  We revisit the holomorphic anomaly equations satisfied by the topological
string amplitude from the perspective of the 4D-5D lift, in the context of
''magic'' N=2 supergravity theories. In particular, we interpret the
Gopakumar-Vafa relation between 5D black hole degeneracies and the topological
string amplitude as the result of a canonical transformation from 4D to 5D
charges. Moreover we use the known Bekenstein-Hawking entropy of 5D black holes
to constrain the asymptotic behavior of the topological wave function at finite
topological coupling but large K\"ahler classes. In the process, some
subtleties in the relation between 5D black hole degeneracies and the
topological string amplitude are uncovered, but not resolved. Finally we extend
these considerations to the putative one-parameter generalization of the
topological string amplitude, and identify the canonical transformation as a
Weyl reflection inside the 3D duality group.
