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This question has four close votes. Back when it had one close vote, I attempted to explain why it was not a duplicate of a similar question. I'd like to ask for some help editing it so it's clearer why it should remain open.

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To start with, in my opinion, much of the question is irrelevant backstory. It might be cleaner to strip that out when thinking about the question. That said, as I will discuss below, I think it is a pretty clear duplicate, so it might be better to use the backstory to highlight why it is a new question.

In your question you ask

How hard would it be to apply to switch over to a math department (either in the same university, or a different one), assuming I had taken sufficient math courses, and done very well in them?

The duplicate question asks

For example, suppose someone has enrolled in a computer science phd program. Can he switch over to math(or physics) phd program in the same school later?

I find it hard to see how your question is not a duplicate. You argument that your question is unique seems to be that you would like to see a specific focus on the math/physics interface. I think this is covered in an answer to the duplicate

Other factors that can mitigate or complicate this process: if there's a standard procedure for doing this, if the two departments historically are comfortable with movement back and forth, if the departments are in the same college/school within the university, and so on.

In my opinion, there is nothing special about the math/physics interface. Math and Physics departments sometimes get along and sometimes do not get along. At that level, the answer is really going to depend on the actual departments.

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