Exact finite algorithm

The supplied script uses a sparse polynomial ring over $\mathbb{Q}$, with independent symmetric jets of $h,A,C$ through order two and formal parameters $t,ϵ,\delta$. It substitutes

in the finite atoms (eq:Gamma)–(eq:ell3). Each product is reduced modulo

This reduction is legitimate for extracting the coefficient of $tϵ\delta$: multiplication cannot reduce a parameter exponent, and spatial total differentiation leaves the parameters constant.

For a generator representing ${\partial }^p{F}_{ab}$, with $|p|<2$, the total derivative rule is

The derivative is extended by the polynomial product rule. Only the derivatives of the Christoffel atoms are required, so no third field derivative is introduced.

The coefficient of $tϵ\delta$ is $D^2{\ell }_3(h)[A,C]$: cubic homogeneity makes it linear in the residual field. Sorting the resulting monomials by the $A$- and $C$-jet labels yields (eq:Lpq) and the coefficient list. No integration by parts changes the density or its jet coefficients.

The supplied output identifies SymPy 1.14.0 and exact $\mathbb{Q}$ arithmetic. It reports:

1. $7560$ Hessian monomials, each linear in one $h$-jet, one $A$-jet, and one $C$-jet, with total derivative order two.

2. $2988$ nonzero ordered coefficients ${L^{pq}}_{ij}$, with slot symmetry checked entrywise.

3. All $45$ allowed ordered derivative placements and $25$ unordered placement labels occur.

4. On $h_{\ast }=(w^1{)}^2\eta$, the constant-slot matrix agrees entrywise with the retained XVI-J table, and the specified mixed constant-slot value is six.

5. Its constant-slot dual-source contraction agrees with the XVI-J decorated tensor.

6. All $45$ background decorated tensors are nonzero as forms in a general formal source jet.

These are computationally verified statements documented by the supplied code and output, not new execution results claimed for this TeX revision.

The monomial and coefficient counts are:

The implementation of the first comparison uses xvi_common.ell3_of on two nonconstant $w^1$-dependent polynomial triples. For each chosen triple, agreement is an exact polynomial identity in $w^1$. The second comparison uses a separately written four-variable density implementation and two quadratic polynomial triples evaluated at a rational point.

These are finite exact test cases. They do not establish coefficientwise equality between independent implementations for every possible field jet. The supplied output explicitly makes that qualification, which is retained here.