Theorem: Simultaneous Ward-compatible extension

The distributions (eq:Fhat)–(eq:Jhat) are real extensions of the regular data (eq:regularstress)–(eq:regularcurrent). They have transverse scaling degrees at most $2,4,3$, respectively. They satisfy

The reference-lowered stress ${\bar{g}}_{\lambda \mu }({\hat{T}}_L{)}^{\mu }{}_{\nu }$ is symmetric.

Replacing $U_L$ by ${U_{L,\epsilon }}^{\rho }$ gives strongly convergent regulated extensions. The errors are bounded by $C{\epsilon }^2{p}_2$, $C{\epsilon }^2{p}_4$, and $C{\epsilon }^2{p}_3$ for field square, stress, and current.

Proof. On $r>0$,

This proves exact regular-part matching.

All derivatives in (eq:That)–(eq:Jhat) are derivatives of a defined distribution. Since

conservation follows without a boundary-flux omission. The mixed trace is

The changes of $a/4$ and $b/2$ under a change from conformal coupling are $nqs$ and $-nqs$, respectively, giving (eq:extendedimprovement). Symmetry is immediate from the block expressions. The scaling-degree and strong-convergence claims follow from Theorem thm:U.