Theorem: Analytic direct subtraction at every regular point

At every $X_0$ with $r_0>0$, choose a sufficiently small Euclidean developing neighbourhood $V_E$ and parallel frame. Then

is jointly real-analytic on $V_E\times V_E$, including its diagonal, Hermitian, and a bisolution of the flat Euclidean Laplacian.

No condition such as $1/(2\alpha )\notin \mathbb{N}$ is required.

Proof. In developing Cartesian coordinates the two terms are fundamental distributions of the same local operator in each variable. Their delta sources cancel. The difference is therefore annihilated, distributionally, by ${\Delta }_X$ and ${\Delta }_{X^{\prime }}$, hence by the elliptic operator ${\Delta }_X+{\Delta }_{X^{\prime }}$ on the product. Elliptic regularity gives smoothness and analytic elliptic regularity gives joint real analyticity MorreyNirenberg,Hormander. Hermiticity follows from the self-adjoint heat kernel.

This treats the total distribution and never differentiates a moving image-sector indicator in isolation.