Tensor basis and propagated outputs

Use Minkowski spacetime

All distributions have complex-bilinear pairings and use flat coordinate densities unless a density conversion is printed.

Let $e_I$ be a basis of covariant symmetric tensors, with algebraic dual $e^I$:

The dual basis is not obtained merely by metric index raising.

One concrete basis uses $I=(i,j)$, $0\le i\le j\le 3$:

Thus off-diagonal component factors are fixed by the duality, not inserted independently.

Define

The notation ${\mathsf{A}}^{\mu \nu }{}_L$ means evaluation on $e_L$; similarly ${\mathsf{b}}_L={\mathsf{b}}^L$. These role labels do not instruct another fiber-metric multiplication.

For tests supported in $L⋐U$, put

Strong distributional convergence means uniform convergence on bounded test sets, which have common compact support and uniform bounds on all derivatives.