**Motivation:** Incompleteness (and various independence statements) is about unprovable statements. One natural way to make an unprovable statement provable is to assume it as a new axiom. But this feels like cheating, so people often look for "natural" axioms to add that will imply their favorite unprovable statement. The question is whether there are any statements that are unprovable in a theory where essentially the only way to get a theory which proves the statement is to assume it.

**Question:**
Let $T$ be a theory (e.g. ZFC). Say a statement $S$ is "minimally unprovable" in/over $T$ if $S$ is not provable from $T$ (and neither is its negation), and such that if $R$ is any statement that is provable from $T \cup \{S\}$ but not provable from $T$ alone, then there is a proof in $T$ that $R$ is equivalent to $S$.

Does every (sufficiently powerful?) theory have minimally unprovable statements?

Are there examples of a(n interesting) theory $T$ together with a minimally unprovable statement $S$?