  The weak value approximation has been in use for thirty-five years, but it
has not as of yet received a truly complete derivation, leaving its
mathematical validity in a state of limbo. Herein, I fill this gap, deriving
the weak value approximation under the von Neumann and qubit probe models. Not
only does this provide a level of mathematical support to the weak value
approximation not attained in previous works, but the techniques demonstrated
in the process might be usable by others to forge similar derivations for
alternative models, thus teasing the possibility of even broader validation in
the future.
