Audio note: this article contains 212 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

Our posts on natural latents have involved two distinct definitions, which we call "stochastic" and "deterministic" natural latents. We conjectured that, whenever there exists a stochastic natural latent (to within some approximation), there also exists a deterministic natural latent (to within a comparable approximation). Four months ago, we put up a bounty to prove this conjecture.

We've been bottlenecked pretty hard on this problem, and spent most of the last four months attacking it. At long last, we have a proof. As hoped, the proof comes with some qualitative new insights about natural latents, and we expect it will unbottleneck a bunch of future work. The main purpose of this post is to present the proof.