its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application,

But he didn't show this though. I skimmed the paper many times. He creates multiple branches of these trees in the last page, so it's not truly a single nested operation.

The formulas are provided in the supplementary information file, as mentioned in the paper. https://arxiv.org/src/2603.21852v2/anc/SupplementaryInformat... You want page 9.

Well, it is still the case, even if not explicitly shown. Personally I think it almost boils down to school math, with some details around complex logarithms; the rest seems to be simpler.