You can turn a sphere inside out without tearing, poking holes, or creating a single crease - and this is not a trick, it is a proven mathematical theorem. You cannot do the same thing to a circle. The reason is a topological invariant called the turning number, and it is the same invariant governing the geometry every LLM embedding space is built on. Stephen Smale proved eversion was possible in 1957. It took seven more years before anyone found a way to actually do it. The mechanism is corrugation. Divide the sphere into guide strips connected by wavy sections that let the surface pass through itself without pinching. Push the caps past each other before a crease forms. Twist them in opposite directions - the same way a belt with a loop untwists when you rotate each end half a turn. Then push the equator through and uncorrugate. Bowls, domes and saddles cancel in a pattern that stays invariant throughout - the same structure that makes high-dimensional transformer geometry behave nothing like 3D intuition. 21 minutes. Bookmark & watch today. The mathematical proof that breaks every intuition you have about surfaces.