Meaning
The exact complete-ground projection projects onto every ground state of the full Hamiltonian.
For a self-adjoint finite Hamiltonian H with lowest energy E₀, P is the exact complete-ground projection when H−E₀I≥0 and Ran P=ker(H−E₀I). Its complement I−P contains the excitations on which a coercive estimate is tested.
Example (illustrative)
For H=diag(0,0,3), the complete-ground projection is diag(1,1,0). Using diag(1,0,0) would incorrectly treat the second zero-energy vector as an excitation, making any positive bound H≥κ(I−P) fail.
Scope and limits
One selected vacuum vector, or the ground of one Hamiltonian block, cannot replace the full kernel. The complete ground need not have rank one.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 14.6.