Support and source choices are not interchangeable

Compact Cauchy data, Schwartz data with a Fourier gap, compact spacetime smearing, and polynomially decaying Newton initial data have different roles. No proof exchanges them.

In particular, a compactly supported ungauged Einstein perturbation need not vanish: compact pure gauges exist. The zero-past uniqueness used above applies to the homogeneous wave equations, not to an unrestricted gauge equation by itself.