> The mass of each stays the same before and after the binding.

Actually not. The best example for this is nuclear fusion. There is a measurable mass difference between the rest mass of a nucleus and the total rest mass of the protons and neutrons.

**If I understand well, your question is about the equivalence of energy and mass.**
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I know it's a bit confusing, but in bound and free systems, the total mass is different. With gravity it's more complicated, as in classical physics the source of gravity is the mass its self. But also in the merging of two black holes there is a mass difference, what is radiated away with gravitational waves (mostly). (I'm not sure if they can measure the mass difference directly, though.) Although in most (types of) processes the mass difference is usually negligable/unmeasurable.

These articles explains this whole stuff in more details:

- [Nuclear binding energy][1]
- [Mass–energy equivalence][2]


  [1]: https://en.wikipedia.org/wiki/Nuclear_binding_energy
  [2]: https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence