But Pi is not the relationship between traits in a physical circle. Measure any physical circle, and you'll get Pi only to a few decimal places and then it will be inaccurate after that. Pi is not the propery of any physical circle.

Fair enough. It's a trait of a random selection of circles.

And it isn't invented, either. We don't decide the sequence of numbers in Pi; it's already "out there" and we explore it, just like we explore a physical landscape.

Agreed. It's a description of reality.

Now, the value derived is done through mathematical simulations of realities, but none-the-less, pi in and of itself is in fact "invented" in the sense that there are other numbers or combinations of numbers that describe the same thing (for instance, we could instead use area, or circumference, rather than the ratio of the radius to the circumference). So, in one sense, as an arbitrary number value, it's invented. In another sense, as a value required for formulas (regardless of how it's expressed) it is a relationship between other values. That relationship is, essentially, a derivation of the laws of physics. That is, you are saying a derivation of formulas for mass of curved things is an example of something existent without mass.

Any rate.... You haven't demonstrated that "pi" isn't an invention, and if it doesn't actually exist, as you assert, then.... It's fictitious.