Evaluated kernel and commutator

Write

IX-E-2 supplies, on $t,s>0$ and $R>0$,

Away from the cone boundaries it is zero for $R<|\tau |$, equals $1/(32\pi R)$ for $|\tau |<R<u$, and is zero for $R>u$. The direct delta contribution is separate; the reflected cone is a jump, not another delta. The Fourier definition governs the spatial apex.

The same input supplies

These identities are not re-evaluated here IXE2.

The compact-detection lemma is also imported: a nonzero compactly supported distribution $fg\mathrm{Im}K$ in the open half-space gives a nonzero actual commutator on the unrestricted configuration space. No transfer to a constrained quotient is asserted.

The strongly separated witness class has exact zero commutator. The observation-spacelike control strictly inside

has, for ${\partial }_t\varphi =1$ and $h_{00}=\alpha >0$,

for the nonnegative tests specified in IX-E-2. That control is not an obstruction witness without both actual ordering certificates.