Lemma: Auxiliary multiplication compatible with free coherence

Wordwise free coherence of ${\star }_c$ is compatible with a symmetric admitted auxiliary multiplication ${\mathsf{M}}_2$, with classical term $VO$, that differs from $V{\star }_{c}O$ on a declared first-stage pair satisfying

Proof. Consider the auxiliary map

viewed as a formal map with zero positive-order coefficients. It is symmetric, depends only on actual arguments, and has admitted values by classical carrier closure. Its definition leaves ${\star }_c$ and every inherited wordwise equality unchanged.

Choose nonzero tests $f,g$ with

Take

Let $F=\mathrm{supp}f$ and $G=\mathrm{supp}g$. Theorem thm:support places the metric support of $B_f$ in $P_F=J^{-}(F)\cap \{t\ge 0\}$ and its scalar support in $F$. Consequently,

Since $F\subset \{0<t<a\}$, every point of $P_F$ has time coordinate between zero and $a$. Thus

The same theorem gives

Causal future-directed curves cannot decrease the time coordinate. Therefore,

Both pair orderings belong to $D_{\mathrm{first}}$.

The locked reverse first stage gives

whereas

The nonzero tensor $g\otimes f$ has a nonzero image under the inherited injective map $q_R$. Therefore ${{\mathcal{Q}}^{\mathcal{R}}}_{g,f}\ne 0$, and the two expressions differ. This proves the asserted compatibility and discrepancy.