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

Abstract

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 $\beta_\mu C_{\gamma\phi}^{(2)}=-55/(96\pi^2)$ and, after an order-reduced frame translation, the contribution $\beta_\mu(r_T-r_\gamma)=-55/(48\pi^2M_{\rm P}^4)$. A finite operator registry gives explicit rank-two incompleteness tests: independent $F^4$ directions split photon polarizations on a magnetic background, while $\mathcal CFF$ 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.