What is verified

The program below uses symbolic rational arithmetic. It contains no floating-point constants.

It verifies the following independent paths:

1. The Hessian is computed from the finite curvature atoms at the residual jet $h=0$, $\partial h=0$, ${{\partial }_1}^{2}h=2\eta$. It is compared with the closed trace polynomial (eq:F).

2. The source is computed both through the four-index coefficient $C_2$ and through the matrix expression (eq:Qmatrix).

3. The decorated tensor is contracted both by direct double polarization and by differentiating the quadratic source after the dual-edge substitution.

The origin-jet calculation suffices for the entire background $h_{\ast }$ because Lemma lem:Llinear proves that the constant-slot Hessian depends only on its constant second derivative. The program is not an additional mathematical assumption: the preceding polynomial proofs establish the tables.

The listing requires Python 3 and SymPy. Its mathematical operations are rational sums, products, symbolic differentiation, and polynomial expansion. No execution of the listing is represented in this manuscript.