Balancing Loop

 Taking physics classes inspired me to try and find a new way to incorporate physics into a paper puzzle. While other genres like Stostone, Aquarium, and Dosun-Fuwari use gravity in the sense of objects settling as far down (or up) as they possibly can, I wanted to go a bit more complex by introducing torque. What better base type to use for balancing than a Balance Loop? I like how the logic turned out; it may feel like heavy bifurcation near the end but I prefer to see it as mathematics.

made on 8/23/24

Solve this Balance Loop such that, assuming gravity is downward and that segments have standard, uniformly distributed mass, the loop would balance on the provided edge at the bottom, without having to fall onto it first. (Gravity exerts torque on each individual edge segment according to the number below it. The sum of torques on the left should equal the sum of torques on the right.)