Proposition: Full-Hessian vertex within the C 2 sector

The selected off-diagonal two-edge distribution is

It contains the full Hessian of the declared density, but only the $C_2$ source. Its overall numerical factor is $2/2!=1$. It is invariant under constant changes of fiber basis.

Proof. Contract two copies of (eq:edgefiber) with the source polarization and the full jet Hessian. The vertex-end derivatives fall on the scalar edge factors at $w$. Lemma lem:sign produces the sign in (eq:apq); the remaining fiber contraction is (eq:Kpq).

Assumption ass:normalization supplies the insertion factor two and the single identical-edge factor $1/2!$. The change of the vertex does not alter either factor. Grouping the two equal ordered terms when $p\ne q$ gives (eq:ak).

For a constant basis change ${\tilde{B}}_i=B_j{A}_{ji}$, the dual vectors transform as ${\tilde{B}}^i=(A^{-1}{)}_{ij}{B}^j$, while each Hessian matrix transforms as ${\tilde{L}}^{pq}=A^{\mathsf{T}}{L}^{pq}A$. The two inverse matrices in the source slots cancel the two matrices in the Hessian. Equivalently, the contraction is defined intrinsically by the fiber tensor (eq:edgefiber).