  Measurability with respect to ideals is tightly connected with absoluteness
principles for certain forcing notions. We study a uniformization principle
that postulates the existence of a uniformizing function on a large set,
relative to a given ideal. We prove that for all $\sigma$-ideals $I$ such that
the ideal forcing $\mathbb{P}_I$ of Borel sets modulo $I$ is proper, this
uniformization principle is equivalent to an absoluteness principle for
projective formulas with respect to $\mathbb{P}_I$ that we call internal
absoluteness. In addition, we show that it is equivalent to measurability with
respect to $I$ together with $1$-step absoluteness for the poset
$\mathbb{P}_I$. These equivalences are new even for Cohen and random forcing
and they are, to the best of our knowledge, the first precise equivalences
between regularity and absoluteness beyond the second level of the projective
hierarchy.
