Actually, there is physical proof of existence. We can prove that black swans exist because we can find one (or several), photograph it, capture it, analyse it, etc. Likewise, we can prove that unicorns exist if we capture one. But we cannot prove the negative, i.e. that unicorns don’t exist. The fact that we haven’t caught one (yet) does not prove anything.
Likewise with gods. If we could “catch” a god, display it, measure it, etc., and verify it is indeed a god by fulfilling the criteria we set up beforehand, we have proven its existence. But as with the unicorns, we can’t physically prove its nonexistence.
So we are left with logical and mathematical proofs. With all the problems that follow, including proving negatives.