Corollary: The evaluation identity cannot be promoted

The identity ${\Delta }_{\xi }\mathcal{O}{|}_{{{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}}=0$ of (eq:ledger) cannot be realized as an identity in a quotient of ${\mathcal{E}}_H$ by the classes of functionals vanishing on ${\mathcal{C}}_W$, compatibly with the inherited brackets and a nonzero formal unit. The auxiliary pairing itself belongs to ${\mathcal{N}}_W\setminus {\mathcal{N}}_{\mathrm{sol}}$: it vanishes on ${\mathcal{C}}_W$ and has nonzero class in ${\mathcal{E}}_H$ by the copied witness. Its vanishing is an evaluation on ${{\mathcal{C}}_{\mathcal{W}}}^{\mathrm{bos}}$, and remains one.

Proof. Such a promotion is a unital homomorphism whose kernel contains $({\mathcal{N}}_W/{\mathcal{N}}_{\mathrm{sol}})[[\hslash ,\kappa ]]$, hence every ${\bar{L}}_q$; Theorem thm:quotient forces $\hslash 1=0$. Membership of ${\mathfrak{B}}_{\xi }$ in ${\mathcal{N}}_W$ is the vanishing on ${\mathcal{C}}_W$ in (eq:ledger); non-membership in ${\mathcal{N}}_{\mathrm{sol}}$ is the value $-8I\ne 0$ at $u_{\ast }\in {\mathcal{C}}_{\mathrm{sol}}$.