I'm assuming that the quadratic parent refers to comes from the increased potential energy per kg of material multiplied by the ability to store more material due to the additional volume.

If you picture a dense weight like a cannon ball on the end of a string you're right, but if you're digging down n meters, encasing n/2 meters worth of dirt and moving it up and down the free n/2 meters of shaft, the energy storage would indeed be proportional to n^2.

I don't know anything about the field and had the same reaction you did, but considering parent is running a startup in it they're either a lunatic that doesn't know the equivalent of FizzBuzz or there's something we missed on first inspection, and we should charitably assume the latter...