometry parameters – and so will the loop’s contribution to the total Casimir energy.

For any given loop, we have to perform a fairly simple computation, which in the end gives a single number. Furthermore, it is perfectly feasible to design the algorithm in such a way that at the end of the computation of the loop’s contribution to the Casimir energy, we still remember all the intermediate values 