Equation ideal and the meaning of a witness

Let $\mathcal{W}$ be the resulting microcausal polynomial Wick carrier. Use its dynamical equation ideal $\mathcal{J}$: the two-sided ideal of admissible slotwise equation images generated by $P_m$ in scalar slots and ${\mathsf{P}}_{\mathrm{gr}}$ in graviton slots, with the usual admissible kernel completion. The Wick matrix is a bisolution.