That’s exactly the distinction I’m asking about. Algebra and coordinate geometry can represent supply and demand, and they can prove what follows if the model’s assumptions hold. They do not prove that the model accurately describes an actual economy.

For example, from assumed supply and demand functions, you can mathematically prove where their curves intersect. But mathematics alone cannot prove that real consumers have that demand curve, that real producers have that supply curve, that the relevant variables were held constant, or that the resulting equilibrium is what caused the observed market price.

“Price rises when demand increases, ceteris paribus” is a model-derived prediction. Whether real markets actually behave that way, under what conditions, and how accurately… all of that is empirical questions. Data can support, falsify, estimate, or justify the model’s applicability but the algebra cannot prove its correspondence to reality.

This is in stark contrast to physics. It makes economics look more like theology. Although, that’s probably still a loaded statement.

Regardless, “nearly every concept in Econ 101 has a mathematical breakdown” doesn’t answer my question. I’m specifically asking for an example where the mathematics proves that the mathematical model itself is an accurate description of the real economy, rather than merely proving consequences within the model.

And if the answer is that economics establishes that correspondence empirically rather than mathematically, then that’s precisely the distinction I was making.