Characteristic-cone running from on-shell residues: Identifiability on the physical EFT quotient

We formulate a data-completeness test for extracting low-energy characteristic-cone running from flat-space on-shell renormalization-group residues. In four-dimensional, parity-even Einstein–scalar–Maxwell effective field theory through four derivatives and four external fields, Wilson directions are first placed on the physical EFT quotient and background propagation is represented by a traceless response on the full massless root sector. The response is identifiable from chosen amplitude coordinates precisely when the amplitude kernel lies in the response kernel. For the published opposite-helicity vector–scalar residue we find ${\beta }_{\mu }{C}_{\gamma \phi }^{(2)}=-55\mathrm{/}(96{\pi }^2)$ and, after an order-reduced frame translation, the contribution ${\beta }_{\mu }(r_T-r_{\gamma })=-55\mathrm{/}(48{\pi }^2{M}_{\mathrm{P}}^4)$. A finite operator registry gives explicit rank-two incompleteness tests: independent $F^4$ directions split photon polarizations on a magnetic background, while $\mathcal{C}FF$ produces an analytic splitting along a principal Weyl axis on a Petrov-I Einstein–scalar background. For the isolated scalar–photon contact, direct Schur reduction finds nonzero raw mixing but no first-order physical off-diagonal entry, by the Maxwell Ward identity. Thus the residue fixes one diagonal cone contribution but does not determine the term-complete characteristic response over the declared admissible coefficient domain.