Physical membership is a substantial conclusion

In the supplied statement, admitting $L$ to the original physical field algebra requires sharply localized reconstructions, all finite mixed products, a common invariant domain containing ${\mathcal{D}}_{\mathrm{old}}$, and compatible affiliation to the original net.

A scalar distribution satisfying $\square L=T^{\mu }{}_{\mu }$ in some larger distribution space would not establish those requirements. A formal solution obtained by a Green operator need not be a local physical field.

The reported proof route uses wedge modular data, half-sided modular inclusions, control of a critical scalar tail, cone modular identification, scalar-wave extraction, and mixed-product reconstruction Dossier. Those operations are recorded here, not reconstructed or independently validated.