I'm not sure I'm intuiting the transformation to and from log-normals. My intuition is that since the mean of a log-normal with location u scale s is e^(u+s^2/2) rather than e^u, when we end up with a mean log, transforming back into a mean tacks the s^2/2 back on (aka we're back to value ~ X rather than value ~ 0). Maybe I'm missing something, I haven't gone through to rederive your results, but even if everything's right I think the math could be made more clear.

Great post! We need to see this kind of reasoning made explicit!