Both internal placements of K 5

A successful I3 construction must supply a map of the form

whose value on the reconstructed six-branch sum is the specified contraction with ${\mathcal{C}}_{\mathrm{cross}}$ and the mixed vertex Hessian, and which:

1. agrees with the canonical composition on its proved domain;

2. is defined on tests meeting $N$;

3. retains the prescribed grouped cross-lines and all tensor coefficients;

4. has a proved complete-graph wavefront bound appropriate to that composition.

The internal Fourier-amplitude module proves none of these requirements for multiplication by ${\mathcal{C}}_{\mathrm{cross}}$. Its estimates are estimates on the internal amplitude before this new operation.

On the inherited microlocally admissible domain $U_{\mathrm{can}}$, the reconstructed sum (eq:presum), composed with ${\mathcal{C}}_{\mathrm{cross}}$ and the specified vertex operators, equals the canonical grouped expression (eq:target).

No equality

follows from this representation.

Proof. Each reconstruction is an equality of already defined tree distributions. Restrict those equalities to the open domain where the subsequent products and pullbacks have been proved. The ordinary distributional operations there respect equality, giving the first assertion.

The internal reconstruction proves no new normal-set condition for that subsequent product, so it cannot identify a larger domain.

The copied opposite-covector audit prevents declaring the ordinary final product theorem automatically applicable at $N$. It is not retried here.

A complete-pair cancellation or a generalized composition could in principle provide (eq:jointmap). Neither is established by the exact internal pole decomposition.

I2 is Constructed for the named internal amplitude map. I3 has a defined common precomposition presentation and a canonical-domain representation.

The joint characteristic operator map (eq:jointmap) on tests meeting $N$ is not constructed. No complete-pair cancellation theorem or alternative composition theorem is supplied.

The ending for the grouped characteristic target is therefore Unresolved at I3, within G2/F2/E1/D2/J2. No incompatibility excluding every representation in Definition def:class is proved.

Proof. Theorem thm:I2 discharges the stated internal amplitude gate. Proposition prop:precomposition supplies the exact internal sum, and Proposition prop:canonical supplies its composition only on the inherited admissible domain.

Those maps do not act on a test crossing $N$ by the required characteristic composition. The uniform frequency estimates control the internal retarded pole, not its product with the grouped cross-lines. No theorem defining that latter operation has been proved. This is precisely the missing I3 requirement.