Theorem: Separation from every quadratic polynomial form

There are real compact $f$ and a normalized symmetric ${\chi }_3\in {C_c}^{\infty }({\mathbb{R}}^9)$, supported away from all three-creation transfers, such that

For every $ϵ>0$, the states

have equal expectations for every defined polynomial form of degree at most two, whereas

Proof. Choose $p_{\ast }\ne 0$, all three signs positive, and $p_1=p_2=p_3=p_{\ast }$. Every pair transfer is $2p_{\ast }\ne 0$. At time zero,

and the total divergence vanishes. Therefore

The ordered response at small positive $\tau$ is

All cyclic terms coincide at this point.

Take a sufficiently small even nonnegative spatial bump $b_0$ with ${\hat{b}}_0(3p_{\ast })>0$, and

where $g_0\ge 0$ is a nonzero smooth bump in $(1,2)$. For sufficiently small $h>0$, ${K_{f_h}}^{+++}(p_{\ast },p_{\ast },p_{\ast })\ne 0$. Fix this $f$. On a small symmetric neighborhood avoiding all transfers, take a nonnegative bump $\beta$ and define

Equation (eq:sector) gives

A degree-at-most-two polynomial cannot connect vacuum and three particles. Its expectations in ${\Psi }_{\pm }$ therefore agree. The displayed cubic difference follows by expansion and symmetry.