I think what you want is the supplementary information, part II "completeness proof sketch" on page 12. You already spotted the formulas for "exp" and real natural "L"og; then x - y = eml(L(x), exp(y)) and from there apparently it is all "standard" identities. They list the arithmetic operators then some constants, the square root, and exponentials, then the trig stuff is on the next page.

You can find this link on the right side of the arxiv page:

https://arxiv.org/src/2603.21852v2/anc/SupplementaryInformat...

Didn't read the paper, but it was easy for me to derive constants 0, 1, e and functions x + y, x - y, exp(x), ln(x), x * y, x / y. So seems to be enough for everything. Very elegant.

I was curious about that too. Gemini actually gave a decent list. Trig functions come from Euler's identity:

    e^ix = cos x + i sin x

which means:

    e^-ix = cos -x + i sin -x
          = cos x - i sin x

so adding them together:

   e^ix + e^-ix = 2 cos x
   cos x = (e*ix - e^-ix) / 2

So I guess the real part of that.

Multiplication, division, addition and subtraction are all straightforward. So are hyperbolic trig functions. All other trig functions can be derived as per above.

last page of the PDF has several tree's that represent a few common math functions.