  This paper presents a simple framework that organizes thin-wall Coleman-De
Luccia instantons based on the Euclidean geometries of their original and
tunneled vacuum patches. We consider all a priori allowed vacuum pairs (de
Sitter or Anti-de Sitter for either patch, Minkowski can be obtained as a limit
of either), and $O(4)$-symmetric thin-wall geometries connecting them. For each
candidate bounce geometry, either a condition under which a solution to the
$O(4)$-invariant equations of motion exists is derived, or the would-be vacuum
transition is ruled out. For the parameter regimes in which a solution exists,
we determine whether expansion/contraction of the bounce supplies a negative
mode in the second variation of the Euclidean action. All results follow from
the monotonicity of a single function.
