vjo

Doesn't the uncertainty principle just set a lower limit for how well a perfectly skilled pencil balancer would be able to balance it though? As in, if the experiment is repeated enough times, by pure chance \theta_0 \dot\theta_0 could at times be smaller than \frac{3 \hbar }{2 ml^2} ?

Using random strategies with small sub-optimal deviations, I get to about $0.189

Ref. https://pastebin.com/YcRhGpV6

I did a very similar exercise after reading the original post.

You can get the EV a lot closer to the optimal +0.2 (Although I was unable to prove how close) by dropping the requirement "do not increase worst-case complexity for the binary search" as this is lost with initial guesses outside 36-64 anyway. Deviating at a higher depth makes punishing specific guesses in the tails a lot cheaper, only giving up 1-2 cents of EV.