Corollary: Smooth compact non-determination

The witness map has nontrivial kernel even among ${C_c}^{\infty }$ kernels, all of which satisfy eq:microcausal.

Proof. Replace both deltas in eq:invisiblekernel by the same nonzero nonnegative $f\in {C_c}^{\infty }(M)$. The resulting nonzero smooth compact kernel is still annihilated by every pure-trace witness. Its wavefront set is empty.