Proposition: Exact bilinear differential operator

For smooth tensor directions $A,C$,

The coefficients satisfy

whenever they are nonzero. They are linear differential expressions in the remaining field $h$. For such a term, the derivative order on that remaining factor is at most $2-|a|-|b|$. Moreover,

No integration by parts is made in this definition.

Proof. The chain rule in the independent jet coordinates gives (eq:Hessian). Equality of mixed partial derivatives gives the symmetry.

Differentiating a cubic jet monomial removes two field factors. The remaining coefficient is linear in one field jet. The original total derivative order was two, so the derivatives on the two varied factors have sum at most two, and the remaining factor has the stated residual order.

The interaction switch is fixed under field differentiation. This proves the last formula without derivatives of ${\chi }_{\mathrm{I}}$.