 Oracle-assisted finite-extension construction

Build an infinite oracle as an increasing union of finite binary strings. With the diagonal halting set as an oracle, decide whether some finite extension makes a specified Turing functional halt with a binary output at a fresh input. In a yes case, preserve that finite computation and commit the opposing set's bit to the opposite value. In a no case, no subsequent infinite extension can yield a characteristic-function value there. Length growth makes the resulting sets computable in the halting oracle.