Decorated-spin occupation insertion ($⟨{\tau }^2⟩=\mathrm{tr}(Q{V}^{L-1})/\mathrm{tr}{V}^L$)

An observable on a spin removed by decimation is inserted before its local sum. For ${\tau }^2$, replace $V_{st}={\sum }_{\tau }{e}^{\beta \tau (J_{1}s+J_{2}t)}$ by $Q_{st}={\sum }_{\tau }{\tau }^2{e}^{\beta \tau (J_{1}s+J_{2}t)}=2\cosh [\beta (J_{1}s+J_{2}t)]$. Contracting the unchanged ring gives the displayed thermal expectation value. In the infinite chain it becomes $v_{+}^{\mathsf{T}}Q{v}_{+}/{\lambda }_{+}$ for normalized dominant eigenvector $v_{+}$.