Theorem: Smooth native subtraction remainder

The construction (eq:Utransport)–(eq:Hnormalized) is a local subtraction parametrix for the actual operator $\mathcal{B}$. For any normalized native Feynman inverse $F_{\mathcal{B}}$ in the same distinguished class,

near the diagonal in $O\times O$. Likewise,

A sufficiently long finite truncation gives a remainder of any preassigned finite differentiability order.

Proof. Applying ${\mathcal{L}}_x$ to the inverse-distance and logarithmic terms gives the transport equations (eq:Utransport)–(eq:Vrecursion). They cancel the successive non-smooth coefficients. The all-order asymptotic summation makes the residual smooth. Multiplication by ${{\mathcal{A}}_y}^{-1}$ gives the correct delta normalization for $\mathcal{B}=\mathcal{A}\mathcal{L}$.

This is a distinguished Feynman parametrix. Uniqueness of distinguished parametrices modulo smooth kernels IS gives the first difference. The corresponding positive-frequency splitting has the same normalized singular coefficients, giving (eq:smoothW). Finite truncation leaves only sufficiently high powers of $\sigma$ times logarithms and their boundary values; their differentiability increases with the number of retained coefficients.