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... the 4-momentum is defined as the invariant mass times
the 4-velocity, with no contribution from potential energy.
... I think I hear you saying that the invariant mass
of a system is independent of any interactions between its
parts. Is that right? If not, please correct. If so, are you
saying, for instance, that the invariant mass of an "atom" of
positronium is precisely twice that of a single electron?
Nope.
... positronium consists of electron + positron + fields. The
fields contribute to the mass of the combined system, no
problem. ... Note that the fields are derivatives of the
potential.