  The offset microlensing degeneracy, recently proposed by Zhang et al. (2022),
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. While the
inner-outer degeneracy expects the source trajectory to pass equidistant to the
planetary caustics of the degenerate lens configurations, the offset degeneracy
states that the same mathematical expression applies to any combination of the
close, wide, and resonant caustic topologies, where the projected star-planet
separations differ by an offset ($s_{\rm A}\neq s_{\rm B}$) that depends on
where the source trajectory crosses the star-planet axis. An important
implication is that the $s_{\rm A}=1/s_{\rm B}$ solution of the close-wide
degeneracy never strictly manifests in observations except when the source
crosses a singular point near the primary. Nevertheless, 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 $(s_{\rm A}-s_{\rm B})^6\ll1$ when crossing outside of
caustics. The extent to which the offset degeneracy persists in oblique source
trajectories is explored numerically. Lastly, 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.
