How do partons' spin/orbital angular momenta contribute fractionally to the nucleon spin structure? Experimentally it is found that the spin and orbital angular momenta of quarks and gluons contribute fractionally to the total nucleon spin $1/2$, as in:
$$\frac{1}{2} =\frac{1}{2}  \Sigma_q + \Sigma_g + L_q + L_g$$
But how do these contributions break into fractional parts of $1/2$ if the individual partons themselves each have quantized angular momenta in units of $1/2$? Or, what is the technical meaning of the 'contributions' in this equation, if it is not so naive?
 A: The quark spin is (almost*) unambiguous, but the other three contributions to the total angular momentum turn out to be gauge dependent.  Except for certain special projections in certain momentum limits, it is not possible to observe the gluon spin, gluon orbital angular momentum, and quark orbital angular momentum separately.  Performing a gauge transformation mixes these terms together.
However, any sum of operators that obey the angular momentum commutation relations itself obeys the same commutation relations; hence, the sum represents a form of angular momentum.  The total angular momentum is thus quantized in the way that all angular momenta are, in integer or half-integer multiples of $\hbar$.  Since the overall strong interaction Hamiltonian is rotation invariant, its eigenstates may be made simultaneous eigenstates of the total angular momentum operator.  A nucleon is a strong eigenstate with total angular momentum $\frac{1}{2}$.  A $\Delta^{++}$ has total angular momentum $\frac{3}{2}$.  A $\pi^{0}$, made up of only two quark field quanta (a quark and an antiquark) has total angular momentum $0$.
*It turns out that quantum corrections (the chiral anomaly) even make the total quark spin scheme dependent.  The spin depends on the extent to which the gluon field is polarized.
A: Here is what a proton really looks like in Quantum Chromo Dynamics, QCD,qualitatively, a snapshot.

All these quark antiquark  quantum numbers add up to the sum of 1, the baryon number of the proton.
Evidently  to be able to calculate that the spin of the proton adds up to 1/2 is a task to be fulfilled by a QCD calculation. The 1/2 of the spin is a boundary condition the same as the 1 for baryon number, but that is easier to prove because there will be an equal number of quarks and antiquarks in the sea except for 3 valence ones, which are also bounded by charge conservation and thus the uud single choice. Angular momenta are a different kettle of fish as the orbital ones will be taking various values that have to be somehow integrated over.
AFAIK lattice QCD has some successes in calculating masses for baryons and are trying to get at the spin.
it says in page 6  of a  of a summary paper ::

so that the question regarding the other ∼50% contribution to the spin of the nucleon still remains open

in the conclusions it states: 

Reproducing the
  nucleon benchmark quantities will open the way for providing reliable predictions for other hadron
  observables such axial charges and form factors of hyperons and charmed baryons. Furthermore,
  appropriate methods to study of excited states, resonances and decays are being developed, with good
  prospect of providing insight into the structure of hadrons and input that is crucial for experimental
  searches for new physics.

It seems work in getting a  consistent calculation of nucleon properties is going on. 
