  We compute non-extremal three-point functions of scalar operators in
$\mathcal{N}=4$ super Yang-Mills at tree-level in $g_{YM}$ and at finite $N_c$,
using the operator basis of the restricted Schur characters. We make use of the
diagrammatic methods called quiver calculus to simplify the three-point
functions. The results involve an invariant product of the generalized
Racah-Wigner tensors ($6j$ symbols). Assuming that the invariant product is
written by the Littlewood-Richardson coefficients, we show that the
non-extremal three-point functions satisfy the large $N_c$ background
independence; correspondence between the string excitations on $AdS_5 \times
S^5$ and those in the LLM geometry.
