On the Validity of the Weak Value Approximation

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.