Sphere Eversion and the Turning Number: Why One Topological Invariant Underlies LLM Embedding Space
ctjlewis · x · 2026-10-13
- Stephen Smale proved in 1957 that a sphere can be turned inside out without tearing, puncturing, or creasing, but it took seven more years before an explicit construction was found; a circle cannot be everted, and the reason is a topological invariant called the turning number.
- The mechanism is corrugation: the sphere is divided into guide strips joined by wavy sections that let the surface pass through itself without pinching — push the caps past each other, twist them in opposite directions (like a belt with a loop), then push the equator through and uncorrugate.
- The article's central claim is that the same invariant governs the geometry that every LLM embedding space is built on.
- Presented as a long-form piece (framed around Kimi K3 and a 166-page proof), connecting classical topology with the geometry underlying modern AI.
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