A Mathematical Treatment of the Offset Microlensing Degeneracy
  The offset microlensing degeneracy, recently proposed by Zhang et al. (2021),
has been shown to generalize the close-wide and inner-outer caustic
degeneracies into a unified regime of magnification degeneracy in the
interpretation of 2-body planetary microlensing observations. The offset
degeneracy states that for two planetary lenses that differ only in the
projected star-planet separation ($s_{\rm A}\neq s_{\rm B}$), the source
trajectory is required to cross their point of equal magnification on the
lens-axis, located at $u=(s_{\rm A}-1/s_{\rm A}+s_{\rm B}-1/s_{\rm B})/2$ from
the primary, to give rise to degenerate observations. Crucially, the above
formalism applies to all three caustic topologies, and implies that the $s_{\rm
A}=1/s_{\rm B}$ solution of the close-wide degeneracy never strictly manifests
in microlensing observations unless $u=0$. However, the offset degeneracy was
proposed upon numerical calculations, and no theoretical justification was
given. Here, we provide a theoretical treatment of the offset degeneracy, which
demonstrates its nature as a mathematical degeneracy. From first principles, we
show that the offset degeneracy formalism is exact to zeroth-order in the mass
ratio ($q$) for two cases: when the source crosses the lens-axis inside of
caustics, and for $(s_A-s_B)^6\ll1$ when crossing outside of caustics. The
extent to which the offset degeneracy persists in oblique source trajectories
is explored numerically. Additionally, it is shown that the superposition
principle allows for a straightforward generalization to $N$-body microlenses
with $N$-1 planetary lens components ($q\ll1$), which results in a
$2^{N-1}$-fold degeneracy. Lastly, we demonstrate that the recently proposed
"$s^\dagger$" heuristic results from the $s\rightarrow1$ limit of the offset
degeneracy, and thus does not correctly unify the close-wide and inner-outer
degeneracies.
