which do exactly that, and we demonstrate the feasibility of the approach by implementing them in a proof-of-principle calculation.

![\[36c\] Ground-state energy per particle as a function of on-site repulsion $U$ for 36 particles on a $6\times6$ lattice. Results of the three-index constraints imposed on clusters of $2\times2$ and $3\times3$ sites compared with the full $\mathcal{PQGT}'$ results and $\mathcal{PQG}$ results.](energy36c.pdf)

![\[34c\] Ground-state energy per particle as a function of on-site repulsion $U$ for 34 particles on a $6\times6$ lattice. Results of the three-index constraints imposed on clusters of $2\times2$ and $3\times3$ sites compared with the full $\mathcal{PQGT}'$ results and $\mathcal{PQG}$ results.](energy34c.pdf)

![\[30c\] Ground-state energy per particle as a function of on-site repulsion $U$ for 30 particles on a $6\times6$ lattice. Results of the three-index constraints imposed on clusters of $2\times2$ and $3\times3$ sites compared with the full $\mathcal{PQGT}'$ results and $\mathcal{PQG}$ results.](energy30c.pdf)

\[extcluster\]Extended cluster constraints
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\[eT1\]The extended $\mathcal{T}_1$ constraint: $e\mathcal{T}_1$
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To simplify the presentation the constraints are derived in a spin-uncoupled form. The actual implementation of the constraints, however, was performed in a spin-coupled fashion. This changes nothing to the generality of the derived conditions or to the quality of the obtained results. As explained in Section \[intro\] 