Implementation inventory

inventory{

1. Choose $e_I,e^I$ satisfying (eq:basis-duality); the concrete unordered-pair basis is one admissible choice.

2. Use $\eta =\mathrm{diag}(1,-1,-1,-1)$, the displayed curvature convention, and symmetrization with factor $1/2$.

3. Represent ordinary field derivatives by $X_{I,\gamma }$; identify commuting mixed derivatives before taking polynomial derivatives.

4. Retain ${\sigma }_{\epsilon },\partial {\sigma }_{\epsilon },{\partial }^2{\sigma }_{\epsilon }$ as fixed coefficients. Do not vary the switch or discard layer terms before localization.

5. Construct $J_3,C_2,M_2$, then the tensor source (eq:actualsource); finally convert its output by (eq:source-conversion).

6. Differentiate the resulting quadratic jet polynomial with respect to the two indicated jet coordinates and evaluate at zero. The $\alpha$-slot belongs to $G_J$ at $z$; the $\beta$-slot belongs to $h_I$ at $a$.

}