The predicted scaling factors of Planck constant correspond to the integers n defining the quantum phases q=exp(iπ/n) characterizing Jones inclusions. A more precise characterization of Jones inclusion is in terms of group
Gb subset of SU(2) subset of SU(3)
in CP2 degrees of freedom and
in M4 degrees of freedom. In quantum group phase space-time surfaces have exact symmetry such that to a given point of M4 corresponds an entire Gb orbit of CP2 points and vice versa. Thus space-time sheet becomes N(Ga) fold covering of CP2 and N(Gb)-fold covering of M4. This allows an elegant topological interpretation for the fractionization of quantum numbers. The integer n corresponds to the order of maximal cyclic subgroup of G.
In the scaling hbar0→ n× hbar0 of M4 Planck constant fine structure constant would scale as
α= (e2/(4πhbar c)→ α/n ,
and the formula for Hall conductance would transform to
σH =να → (ν/n)× α .
Fractional quantum Hall effect would be integer quantum Hall effect but with scaled down α. The apparent fractional filling fraction ν= m/n would directly code the quantum phase q=exp(iπ/n) in the case that m obtains all possible values. A complete classification for possible phase transitions yielding fractional quantum Hall effect in terms of finite subgroups G subset of SU(2) subset of SU(3) given by ADE diagrams would emerge (An, D2n, E6 and E8 are possible). What would be also nice that CP2 would make itself directly manifest at the level of condensed matter physics.
For more details see the chapter Topological Quantum Computation in TGD Universe, the chapter Was von Neumann Right After All?, and the chapter Does TGD predict the Spectrum of Planck Constants?.