Section on Theorems in specific approaches

Theorems can have at least two central roles in quantum gravity approaches: motivate the specific approach in a context prior to the specific approach (say, in the context of GR), or act as a solid orientation point within the theory.
\subsection{Loop quantum gravity}
There are two central theorems of LQG at this stage: \begin{enumerate} \item \begin{quote} \textbf{Theorem (Ashtekar–Lewandowski Representation Theorem).} \\ Let $\mathcal{A}$ be the space of smooth SU(2) connections on a principal bundle over a 3-manifold $\Sigma$ , and let $\overline{\mathcal{A}}$ be its suitable completion (the space of generalized connections). Let $\mathcal{H}_{\text{kin}} = L^2(\overline{\mathcal{A}}, d\mu_{\text{AL}})$ , where $\mu_{\text{AL}}$ is the Ashtekar–Lewandowski measure. Then, under the assumptions of: \begin{enumerate} \item background independence, \item diffeomorphism invariance, \item irreducibility of the representation of the holonomy-flux algebra, \end{enumerate} there exists a unique (up to unitary equivalence) cyclic representation of the holonomy-flux algebra:

$\mathcal{H}_{\text{kin}} = L^2(\overline{\mathcal{A}}, d\mu_{\text{AL}})$

This is called the \emph{Ashtekar–Lewandowski representation}. \end{quote} \item \begin{quote} \textbf{Theorem (Discrete Spectra of Area and Volume Operators).} \\ Let $\mathcal{H}_{\text{kin}}$ be the kinematical Hilbert space of Loop Quantum Gravity based on spin networks, and let $S \subset \Sigma$ be an oriented 2-surface and $R \subset \Sigma$ be a compact region.
Then, the corresponding area and volume operators:

$\hat{A}(S), \quad \hat{V}(R)$

defined on $\mathcal{H}_{\text{kin}}$ , are essentially self-adjoint and have purely discrete spectra. In particular, the spectrum of the area operator is given by:

$A_S = 8\pi\gamma \ell_P^2 \sum_{i} \sqrt{j_i(j_i + 1)},$

where $j_i$ are the SU(2) spin labels of spin network edges intersecting $S$ , $\gamma$ is the Immirzi parameter, and $\ell_P$ is the Planck length. \end{quote} \end{enumerate}
One is a central signpost about that the quantisation as such as well-defined; the other is a central signpost about what to expect from quantisation. Both theorems are very helpful as they come in at decisive joints of the theoretical proposal; they thus help ground the approach overall in robust pillars.
LQG does not feature a central motivational theorem as such. It does involve rejecting the Entropy argument.
\subsection{Causal set theory}
A central theorem, and a central conjecture in the context of CST are:
\begin{enumerate} \item \begin{quote} \textbf{Order + Number = Geometry:} \\ The causal order of a spacetime, together with its volume information (i.e., the number of elements in a causal set approximation), uniquely determines its Lorentzian geometry up to local conformal transformations. \end{quote} \item \begin{quote} \textbf{Hauptvermutung of Causal Set Theory:} \\ If a causal set $C$ can be faithfully embedded into two Lorentzian manifolds $(M_1, g_1)$ and $(M_2, g_2)$ , then $(M_1, g_1)$ and $(M_2, g_2)$ are approximately isometric at scales larger than the discreteness scale. \end{quote} \end{enumerate}
Causal set theory is developed rather formally. The Malament theorem allows for motivating CST theory. Central results or expected results of CST are formulated in theorem form (such as the conjecture on how causal sets give rise to spacetimes, the Hauptvermutung).