I’m sure a lot of math is used for quantum-mechanics research. Is math useful to avoid making mistakes in research?

For example, according to this answer, it turns out to be true that it’s consistent that the laws of special relativity including energy and momentum conservation laws hold. If a different published result like that that’s useful if it’s true turns out to be false, could people who are good at math do the math to see which ones are legit and which ones are a mistake so that the correct results can actually be used for research without the incorrect ones also getting used by mistake?

Could some research be so hard that people doing research on that topic need to be really good at math and work in collaboration with each other?

Sometimes, people use previously published mathematical results to prove new results and use the axiom of choice to prove them. The axiom of determinacy contradicts the axiom of choice. I’m wondering if people are also publishing results that depend on the axiom of determinacy in the same way and only use them if they’ve either done the math to check that the result doesn’t depend on the axiom of choice or the published paper says the result doesn’t depend on the axiom of choice.