`I'm sorry that I haven't had the patience to read the entire flamefight before posting, so I hope this is not redundant. It should be obvious from reductio ad absurdum that the entire mass gain from raising a kilogram above the Earth cannot in principle reside completely in the kilogram in question. Imagine that the Earth weighed one kilogram, or rather, that you are somehow sitting between two identical kilogram objects somewhere in space. If you ``raise`` one of them by separating it from the other, where does the change in mass go? There's no way of choosing. Now, if you make one of them heavier, maybe you could argue that the majority of the mass change goes only to one of them. In the limit of infinite mass, you could say that all the mass change belongs to one object. However, the Earth does not have an infinite mass, although it may be a good approximation in practice when compared with one kilogram. I have the impression that Greg's point of view has a implicit geocentric assumption that should be made explicit to clarify why he assigns the change in mass to the kilogram. Leaving mass aside for a moment and going back to classical physics, it is basic knowledge that the potential energy change does not ``belong`` to the kilogram but to the Earth-kilogram interaction. Therefore the statement made in the footnote that ``the potential energy — and mass — the kilogram gains by rising out of Earth’s gravity well is independent of its position relative to an observer`` is simply not correct in principle. The kilogram does not gain potential energythe system does.    `