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I think what you're describing is called the stress-energy tensor, which as I understand it is a generalization of mass.

https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor

Since matter is basically precipitated energy from the near infinite energies released by the Big Bang, makes sense to me that it has fairly high amounts of energy compacted.

If nothing else, nuclear bombs made this blindingly obvious.

It is true in general that the "binding energy" of a nucleus is reflected in the measured mass or atomic weight, exactly as mc^2.