Proposition: The precise constant-metric flatness statement

If the components of a nondegenerate metric are constant in all spacetime coordinates, its connection coefficients and curvature tensors vanish. Positive constant rescaling preserves this conclusion. A family of such constant metrics has zero linearized Einstein tensor for its parameter variation.

Proof. Every spacetime derivative of every component vanishes, so

The curvature formulas then vanish. Constant rescaling and differentiation with respect to an external family parameter retain spacetime-constant components.