In the paper Separable Hilbert space in Loop Quantum Gravity carlo rovelli explains why it's not yet known whether lqg is background-independent and discrete. The problem is that isotopic graphs constructed in lqg from spin-networks aren’t in general diffeomorphic, so the isotopy-equivalence classes of diffeomorphic graphs that must as a consequence be dealt with form uncountably infinite sets whose members are hence indexed by moduli (parameters taking values in a continuous set). These moduli can be viewed as a kind of residual background-dependence that ruins discreteness and thus the convergence properties of the theory. Clearly, one must prove that these moduli don’t affect the physics of lqg. As evidence of this, rovelli argues that the spectra of the area and volume operators are unaffected. But what needs to be shown is that the same holds true for the hamiltonian, but nobody knows how to do this and in fact this problem is one aspect of the central unresolved – possibly unresolvable - problem of the lqg program, namely the solving of the hamiltonian constraint. The thing is that the example from QFT rovelli gives for motivating his approach in a very real sense undermines it instead. He points out that a naïve construction of hilbert spaces in QFT yields an uncountable basis, so to obtain finite answers we must assume that only the countably infinite sub-basis – called a fock space - contribute. Notice that this is a condition that is imposed from the outside. This is okay if we believe that QFT is only an approximation to a deeper theory, but we should expect that in a truly fundamental theory such a condition would emerge dynamically. In this “spirit”, rovelli’s suggestion to deal with the moduli involves invoking an extension of the diffeomorphism group to include functions which are almost smooth. This may seem pragmatic in the same sense that the fock space idea is, but it’s also imposed from the outside, which is troubling since LQG is supposed to be a fundamental theory. This is just one of many examples of the contrived nature of the whole lqg endeavor.