Ordering data and sign discipline

Use a free scalar $\mathbf{\varphi }$ satisfying $P_m\mathbf{\varphi }=0$ and a free tensor $\mathbf{r}$ satisfying ${\mathsf{P}}_{\mathrm{gr}}\mathbf{r}=0$. These are not the classical coefficients $h,k$.

Fix scalar Hadamard bisolutions $H_m,H_0$ with one common microlocal orientation, and use zero cross blocks and the stipulated product-kernel normalization

Multiplication by a constant real matrix does not reverse a wavefront orientation. The product and commutators are defined by this frozen matrix:

Only finitely many terms occur for polynomials.

The Green inverses remain (eq:Delta). The action Hessian, equation operator, causal antisymmetric kernel, and product kernel are not interchanged by a sign convention. In particular, no additional incompatible antisymmetric-part condition is imposed on (eq:Wickmatrix). An identification with a separately signed Peierls convention requires its own convention dictionary; it is not needed for the tree proof.

This explicit choice is an algebraic free carrier, not a positive physical graviton state. No strong de Donder constraint quotient is added. The standard free microcausal Wick construction is used with its actual wavefront hypotheses FR,Hormander.

For $n\ge 2$, define

Covectors are identified by the flat completion coordinates. A compact kernel with wavefront in $Z_n$ satisfies the microcausal condition, because co-oriented causal covectors cannot have zero sum unless all are zero.

Let $\mathcal{W}$ be the resulting microcausal polynomial carrier, and $\mathcal{J}$ its closed dynamical equation ideal. It is generated by scalar $P_m$-images and graviton ${\mathsf{P}}_{\mathrm{gr}}$-images, with closure on the admissible compact microlocal kernel stages. The bisolution property makes these dynamical relations compatible with ${\star }_H$.