On graphs of total projective functions

Abstract

It is well known that the graph of a total $\hat{\mathbf{I}}{\pounds }_n^1$-function is ${\hat{\mathbf{I}}}_n^1$. We prove the consistency of the dual assertion at the third projective level: there is a model of $\ZFC$ in which the graph of every total ${\hat{\mathbf{I}}}_3^1$-function is $\hat{\mathbf{I}}{\pounds }_3^1$. This principle is incompatible with ${\hat{\mathbf{I}}}_3^1$-uniformization and hence with the usual projective-determinacy picture. The construction also repairs the final step of the failure-of-uniformization argument from~\cite{HOFFELNER2023103292}.