Lemma: Computational verification of L.1

The script XVI-L-artifacts/source_full.py implements (eq:L-fc)–(eq:L-M2) atom by atom on independent jet coordinates (order $\le 2$) of $\gamma$, of the two slots $X,Y$, of $\sigma$, and of a scalar $\psi$, in exact rational arithmetic (sympy sparse polynomial ring over $\mathbb{Q}$); it exits with status $0$, and every check raises an error on failure. Its output XVI-L-artifacts/source_full_out.txt records:

Symbolic identities (valid for all smooth fields).

1. (S1)

   For all ten components, ${\mathcal{E}}_{\gamma }(\sigma {\ell }_3{)}^{\mu \nu }=\sigma {J_3}^{\mu \nu }+2{C_2}^{\mu \nu \alpha \beta }{\partial }_{\alpha }{\partial }_{\beta }\sigma -2{M_2}^{\mu \nu \rho }{\partial }_{\rho }\sigma$, as a polynomial identity in the independent $2$-jets of $\gamma$ and $\sigma$. Here ${\mathcal{E}}_{\gamma }$ is the Euler operator ${\sum }_a(-D{)}^a\partial /\partial ({\partial }^a{\gamma }_{\mu \nu })$ in the symmetric convention $\delta \int L=\int {\mathcal{E}}^{\mu \nu }\delta {\gamma }_{\mu \nu }$ (a factor $\frac{1}{2}$ for $\mu \ne \nu$), and ${\ell }_3$ is (eq:L-ell3). In particular ${\mathcal{E}}_{\gamma }({\ell }_3)=J_3$ of (eq:L-J3), with sign $+$ in this convention. The Euler computation never required a third-order jet.

2. (S2)

   At $\gamma =\psi \eta$: $C_1=C_2=0$, ${\Gamma }_2=-\psi {\Gamma }_1$, and both identities (eq:L-trace) hold.

3. (S3)

   ${\mathfrak{s}}_{C_2}[X,Y]$ equals XVI-K's routine xvi_common.polar_s$(X,Y,\partial \partial \sigma )$ identically in symbolic $X$, $Y$, $\partial \partial \sigma$; and on XVI-K's dual-slot tensors $X=-\mathcal{P}{B}^i$, $Y=-\mathcal{P}{B}^j$ (all $55$ pairs $i\le j$) it equals XVI-K's $\mathfrak{s}(-\mathcal{P}{B}^i,-\mathcal{P}{B}^j;S)$.

Coefficientwise structural checks.

1. (C1)

   Unpolarized: every monomial of ${\mathfrak{s}}_{C_2}$ (546 monomials), ${\mathfrak{s}}_{M_2}$ (1152), ${\mathfrak{s}}_{J_3}$ (1068) has $\gamma$-degree $2$, $\sigma$-jet order exactly $2$, $1$, $0$, and exactly $0$, $1$, $2$ derivatives on $\gamma$; no foreign generator occurs. Every monomial of ${\mathcal{J}}_{\mathrm{D}}$ has total derivative weight $2$. $J_3$ contains both monomial types $\gamma \partial \partial \gamma$ (534) and $\partial \gamma \partial \gamma$ (534), and no other.

2. (C2)

   The two polarization routes (the defining formula, and monomial polarization) agree for every summand and component, and ${\mathfrak{s}}_A[X,X]={\mathfrak{s}}_A(X)$.

3. (C3)

   Every monomial of ${\mathfrak{s}}_A[X,Y]$ is exactly linear in the $X$-jet and in the $Y$-jet; ${P^{ab}}_{A,ij}={P^{ba}}_{A,ji}$ entry by entry; $J_3$, $2C_2\partial \partial \sigma$, $-2M_2\partial \sigma$ and $R_1$, $R_2$ are symmetric in $(\mu ,\nu )$.

4. (C4)

   Nonzero ordered coefficients ${P^{ab,\mu \nu }}_{A,ij}$ ($\mu \le \nu$), by $(|a|,|b|)$: $C_2$: $(0,0)$: 688; $M_2$: $(1,0)$: 1152, $(0,1)$: 1152; $J_3$: $(2,0)$: 534, $(1,1)$: 996, $(0,2)$: 534. The ordered slot placements $(a,b)$ that occur are: $1$ for $C_2$; all $8$ with $|a|+|b|=1$ for $M_2$; all $36$ with $|a|+|b|=2$ for $J_3$. Each $P_A$ is a rational linear form in the $\sigma$-jet of the order of Proposition prop:jet(4) and nothing else. The full table (5056 entries) is written to XVI-L-artifacts/source_full_table.txt.

Finite exact test (not a coefficientwise equivalence).

1. (F1)

   The density ${\ell }_3$ of this script equals XVI-K's xvi_common.ell3_of on two random rational fields $\gamma (x^1)$ quadratic in $x^1$ (exact polynomial identity in $x^1$).

Proof. These are the printed lines of the output file; Appendix app:out reproduces them.