Let $M$ be a countable transitive model of (enough of) ZFC. Mostowski's Absoluteness Theorem says that $\Pi^1\_1$ statements are absolute between $M$ and larger models, in particular, between $M$ and the universe $V$.


For general $M$, this cannot be extended to $\Sigma^1\_2$ statements, see Andrés Caicedo's answer here: [Failure of Shoenfield's Absoluteness](https://mathoverflow.net/questions/150036/failure-of-shoenfields-absoluteness?noredirect=1&lq=1).


My question is: Can we have absoluteness of $\Sigma^1\_2$ (and beyond) statements between $M$ and $V$ for *some* countable transitive models $M$? (Possibly at the expense of some local or global large cardinal assumptions.)


Or, is it the case that for *any* countable transitive model $M$, there is a $\Sigma^1\_2$ (lightface) statement true in $V$ which fails in $M$?



Buying more absoluteness for countable transitive models?
First, it is consistent that there is a ctm but no $\Sigma^1\_2$-correct ctm, for example if $V$
 is the minimal model with a ctm. If there is a model of ZFC $M$
 containing all the reals (e.g., if there is an inaccessible), then there is a projectively correct ctm: take the transitive collapse $H$
 of a countable elementary substructure of $M$. Since reals don't move in the collapse, $H$ is projectively correct with real parameters. (By this argument there is a projectively correct model of ZFC if and only if there is a projectively correct ctm.)


As Andrés points out, one can also obtain correct models from determinacy hypotheses, though these hypotheses are much stronger than the existence of such models. E.g., if $\Pi^1\_1$-determinacy holds then there is a $\Sigma^1\_2$-correct ctm (e.g., $L\_\kappa$ where $\kappa$ is the least Silver indiscernible).


