Proposition: Finite linear observation records do not fix the source

Let finitely many linear observation-test records of smooth compact metric configurations be prescribed. There are two configurations with identical such records but different source functionals of $j=\square h$. In particular, an installed calibration table without a field-acquisition map cannot determine that source.

Proof. If there are $N$ records, choose $N+1$ nonzero scalar bumps $u_j$ with mutually disjoint compact supports in $V$. Use configurations with only $h_{11}$ nonzero and in the span of those bumps. The record map from this span to ${\mathbb{C}}^N$ has a nonzero kernel; let $h_{11}=u\ne 0$ be in it.

A nonzero compact smooth solution of $\square u=0$ cannot exist with zero data before its time support, by local uniqueness. Thus $\square u\ne 0$. The zero configuration and this configuration have the same records, but the source functional with scalar test $b=\overline{\square u}$ has difference

Real configurations and real tests may be chosen when the original records are real linear, by using the corresponding real finite-dimensional argument.