Theorem: The free-shell class of the cubic kernel is obstructed

Fix $m\ge 0$, the frozen $g$ and ${\sigma }_{\epsilon }$, and F1 data $(f,{\varphi }_m)$. Let $u$ be the frozen kernel of ${\mathcal{C}}_3(f,g)$. Then

Equivalently: no $F\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ satisfies $F{|}_{{\mathcal{C}}_{\mathrm{sol}}}={\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}$, and ${\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}$ is not in the image of XI-E's admission map $Q_H$ for any choice of representative. Since F1 data exist for every frozen $g$ and every $m$, no construction admitting ${\mathcal{C}}_3(f,g)$ for all admissible $f$ exists in ${\mathcal{E}}_H$.

Proof. Suppose $v\in [u{]}_{\mathrm{sol}}$ satisfies (eq:mc). By Definition def:class, extended to complex solutions by bilinearity, $v$ and $u$ have the same pairing with ${{\gamma }_1}^{\lambda }\otimes {{\gamma }_2}^{\lambda }\otimes {\varphi }_m$ for the plane waves of Proposition prop:witness, and this pairing equals $⟨v_{{\varphi }_m},{{\gamma }_1}^{\lambda }\otimes {{\gamma }_2}^{\lambda }⟩$. The covectors $k_{a♭}=(1,\mp \mathbf{n})$ are null with $k_{a,0}=1>0$, hence both in ${\tilde{V}}_{+}$. Lemma lem:decay gives $O({\lambda }^{-1})$; Proposition prop:witness gives $\ge \lambda |{\Lambda }_f|-C$ with ${\Lambda }_f\ne 0$. Contradiction. The equivalent formulations are Lemma lem:reduction, and the existence of F1 data is XI-F's Theorem F1.