Proposition: Complete blind identity

At ${\beta }_0$ and its normalized version,

The complete response vanishes there for every $d,\theta$.

Proof. The spatial reflection

preserves $\eta$, fixes $k_1,k_2,t_0,A$, interchanges $k_3,k_4$, and preserves $B$. It negates the blind gauge vector. After reflecting, exchange the complete data of legs three and four. Their species and physical tensors agree. The original label is recovered with $G$ negated.

Every raw contact is reflection invariant. Its differentiated response is symmetric under simultaneous external permutation. The Einstein vertices and propagator have the same covariance, and the complete tree contains all three exchanges. Each scalar response therefore equals its own negative. Multilinearity gives zero. The same argument survives polarization rescaling.