Theorem: Cubic symbol and one-line contraction

The functional $\mathcal{T}(f,g)$ is a continuous homogeneous cubic polynomial on smooth scalar configurations. Its polarized kernel is compact and obeys (eq:microcausal). Hence it belongs to the classical cubic symbol space ${{\mathcal{P}}_{\mathrm{mc}}}^{(3)}$. The construction requires no Hadamard covariance.

Proof. For smooth $\varphi$, the compact smooth stress source in (eq:metricresponse) has a unique smooth retarded response. Applying $L_{\epsilon }$ gives a smooth coordinate response. Multiplication by the smooth scalar derivative and integration against $f$ therefore defines a number. Smooth Cauchy continuity on fixed causal compacts proves continuity. Homogeneity is quadratic in the stress and linear in the remaining scalar factor.

For the distributional kernel, compose the coordinate kernel and the graviton inverse:

Before integrating $z$, each propagator has endpoint covectors summing to zero. The two factors share only $z$. A violation of the product criterion would force the unshared $x$ and $y$ covectors to vanish, and then force the shared covector to vanish as well. The product is therefore defined. Proper integration over the bounded causal intermediate region gives a distribution with opposite $x,y$ covectors.

Attach the local stress kernel at $y$, using independent scalar slots $u,v$, and the final scalar slot at $x$. The stress diagonal kernel has total covector sum zero. Each attachment shares only one slot with the preceding kernel, so the same product argument applies. Derivatives do not change the zero-sum property.

After the proper pushforwards, every external covector tuple in the cubic kernel satisfies $p_1+p_2+p_3=0$ in the fixed Cartesian trivialization. A nonzero tuple with all entries future causal or all entries past causal cannot satisfy this identity. This proves (eq:microcausal). Permuting and symmetrizing scalar slots preserves the conclusion.

The support cutoffs $f$ and $g$, and retarded propagation inside the finite slab, make all these pushforwards proper. The distributional operation criteria are used with their support requirements, not merely as formal kernel multiplication BDH.