A well-defined quantum theory is clearly presented by Rovelli in the 2011 Zakopane lectures: http://arxiv.org/abs/1102.3660
It definitely satisfies your criterion A, easily seen to heuristically give B, and I do not know personally what is the status of C, but I know that a graviton propagator is definable and computable, which might be sufficient.
Personally, I believe there is a lot of underlying commonality with your own work (which I follow with a dilettantish interest). In particular, Rovelli has also introduced fermion and gauge theory coupling by means of lattice field theory living on a quantum graph, which to my mind resembles string-nets: http://arxiv.org/abs/1012.4719
There are also a nice set of recorded lectures at the Perimeter (perhaps you've already seen them in person, however), which contains a lot of colloquial talking which helps to fill in between the lines, and which I think expresses Rovelli's personal view of the state of the research much better than his written work: http://pirsa.org/C12012
This post imported from StackExchange Physics at 2014-04-04 16:49 (UCT), posted by SE-user genneth