Theorem: Contact-independent nonlocal response

For $p^2>a$,

This discontinuity is unchanged by every permitted contact. It is nonzero, so no permitted contact can make the response local or identically zero.

Proof. The advanced form reverses the boundary value, while the real local polynomial is unchanged. Lemma lem:dispersion gives (eq:absorptive). The positive densities and nonzero rest-frame projectors show nonvanishing.

A distribution supported only at zero has polynomial Fourier transform and hence no spectral discontinuity. Thus the nonzero discontinuity excludes locality and cannot be cancelled by the polynomial family.