>Basically Paul models an atmospheric phenomenon with some math and physics theory, then manually or via software fits a differential equation and then solves it, this the foundation of all known physical theories these days.

Ideally you "find a differential equation" or other mathematical descriptions via physical principles and insert missing parameters via measurements. You do that because in particular you want to understand "why" things work in a certain way, it seemed to me that Paul was using an inequality and interpreted it freely, without much regard for the original intention.

And there are also parts of mathematics which enter the foundation of physical theories, not just differential equations, so differential equations are "a" foundation

but not "the" foundation, but maybe you wanted to say that, that is you seemed to have been in a haste when writing - even the "is" in front of the foundation is missing.

Ideally you "find a differential equation" or other mathematical descriptions via physical principles and insert missing parameters via measurements. You do that because in particular you want to understand "why" things work in a certain way, it seemed to me that Paul was using an inequality and interpreted it freely, without much regard for the original intention.

And there are also parts of mathematics which enter the foundation of physical theories, not just differential equations, so differential equations are "a" foundation

but not "the" foundation, but maybe you wanted to say that, that is you seemed to have been in a haste when writing - even the "is" in front of the foundation is missing.