Does the strong force explain the binding energies of nuclei? Does the current theory of the strong force accurately predict the Nuclear Energy Binding Curve?
See definition of this curve on Wikipedia.
 A: In principle, yes. There is no reason to doubt that QCD and QED predict the curve of binding energy.
In practice, no. Any practical calculation involves two stages, based on the large separation of scales between the proton mass and nuclear binding energies. The first is the calculation of the nucleon-nucleon force using QCD, the second is nuclear many body calculations of the binding energy.
At this point, large scale numerical simulations of QCD are not yet accurate enough to constrain the nucleon-nucleon force at the level needed to make accurate predictions for nuclei. Also, first principles calculations of nuclei based on empirical nucleon-nucleon interactions have been extended to medium mass nuclei (calcium, for example), but not to heavy nuclei such as lead, gold, or uranium. There are very accurate fits to heavy nuclei based on empirical density functionals.
A: Yes, it is called quantum chromodynamics.
You can read about it here
https://en.wikipedia.org/wiki/Quantum_chromodynamics
