Theorem: Theorem 11, conditional massless obstruction

Assume the inherited seed, the reduction identities as re-checked and not re-proved, Route A, the plateau and root-cutoff requirements, Lemma lem:support, and Lemma lem:field-only. For the specified massless special-family instance of Proposition prop:nonvanishing, no regular, seed-preserving, antifield-linear completion $I_t=F_t+A_0+\kappa A_1$ satisfies $Q_{\mathrm{inv}}{I}_t=O({\kappa }^2)$.

Proof. On the witness, with $p^{\prime }>0$ on $\mathrm{supp}\xi$ and $0\le {\xi }^2\not\equiv 0$, equation (eq:s0Ft) gives $s_0{F}_t>0$. Closure at field-only degree and order ${\kappa }^0$ requires $s_0{F}_t+P_0{Q}_0{A}_0=0$. Lemma lem:field-only gives $P_0{Q}_0{A}_0=0$ on the free shell, so $s_0{F}_t=0$, a contradiction. The higher term $A_1$ cannot repair a contradiction already present at order ${\kappa }^0$.