Assumption: Admissible one-point extensions

For each fixed $(\alpha ,\beta ,\xi )$, consider real distributional extensions $(F,T,J)$ of the displayed regular data satisfying:

1. Translation invariance along $\mathcal{W}$, covariance under longitudinal Lorentz transformations, and transverse rotations. The reference-lowered stress is symmetric.

2. Charge conjugation sends $\beta$ to $1-\beta$: $F,T$ are even and $J$ is odd. Transverse reflection reverses the meridian and therefore also sends $\beta$ to $1-\beta$. These requirements apply to the family of sectors.

3. The componentwise transverse scaling degrees are at most $2,4,3$ for $F,T,J^i$, respectively, and the extensions are almost homogeneous of logarithmic order at most one:

   $$(E+2{)}^{2}F=0,(E+4{)}^2{T}^{\mu }{}_{\nu }=0,(E+3{)}^2{J}^{\mu }=0.$$

   These are equations for the stated Cartesian component distributions.

4. The Ward representative is Definition def:Ward, and the required identities are

   $$\mathfrak{D}T=0,\mathfrak{d}J=0,\mathfrak{t}T=3q\Delta (nF).$$

For a coupling-dependent family, $F,J$ are required to be independent of $\xi$, and the improvement identity is required with this same $F$.