The metric is an input, not an unknown solved by this certificate

Every derivative, trace, conformal Killing equation, support comparison, and improvement formula above uses the declared $\eta$. The certificate does not determine a metric from arbitrary finite Ward values. It verifies identities relative to that supplied metric.

This does not deny that geometric data can be encoded in sufficiently rich physical observations under additional reconstruction hypotheses. No such inverse problem or uniqueness theorem is part of this paper.