Proposition: Products proved by the direct sheets, with their stated domains

The retrieved passages of XIV-H, XIV-I, XIV-J prove the following products, with the domains they state:

1. XIV-H Hkernel: the internal products $R(b-a)R(a-z)$ and $R(b-z)R(z-a)$ exist on all of $W$ as pullbacks, including at $a=z$. Internal; no cross-line.

2. XIV-H HH2: exact kernel maps for $K_8$, $K_9$ with regular conormal amplitudes on the retained separated-endpoint patch. Internal; a support restriction.

3. XIV-I II3: on $U_{\mathrm{can}}$, the reconstructed sum composed with ${\mathcal{C}}_{\mathrm{cross}}$ and the specified vertex operators equals the canonical grouped expression. The domain is described as “the open domain where the subsequent products and pullbacks have been proved”; no list of those products is in the passage.

4. XIV-J Jreference: ${\mathcal{C}}_{\mathrm{cross}}{}_D{K}_{\mathrm{supp}}$ exists on $U\setminus N_U$ with the normal form ${t^{\circ }}_D$ and the wavefront bound (eq:referenceWFref); a local reference for $K_{\mathrm{supp}}$, not for $K_{\mathrm{ret}}$. The relation of $U\setminus N_U$ to $U_{\mathrm{can}}$ is not stated.

Hence the list of cross-line products proved on $O$ with an operational description of $O$ is empty: (L1), (L2) are internal; (L3) describes $O$ by the products themselves; (L4) has an operational domain not related to $O$ in the sheets. XIV-E's phrase “complete specified graph products” is not used to fill the list.

Proof. The cited passages. For (L4), the bound recalled from the patch is