= Private-witness diagonalization for incomparable enumerable sets
To prevent a <Turing functional> $\Phi_e^B$ from computing the <indicator function> of an enumerated <set> $A$, reserve a permanently unique fresh witness $x$. On observing a convergent computation with <oracle use> $u$, enumerate $x$ into $A$ if the answer is zero; otherwise keep it out. Restrain subsequent lower-priority enumerations into $B$ below $u$. Higher-priority actions may initialize this strategy and retire its witness, but witnesses are never reused. In a <finite-injury priority construction>, <mathematical induction> on priority shows that the final computation, if observed, remains correct as a computation from the final oracle and disagrees with $A$ at $x$. If no computation is ever observed after the final initialization, the final oracle computation diverges, since any convergent computation uses only finitely many eventually stable oracle bits.
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