Compatible Hadamard product data

For a concrete normal-ordering presentation, choose flat scalar product kernels $H_{\mu }$ of a common Hadamard orientation, normalized by

No positivity is required. A compatible translation-invariant multiplet is

All unspecified blocks vanish.

The blocks are bisolutions, their graded antisymmetric parts are $i\Delta$, and

The reversed identity follows by slot interchange. Thus the free differential is also a derivation of the compatible polynomial Hadamard product.

For compact microcausal polynomial kernels this product is the finite graded contraction expansion; the wavefront criterion is unchanged by the finite field matrix and Koszul signs FR.

The notation in (eq:Hadamard) distinguishes the scalar equation kernel from the scalar action-Hessian kernel. A prefactor alone does not reverse a wavefront orientation. Consequently no sign change from $-4$ to $+4$ is justified merely by comparing two differently normalized scalar symbols.

If inherited Hadamard data are not translation invariant, the auxiliary blocks must be defined by their actual slot derivatives and the compatible presentation transported by the smooth ordering isomorphism. An unchanged component formula is not silently asserted in a different ordering coordinate.