It's not that different, when a human proposes a better definition vis-a-vis a competing one for example, they would defend this by certain desiderata.

Often a mathematician or physicist will use their intuition to speed up the naive brute force of candidate well formed formula variations so that the desired properties emerge, postulating the existence of an intersection on multiple desiderata can in itself be viewed as a novel conjecture, to be proven or disproved.

A very basic (unimpressive) example for an example desideratum is regularity or compactness. the tau=2 * pi substitution does make a whole bunch of expressions more slightly more regular and compact. That is something objective and measurable on a system of theorems.

There is no mathematician's moat vis-a-vis machine learning at a fundamental level. There can be artificially sustained moat, if AI powers limit the distribution of say cryptographic advance capable models, in jurisdictions outside such AI powers, but even that would be expected to be fleeting and temporary...