Following the explanation of radial quantization of Slava Rychkov in link the states are defined on the spheres with which we foliate the space-time and the generator that moves the states from one surface to another is the dilations generator $D$ which plays the role of the Hamiltonian.
He says that the states (for simplicity scalar states under $SO(D)$ rotations) living on the spheres are classified according to their scaling dimension, that is
$$ D |\Delta> \ = i \Delta |\Delta>. $$
Then, in section 3.1.3 the author explains how to construct states living on the sphere by inserting operators inside the sphere.
When we insert an operator at the origin, we generate an eigenstate of the dilations generator $D$, namely
$$ D |\Delta> \ = D\ \Phi_\Delta(0)\ |0>= i \Delta |\Delta>. $$
Then, if we insert an operator $ \Phi_\Delta(x)$ with $x\neq 0$ the resulting state $|\Psi> =\Phi_\Delta(x) |0> $ in not an eigenstate of $D$, whereas it is a superposition of eigenstates. Moreover, when he start to discuss the OPE in Sec. 3.3, above the equation (3.72) he says that $|\Psi>$ lives in the surface of the sphere.
The question is: Since $|\Psi>$ is not an eigenstate of $D$, we cannot assign an eigenvalue to this state. But in this picture the states living on the sphere are labelled by their eigenvalue respect to $D$. It seems there is a contradiction, can anyone clarify the problem?