You went into way more detail then I ever could but my take on this jewy article was that instead of actually solving the problem they threw out the third body of mass and declared victory.
And then got published for their fine work.
I think what they did (at least as described) is used statistical mechanics to get a numerical answer in the form of an interval estimate. An interval estimate is basically an average and error band dependent on how sure you want to be (under a bell curve).
I don't think they "solved" anything or do any algebra for their solution. Hence they'd get an "exact" answer in the form of a formulaic expression.
And then got published for their fine work.
They own all the publications. All the major one's anyway. Not sure who the editors are on Nature though, but it's high impact factor. I don't have access to it so I can't read the original article, else it would be worth checking it out.
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