Proposition: Structure of the full-Hessian vertex

For the declared unintegrated density, there are unique coefficients ${L^{pq}}_{ij}(w;h)$ such that

The identity is an identity in independent field jets, before any integration by parts. Its properties are:

1. Each ${L^{pq}}_{ij}$ is linear, with constant rational coefficients, in residual jets of order exactly $2-|p|-|q|$.

2. Every nonzero monomial of the full Hessian has total derivative order two over $h,A,C$.

3. The slot symmetry is ${L^{pq}}_{ij}={L^{qp}}_{ji}$.

4. The constant-slot coefficient is

   $${L^{00}}_{ij}(w;h)=D^2{\ell }_3(h)[B_i,B_j](w)=L_{ij}(w;h),$$

   the constant-slot form retained by XVI-J.

5. The original vertex decomposition contains no derivative on ${\chi }_{\mathrm{I}}$.

These structural claims do not depend on a numerical coefficient list.

Proof. The atoms $u_i,b_i$ have field degree $i$ and derivative order zero. The atom ${\Gamma }_n$ has field degree $n$ and total derivative order one. Both $\partial {\Gamma }_n$ and the products in $R_n$ have field degree $n$ and derivative order two. Hence $r_n$ has field degree $n$, and each term of $r_3+b_1{r}_2+b_2{r}_1$ is cubic with total derivative order two.

The mixed variation of a cubic monomial replaces two field-jet factors by the corresponding $A$- and $C$-jets and leaves one residual $h$-jet. It preserves the total derivative count. The residual jet therefore has order $2-|p|-|q|$, proving the first two statements and the range in (eq:Lpq).

Uniqueness follows from independence of the symmetric field components and their multi-index jets. Symmetry of the mixed variation gives the third statement. Constant increments eliminate all terms with $|p|+|q|>0$, proving the fourth. The fifth follows because the switch multiplies the density and is not differentiated in this representation.