Theorem: Existence and on-shell descent

Equation (eq:star) defines an associative unital graded algebra. All its polynomial contractions exist. The product descends to the free shell and has the causal relations (eq:antisym).

Proof. Every nonzero wavefront pair of $H$ has the same orientation: one endpoint is in one causal cone and the other in the opposite cone. Derivatives and constant field matrices do not alter this fact.

In a contraction, cancellation of the propagator covectors would require the contracted slots of the first kernel all to point in one cone, and those of the second all to point in the opposite cone. If every slot is contracted, this is excluded by (eq:mc). With uncontracted slots the same argument excludes the dangerous zero covector in a pushforward. It also excludes an output wavefront tuple wholly in either causal cone. Compact kernel supports make the integrations proper. Thus contraction gives another compact kernel satisfying (eq:mc).

Associativity follows from the exponential formula: on three factors the contraction operators ${\Gamma }_{12},{\Gamma }_{13},{\Gamma }_{23}$ have constant coefficients and commute with the graded signs already included. Both bracketings give their common exponential followed by multiplication.

For descent, the ranges of $H$ satisfy the free equations in both slots. On the reduced variables $(\gamma ,\varphi ,c,\bar{c})$, with $b=D\gamma$, all contraction directions are therefore tangent to the linear solution space. A polynomial vanishing on that space has all tangential derivatives zero there. Its contractions consequently vanish there as well. The kernel version follows by the same distributional contractions just established. Hence the vanishing ideal is a two-sided star ideal.

Finally, the graded commutator of linear generators is $\hslash$ times (eq:antisym).