Separation of number sets in an arithmetic theory ($n\in A\Rightarrow S⊢H(\bar{n}),n\in B\Rightarrow S⊢\neg H(\bar{n})$)

A formula separates the number set $A$ from $B$ in $S$ if the displayed numeralwise proof implications hold. No decision is required outside $A\cup B$. In a consistent theory the sets must be disjoint. A diagonal sentence asserting the negation of its membership in a separator of provable from refutable codes is undecidable by the two direct contradiction cases.