Proposition: Constant rescaling of the lower-index comparator

For $c>0$,

The first equation corresponds to consistently rescaling both the background and its metric variation.

Under an invertible affine coordinate change, the comparator transforms as a covariant two-tensor, provided its background and metric jets are transformed together.

Proof. Every term in the linearized Ricci formula contains one inverse background factor and one metric-variation derivative. Rescaling both cancels those factors, leaving ${R^{(1)}}_{\mu \nu }$ unchanged. The scalar contraction gains $c^{-1}$, while multiplication by the background in the Einstein trace term gains $c$. This proves the first identity.

With the variation held fixed, all Ricci terms instead gain $c^{-1}$, and the trace term has the same overall factor. This proves the second identity.

For an affine coordinate change, its Jacobian is constant. Derivative slots transform covariantly, inverse-background factors contravariantly, and all contracted Jacobian factors cancel. The two uncontracted tensor indices give the stated covariance.