Theorem: Polynomial product

Every contraction is defined. Its support is compact, and its positive-degree kernel satisfies (eq:microcausal). A complete contraction is a scalar. The product is associative, unital, even, and ghost-number preserving. It preserves the regular subalgebra and extends coefficientwise to formal series.

Proof. A forbidden product cancellation would require zero covectors in all unpaired variables and $q\in {V_{-\sigma }}^k$, $q^{\prime }=-q\in {V_{\sigma }}^k$ in the paired variables. The corresponding nonzero input wavefront tuple would lie wholly in a forbidden causal cone, contrary to (eq:microcausal). Hence the distribution product exists.

The projection is proper on the compact support of $u\otimes v$. Suppose a nonzero output tuple were wholly future causal. If $\sigma =+$, the second input tuple, consisting of its future external entries and $q^{\prime }$, would be wholly future causal. It must therefore be zero. This forces $q^{\prime }=q=0$ and the second external tuple to vanish. The first tuple is now wholly future causal and must also vanish, a contradiction. For $\sigma =-$, start with the first input. Past-causal output tuples are excluded by the reversed argument. Symmetrization preserves the exclusion. When no variables remain, proper pushforward gives a scalar; there is no degree-zero wavefront claim.

On three factors the even contraction operators ${\Gamma }_{12},{\Gamma }_{13},{\Gamma }_{23}$ commute: their coefficients are independent of the configuration variables and the graded signs are included. Both bracketings equal

The required multi-contractions satisfy the same wavefront cancellation argument, so this identity involves defined distributions, not merely formal symbols. Constants give the unit. Finite-rank kernels contract to finite-rank kernels. The grading assertions follow from Definition def:admissible. For a fixed formal coefficient only finitely many orders of contraction and input coefficients contribute.