Meaning
A finite Gauss certificate records the constraint-defined physical space and how observables act on it.
It specifies an extended carrier, finite gauge variations, constraint operators G(ξ) or an averaging projector P_G, and the identification H_phys=∩ξ ker G(ξ)=Ran P_G. It also records whether the observables preserve that space and whether locality and boundary/seam matching survive the reduction.
Example (illustrative)
For the toy constraint G=diag(0,1), the physical space is span(e₁). The swap operator sending e₁ to e₂ fails to preserve it. A constraint projector alone therefore does not certify that every proposed observable is physical.
Scope and limits
The Gauss physical projection is not automatically the complete-ground projection. This certificate also does not assume a particular downstream implementation of the constraints.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 8.4.