The original hoped theorem was stated and explained in https://www.lesswrong.com/posts/Gd36HT7qLr684SYuQ/stochastic-natural-latent-implies-deterministic-natural. Recently, David Lorell used LLMs to find a proof https://www.lesswrong.com/posts/TgboJpeN95bs84odk/redux-stochastic-natural-latent-implies-deterministic. This states that there's a universal constant C such that for any distribution over two observable variables X,Y, there exists a deterministic natural latent (given as a function the observables) whose approximation errors are at most C times the approximation errors of any stochastic natural latent.
C in the original proof was shown to be provably at most 1771. There's apparently a proof (with a PR to the GitHub) that it's <994. Empirically, the true bound looks to be a bit less than 2, and there's supposedly a proof that it's above 1.96.
Will be resolved to the range of the lowest proven valid constant value before Oct 1 00:00:00 UTC. The proof needs to be publically available before that date, but it's okay if it takes longer for it to be acknowledged correct. If it doesn't seem acknowledged correct by Oct 15th 00:00:00 UTC, and if I don't decide to check it myself to resolve the market, then it won't count.
Fine print: If a bound is proven for an altered theorem statement, then I'll only count any implied bound for a universal constant for the original theorem statement in the bounty post (which is weaker than the theorem that David proved). If proof of the implication is judged by me to be "easy" and is made available to me then I'll give it the 2 week grace period like with proof verification - otherwise it counts as a necessary part of the proof (and so needs to be made available before the end of September).
If something weird happens I will either try to resolve sensibly according to my judgement, or N/A it.
I give myself permission to bet in this market - you'll just have to trust my honesty if I need to make a judgement call.
1,000