On a local variant of the 12th Delfino problem -- the $\Pi$-side

Abstract

Assume that \(M_n\), the canonical inner model with \(n\) Woodin cardinals, exists. We force a model with continuum \(\aleph_2\) in which every \(\boldsymbol{\Sigma}^1_{n+2}\) set of reals is Lebesgue measurable and has the Baire property, the \(\Sigma^1_{n+2}\)- and \(\Pi^1_{n+3}\)-uniformization properties hold, and the reals admit a \(\Delta^1_{n+3}\)-definable well-order. Thus regularity up to a fixed finite projective level, together with a definable well-order of the reals at the adjacent level, does not force the determinacy strength which would normally explain that regularity, even when this package is strengthened by adjacent \(\Sigma\)- and \(\Pi\)-uniformization. In particular, this gives a negative answer to a local form of Woodin's twelfth Delfino problem asked by Friedman-Schindler.