Theorem: Vanishing on the entire witness class

For every $f,g$ satisfying (eq:strong-separation),

The first equality is distributional and the second is an identity of actual functionals on every allowed configuration.

Proof. On $\mathrm{supp}f\times \mathrm{supp}g$,

Hence neither direct-cone delta meets the observation support. Moreover,

there, so the cosine contributions cancel.

The inequalities have a positive uniform margin on the compact observation support. Even if a time support meets $\Sigma$, there is an ambient open neighbourhood on which

On that neighbourhood the global Fourier identity gives $\mathrm{Im}K=0$. Multiplication by $f(x)g(y)$ therefore gives the zero distribution, with no boundary ambiguity. Equation (eq:commutator) then vanishes for every smooth configuration.