Metric variation and source order

A flat-background field comparison would require an independently constructed family

on one identified differentiable carrier. The remainder topology must control the derivatives used in the equation.

The zero-order source is zero in this particular flat vacuum comparison. A nonzero background source, cosmological term, or curved background requires a different declared linearization. This note does not derive the physical absence of those terms.

The parameter $\lambda$ is a perturbation parameter, not an experiment identifier, physical clock, or Planck constant. A family of operational data $D_{\lambda }$, if used to construct (eq:field-expansion), needs its own comparison maps and differentiability theorem.