Proposition: The extension E 0

Use ${\mathsf{E}}_0$ in one deficient mode $n$, with $0<\nu ={\nu }_n<1$, and Friedrichs in the other modes. Then $B\ge 0$, and

This is a global distribution identity. The correction is supported in $t\ge {\rho }_0$, so this extension also has finite propagation speed. For $m=0$, the correction at its smooth points is

Proof. The homogeneous endpoint choice $b=0$ is diagonalised by the Bessel transform of order $-\nu >-1$, with regularised eigenfunctions $J_{-\nu }(kr)$. The corresponding half-line self-adjoint transform is part of the homogeneous inverse-square analysis DerezinskiRichard. Weber's heat integral replaces $I_{\nu }$ by $I_{-\nu }$. Now use

Watson,DLMF. The angular normalization gives the coefficient in (eq:E0). The heat integral converges superexponentially in $u$, and Laplace inversion as in Theorem thm:closed gives the wave identity. After compact time testing the $u$-range is bounded, so the correction is globally well defined. Its support is $t\ge {\rho }_0\ge d(x,x^{\prime })$. Positivity follows from Theorem thm:positivity.