Factorial conventions and an explicit implementation dictionary

The convention

is used throughout. For a general second-order field jet,

Thus the second-order field data $k,\psi$ in the full lapse are not additional quadratic free-field factors.

The classical implementation dictionary is:

1. Evaluate $a,H,v,\dot{v},N_0,{\dot{N}}_0,{\ddot{N}}_0$ from (eq:background)–(eq:vH).

2. Compute $u_1,u_2$ from (eq:u).

3. Compute $q_1,q_2$ from (eq:q1)–(eq:q2). For the quadratic lapse, only $q_1$ is needed in its spatial rod term.

4. Form $Q_1,Q_2,b_1,b_2,A_1,A_2,P_1,P_2$ from (eq:b)–(eq:P2).

5. Apply the fixed decaying inverse (eq:Coulomb) to obtain $Y_1,Y_2$ using (eq:Y).

6. Raise spacetime indices with ${g_0}^{00}=-1$, ${g_0}^{0i}=0$, ${g_0}^{ij}=a^{-2}{\delta }^{ij}$. Evaluate (eq:W1)–(eq:N).

7. Assemble $n_1,n_2$ using (eq:n1)–(eq:n2).

8. Multiply by the fixed background test density and integrate. No configuration-dependent volume factor is added to (eq:observable).

This dictionary determines the classical coefficients without a scalar-vector-tensor decomposition or unspecified component maps.