Weighted atomic integral

The passive $T$-cone is weighted by $e^{-T}$, and ${{\int }_0}^{\infty }T{e}^{-T}\mathrm{d}T=1$. The angular factor on that cone is the sphere probability. The dynamical integral is

with $c_{\delta }$ the translated atomic profile and $U=U_{\ast }$. The site is $U_{\ast }=(1,1,0,0)$, $V_{\ast }=0$, so $b-a$ is future null. Route B is a Gauss rule on the cap containing the mollifier support: $24\times 16\times 24$ nodes. Route A is a scrambled Sobol rule of length $2^{12}$ on the same cap and does not call Route B.

Absolute gaps are $5.45\times 10^{-3}$ and $3.36\times 10^{-2}$. Relative gaps are about 0.11\% and 0.14\%. A coarser Gauss rule, $12\times 8\times 12$, differs from a $20\times 12\times 16$ rule by 1.5\% relative. This is an independent quadrature check of a closed-form integrand. It is not the seven-row pairing: the vertex and the component adjoint are not in $I$.