First-Order Quantum Correction in Coherent State Expectation Value of
  Loop-Quantum-Gravity Hamiltonian: I. Overview and Results
  Given the Loop-Quantum-Gravity (LQG) non-graph-changing Hamiltonian
$\widehat{H[N]}$, the coherent state expectation value
$\langle\widehat{H[N]}\rangle$ admits an semiclassical expansion in
$\ell^2_{\rm p}$. In this paper, we compute explicitly the expansion of
$\langle\widehat{H[N]}\rangle$ on the cubic graph to the linear order in
$\ell^2_{\rm p}$, when the coherent state is peaked at the homogeneous and
isotropic data of cosmology. In our computation, a powerful algorithm is
developed to overcome the complexity in computing $\langle \widehat{H[N]}
\rangle$. In particular, some key innovations in our algorithm substantially
reduce the computational complexity in the Lorentzian part of
$\langle\widehat{H[N]}\rangle$. Moreover, the algorithm developed in the
present work makes it possible to compute the expectation value of arbitrary
monomial of holonomies and fluxes on one edge up to arbitrary order of
$\ell_{\rm p}^2$.
