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The author uses "implement halts using X" in the sense of reducing the halting problem to the problem of X (technically, using Turing reduction).

Not in the sense of solving the halting problem with a function which has an occurrence of X.

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Why would this distinction matter for the argument?

It does not matter at a high level but I think the distinction is that there should be only one black box in the proof, which is precisely the thing being reduced. Every other instruction/call used n the algorithm must be known to be computable (in this case, addition).

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