Proposition: General-field coefficient construction and grading

For every assembled switch-free coefficient ${\widehat{a^{A\mu \nu }}}_{\lambda }(y,w;h)$:

1. it is linear in the residual-metric jet of order $2-v$, where $v=|p|+|q|$;

2. it is linear in the preparation jet of order $2-s$, where $s=|a|+|b|$ ($s=0,1,2$ for $C_2,M_2,J_3$);

3. $N=s+v$, and every monomial is one preparation-jet symbol times one $h$-jet symbol, with a rational coefficient;

4. it is symmetric in $(\mu ,\nu )$;

5. it is consistent under edge interchange, $(A;a,b;p,q)\leftrightarrow (A;b,a;q,p)$;

6. no unintended generators occur, and the arithmetic is exact rational.

The supports satisfy ${{\Lambda }_{\mathrm{graph}}}^{\mathrm{background}}\subseteq {{\Lambda }_{\mathrm{graph}}}^{\mathrm{formal}}\subseteq {\Lambda }_{\mathrm{adm}}$ with cardinalities $235\le 255\le 255$ (sectorwise $C_2$$25/25/25$, $M_2$$84/84/84$, $J_3$$202/250/250$).

Proof. (1)–(3) and (6) are inherited from XVI-K, prop:hess, and XVI-L, prop:jet, through Definition def:aGeneral. (4) holds because $P_A$ is symmetric in $(\mu ,\nu )$. (5) holds because the defining sum is symmetric under simultaneous interchange of the two edges. The supports, and a coefficientwise confirmation of (1)–(6), are the reported computation of Lemma lem:M1comp.