Fixed-vector transverse angular coefficient $8\pi \mathrm{/}3$

Given: The unit vector $e$, the scalar $\lambda \mathrm{\ne }0$, and the transverse projection and angular integral in Logarithmic coherent infrared tail: finite energy, no Fock limit.

Logarithmic coherent infrared tail: finite energy, no Fock limit uses $P_T(n)=I_3-n{n}^{\mathrm{\top }}$ and $C_b={\int }_{S^2}\mathrm{\mid }b(n){\mathrm{\mid }}^{2}d\mathrm{\Omega }$ for a measurable transverse $b\in L^2(S^2;{\mathbb{C}}^3)$.

$e\in {\mathbb{R}}^3$ is a unit vector and $\lambda \in \mathbb{R}\setminus \{0\}$, as stated in the Example.

Agreement with the existing polar sphere measure: the Euclidean chart measure on $S^2$ equals polar surface measure and is invariant under orthogonal transformations.

Surface integration on compact C1 hypersurfaces: surface integration is the Euclidean chart integral with density $\sqrt{\det (D{X}^{T}DX)}$.

A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in ${\mathbb{R}}^n$: coordinate hyperplanes and singletons have Lebesgue measure zero, hence chart seams and poles have zero area when their coordinate parameter sets are null.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): assume the same Countable Choice carried by the coherent-tail and surface-measure suppliers; no stronger choice principle is used.