The frozen Hadamard matrix

Put $h_{\mu }=-W_{\mu }$, where

These distributions are bisolutions and satisfy $h_{\mu }-{h_{\mu }}^{\mathsf{T}}=i{E}_{\mu }$. Their wavefront relation has one fixed orientation:

The relation includes the null-geodesic condition. The same orientation is used for every species. This is product data; no state or positivity condition is selected.

The full matrix is

All other blocks vanish; the ghost blocks include the vector identity. Slot differentiation includes the corresponding tensor index contraction.

For the translation-invariant frozen data,

These identities follow from ${\partial }_y=-{\partial }_x$, ${\mathcal{P}}^2=\mathrm{id}$, and $D\mathcal{R}=\frac{1}{2}\square$. A replacement matrix need not have zero $bb$ block.