Selected lifts versus a universal operation on classes

The constructed value has a class in ${\mathcal{E}}_H$. This does not establish a binary map on arbitrary representatives of the quotient. A simple equation-functional check explains why.

For a compact scalar test $a$, let $E_a(\Phi )=\int a{P}_m\varphi$. It vanishes on ${\mathcal{C}}_{\mathrm{sol}}$. The tree response to this alternative observable lift is

Indeed $P_m{E}_m^{\mathrm{R}}=\mathrm{id}$ after transposition on the compact test. The right side need not vanish on the free shell.

Thus retarded contraction is not automatically invariant under adding free equation functionals to an entry. Here the prescribed lifts are frozen, so (eq:representative-test) does not invalidate Theorem thm:tree. It prevents the stronger, unproved claim that we have constructed a representative-independent operation on all of ${\mathcal{E}}_H$. No new ideal is introduced to hide this distinction.