Supplied-data audit and decision

The logical status is now precise.

The copied adjoint specifies how an already determined $b={\mathsf{E}}_0^{\ast }a$ enters the derivative. It does not give $b{|}_{\Sigma }$. The copied upper bound likewise does not give that trace. Finally, a mixed covariance formula is not, without a stated covariance-to-Green-operator identification, a formula for ${\mathsf{E}}_0^{\ast }$ on every source $-f\mathrm{d}\varphi$.

The flat realization proves that the proposed universal solid-ball assertion is not a safe substitute for this missing information. It also proves that solid-ball support is unnecessary: the null-sphere certificate is sufficient.