There is no degree invariant half-jump

We prove that there is no degree invariant solution to Post's problem that always gives an intermediate degree. In fact, assuming definable determinacy, if $W$ is any definable operator on degrees such that ${\mathbf a<}W({\mathbf a)<{\mathbf a}}^{\prime }$ on a cone then $W$ is low$_2$ or high$_2$ on a cone of degrees, i.e., there is a degree ${\mathbf c}$ such that $W({\mathbf a)}^{\prime \prime }={\mathbf a}^{\prime \prime }$ for every ${\mathbf a\geq {\mathbf c}}$ or $W({\mathbf a)}^{\prime \prime }={\mathbf a}^{\prime \prime \prime }$ for every ${\mathbf a\geq {\mathbf c}}$.