Point-group filtering in unitary coupled-cluster ansätze: exact criticality under the Abelian filter, freeness measured near equilibrium under the full group
Filtering a unitary coupled-cluster ansatz by molecular point-group symmetry has no settled variational cost for groups with degenerate irreducible representations. We prove that the symmetry-adapted subvariety of the amplitude space is a critical subvariety of the energy, for irreducible representations of any dimension. The gradient along every removed direction vanishes there, so a quasi-Newton optimisation with exact gradients started at the reference determinant cannot distinguish the filtered from the unfiltered ansatz. Every operator invariant under the full point group already lies in the span of the pool retained by an Abelian subgroup, so the filtered pool reaches at every geometry an energy no higher than the fully invariant one reaches. A Hamiltonian-informed pool lies inside the set the equivariant pool touches, and in a symmetry-adapted basis inside the symmetry-filtered pool, which makes a published pair of parameter counts a certificate for the orbital basis. Across six molecules and two basis sets, the full-group filter changes the classical coupled-cluster correlation energy by at most $3.4 \times 10^{-12}$ millihartree in the larger basis where the point group is finite. The parameter count of the unitary ansatz falls from 75 to 30 for ammonia and from 65 to 21 for methane, measured against the Abelian filter current practice uses. The freeness measured for the full-group filter holds near equilibrium and degrades into the static-correlation regime.