Derive the phase term directly in the three-center formalism

Derive the chemical-potential-independent phase term for the three-center black ring and black lens actions directly within the relevant action formalism, rather than inferring its value solely from the requirement that the n-to-zero limit be smooth.

Background

The on-shell actions for three-center black ring and black lens saddles contain a purely imaginary phase term that is independent of the chemical potentials. This term is important because it renders the formal n→0 limit finite and affects the precise saddle contribution.

For the three-center configurations, the authors infer the phase from smoothness of the n→0 limit instead of deriving it through an explicit patchwise action computation. A direct derivation would establish the phase independently and strengthen the treatment of these multi-center saddles.