One could also imagine a huge spin network with lots of vertices and edges, all with small quantum numbers, distributed homogeneously, evenly throughtout the graph. It should look like a flat space when viewed at large scale.

If you use a ball to intersect the spin network, you will get about the same numbers of punctures on the surface and same numbers of vertices contained inside the sphere, whereever you put the ball. Therefore you see the metric is flat.

Is my argument reasonable? Why are people searching for coherent states and weaves, etc?