I don't think this is ever making it past the editor of any journal, let alone peer review.

Elementary functions such as exponentiation, logarithms and trigonometric functions are the standard vocabulary of STEM education. Each comes with its own rules and a dedicated button on a scientific calculator;

What?

and No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, √ , and log has always required multiple distinct operations. Here we show that a single binary operator

Yeah, this is done by using tables and series. His method does not actually facilitate the computation of these functions.

There is no such things as "continuous mathematics". Maybe he meant to say continuous function?

Looking at page 14, it looks like he reinvented the concept of the vector valued function or something. The whole thing is rediscovering something that already exists.

The principal result is "all elementary functions can be represented by this function and constant 1". I'm not sure if this was known before. Applications are another matter, but I suspect interesting ones do exist.

This preprint was written by a researcher at an accredited university with a PhD in physics. I'm sure they know what a vector valued function is.

The point of this paper is not to revolutionize how a scientific calculator functions overnight, its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application, analogous to how a NAND or NOR gate creates all of the discrete logic gates. Hence, "continuous mathematics" as opposed to discrete mathematics. It seems to me you're being overly negative without solid reasoning.