dimensionless energy eigenvalues $ER/2\pi$ at large $R$ asymptote to constant values that give scaling dimensions in ${\rm CFT}_{\rm IR}$. Moreover the asymptotic eigenvectors that correspond to scaling states $|v_{i}\rangle$ in ${\rm CFT}_{\rm IR}$ are obtained as vectors in the truncated subspace of ${\cal H}_{0}$. The corresponding components of $|v_{i}\rangle\in {\cal H}_{0}$ give us pairings \[TCSA\_pairing\] i | v\_[j]{} between scaling states in both CFT’s.

Pairings of this type are also associated with conformal interfaces. A conformal interface ${\cal I}$ between ${\rm CFT}_{\rm UV}$ and ${\rm CFT}_{\rm IR}$ can be described (via the folding trick [@BBDO]) as a conformal boundary condition in the tensor product ${\rm CFT}_{\rm UV} \otimes {\rm CFT}_{\rm IR}$. If $\phi_{i}^{\rm UV}$ and $\phi_{j}^{\rm IR}$ are scaling fields in the two theories then we can define a pairing \[conf\_pairing\] \_[UV]{}i |j\_[IR]{} = \_[i]{}\^[UV]{} \_[j]{}\^[IR]{} \_[I]{} as a disc one-point function with boundary condition ${\cal I}$. Such pairings are canonically normalised by means of Cardy constraint [@Cardy].

In TCSA there is no canonical way to fix normalisation of the eigenstates $|v_{j}\rangle$ so the natural observables that are free from this ambiguity are component ratios \[gen\_rat\] \^[j]{}\_[i,k]{} = . In (untruncated) quantum field theory perturbed states more often than not 