Abstract

Landmark XI-F proved that the frozen kernel $u$ of the cubic Einstein insertion ${\mathcal{C}}_3(f,g)={\mathcal{A}}_1[u_3;\varphi ]$ carries the timelike conormal $\pm \mathrm{d}t$ at the initial surface, and left open whether the free-shell class of ${\mathcal{C}}_3$ contains some other representative in XI-E's polynomial microcausal class ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$. We decide that question at fixed regulator width, with ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ and the frozen prescription unchanged.

Separation. For compact kernels on the ambient Minkowski space, vanishing on all triples of free solutions is equivalent to membership in the bulk equation ideal ${\square }_1{\mathcal{E}}^{\prime }+{\square }_2{\mathcal{E}}^{\prime }+P_{m,3}{\mathcal{E}}^{\prime }$, and the free-shell class is encoded faithfully by the trisolution ${\mathsf{W}}_u=({\mathsf{E}}_0\otimes {\mathsf{E}}_0\otimes {\mathsf{E}}_m)u$. The class meets ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ if and only if $\mathrm{WF}({\mathsf{W}}_u)$ avoids ${\tilde{V}}_{+}^3\cup {\tilde{V}}_{-}^3$. Pairings with exact null plane waves are class invariants.

Obstruction. For XI-F's admissible observation data, pairing with the pure-trace counter-propagating plane waves $\eta e^{i\lambda k_a\cdot y}$, $k_1+k_2=2\mathrm{d}t$, gives $-i\lambda {\Lambda }_f+O(1)$ with ${\Lambda }_f\ne 0$, whereas every compact kernel in ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ decays rapidly in that direction, because $(k_1,k_2)$ lies in ${\tilde{V}}_{+}^2$. Hence $[u{]}_{\mathrm{sol}}\cap {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}=\varnothing$: every distribution reproducing ${\mathcal{C}}_3(f,g)$ on ${\mathcal{C}}_{\mathrm{sol}}$ has a wavefront point with two co-oriented null covectors, one in each metric slot and zero in the scalar slot. These are the two halves of XI-F's $\pm \mathrm{d}t$. The leading coefficient is the Cauchy pairing $B_{\Sigma }$, computed explicitly on the pure-trace sector. Transverse (Einstein-shell) pairs are blind to the $00$-component by the Bianchi identity; the witness lies on the gauge-fixed shell and off the Einstein shell.

The cubic source is not ramped, the Cauchy surface is not moved, no interacting differential, retarded quantum product, loop, state, or sharp-width limit is constructed, and the empty Einstein jet is not repaired. The classical ledger $-I+I+0=0$ of XI-F is untouched.