When connections are fundamental objects, is LQG the only possible theory? In an interview Abhay Ashtekar claims:

if you have a background independence and if you want to have Einstein connections this (Loop Quantum Gravity) is the only thing you can write down.

Where the connection is the fundamental object. Can someone provide the proof of uniqueness of this mentioned claim?
 A: What he's probably referring to is called the LOST theorem:

There's only one cyclic representation of the holonomy-flux algebra that has a diffeomorphism-invariant cyclic vector (the Ashtekar-Lewandowski vacuum).

That representation is given by cylindrical functions.
Unfortunately I don't know of any good sources with a self-contained proof of this statement, but I know that it's generally accepted within the LQG community. It is frequently compared to the Ston-von-Neumann theorem from QM, because it also claims that a certain representation is unique up to unitary transformations if certain criteria are met.
LQG is constructed from that representation by "schedding excessive size": the cylindrical functions form a huge nonseparable space, but most of that space is actually just the gauge group of spatial diffeomorphisms. In fact one of the constraints that we must apply in LQG is the constraint of spatial diffeomorphism invariance. After applying that constraint, you're left with a separable Hilbert space so you can start doing quantum physics.
