Proposition: D4: the simultaneous zero-contact prescription is incompatible

Under Assumption ass:source, the additional prescription ${H^{\mathrm{TO}}}_{jh}=0$ is incompatible with the stated source presentation and differential identity.

Proof. Differential compatibility gives

In the scalar case, choose a nonzero real compact test $q$ and pair with $q\otimes q$. The result is

For a bundle-valued identity, select the nonzero component of the declared diagonal identity and the corresponding bundle test. The zero-kernel prescription gives zero on the same test, which is a contradiction.