> This clearly solves the halting problem (we assumed the existence of halt as the first step of our proof by contradiction).

Implementing the halting problem using addition does not assume the existence of halt. It assumes the existence of addition, and you use that to show that halt must also exist. And that's not possible to do.

To implement halts using addition, you need to:

- Take the input to halts, namely the source code to a program.

- Convert that into an input to addition, namely two numbers.

- Call into addition, which adds the numbers.

- Use the result of that addition to infer whether the original program halts or not.