= 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.
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