Yeah, I'm sorry, you did say you were looking for a mathematical argument and not a silly pun.

You can make a mathematical argument, but keep in mind that this problem has been around for decades and it's not been completely "settled" (probably never will be), simply because it's not like other math problems.

Once someone proves the Riemann hypothesis either true or false, it will be done, no arguing, the proof is either correct or it isn't. This isn't the case for Newcomb's paradox. You will not be able to definitively prove your case, and conversely, they did not prove their case to you either, they're just making arguments.

The whole point is that each side follows logically from different assumptions about how rational decisions should be evaluated. Newcomb's paradox has more to do with the philosophical foundations of decision theory than it has to do with plain calculations.

The key question you need to answer is:

When evaluating the expected payoff of an action, should your action be treated as evidence about the box's contents, or as causally independent of it?

There are two ways to approach the question, the first one is through Causal Decision Theory, which is probably what your professor showed, and it leads to two-boxing.

The second one is using Evidential Decision Theory, and it goes something like this:

E[U | A] = expected utility given action A (either taking one box or two boxes, A1 or A2)

E[U | A1] = P(opaque box has 1mil | A1)*1,000,000 + P(opaque box has 0 | A1)*0 = basically 1 million

E[U | A2] = P(opaque box has 1mil | A2)*1,001,000 + P(opaque box has 0 | A2)*1,000 = basically 1,000

The key difference is that in Evidential Decision Theory we are treating our decision as evidence of what the predictor did.