One trick for proving things is to proceed with your argument and, every time you hit a snag, just figure out what you need to get over that hurdle and then assume it as part of your premise. Then, when you're done, see if the condition can be relaxed without compromising the theorem. If it can, yay for that. Otherwise, at least you have something for a more limited case.
I'm still working on the proof that the optimal stratification yields strata variance inversely proportional to the number of blocks. I think I may have a way forward, but I've had to add a couple conditions. Most notably, I need the density function to be continuous and monotonic. The first is really just a convenience. The second seems like it will help quite a lot. Since I'm looking at partitioning unimodal distributions, the monotonic condition is no problem except that I need to split the distribution in two at the mode to get two disjoint monotonic densities.
All this is completely unnecessary and not even particularly useful since the distribution changes with every query so trying to find optimal partitioning is a bit of a fools errand. However, it could be a really pretty proof and, as with everywhere else, there's a place in math for aesthetics.
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