Proposition: Class-wide exclusion criterion

If there are $\xi ,\zeta \in {\mathfrak{g}}_W$ with ${\bar{\mathfrak{B}}}_{[\xi ,\zeta ]}\ne 0$, then no member of ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ satisfies all the representation, descent, and cancellation interfaces.

More directly, any parameter $\theta$ with ${\bar{\mathfrak{B}}}_{\theta }\ne 0$ excludes the target (eq:target) for every shell-descending rule in the class.

Proof. The first assertion combines Theorem thm:triviality with Lemma lem:cocycle. For the second, the left side of (eq:target) is zero on the shell by (eq:Y-zero), whereas its required right side is nonzero.

Likewise, ${\bar{\delta }}_{\theta }\bar{F}=0$ for every fixed compensating functional $F$. Adding one such functional cannot change the conclusion.