M.7 Covariance and Ward-compatibility constraints

This section treats prescription covariance of an extension ${\mathfrak{R}}_{\mathrm{M}}={\mathfrak{R}}_{\mathrm{L}}+\Delta {\mathfrak{R}}_{\mathrm{L}}$ on the flat background. It is kept distinct from interacting-sector Ward/BV symmetry (M.8–M.9; Target II). Gate S-bv is certified (Theorem thm:wpA-Sbv), so the (S-bv) dependence of this section is discharged; M-C${}_{\mathrm{curv}}$ is Proved obstructed (graph-specific), broad M-C is not achieved (Corollary cor:wpB-MC), and M-D is not claimed. The audited M.7 revision repaired three defects of the previous M.7 computation (Remark rem:M7-audit).

-background Lorentz covariance conditions (LC0–LC3).

1. (LC0)

   Lie algebra $\mathfrak{so}(1,3)\subset \mathfrak{gl}(4)$, $X=(X^{\mu }{}_{\nu })$ acting on coordinates $z^{\mu }$, with $X^{\mathsf{T}}\eta +\eta X=0$; generators $\{J_{12},J_{13},J_{23},K_{01},K_{02},K_{03}\}$ ($K_{0i}$ symmetric, $J_{ij}$ antisymmetric as matrices).

2. (LC1)

   Coordinate polynomials. On functions of the difference variable, $(X\cdot f)(z)=-(Xz{)}^{\mu }{\partial }_{\mu }f(z)$. By Theorem thm:wpA-Sbv, ${\mathcal{H}}_{4+N}\cong {\mathrm{Poly}}_N$ via numerators ($Q$ is invariant), inducing matrices ${\rho }_N(X)$ on the selected basis $B_N$.

3. (LC2)

   Contact distributions (repaired). $W_N=\mathrm{span}\{{\partial }_{\gamma }\delta :|\gamma |\le N\}$, where ${\partial }_{\gamma }={{\partial }_0}^{{\gamma }_0}\cdots {{\partial }_3}^{{\gamma }_3}$ are lower-index coordinate derivatives in $z$ (the same ${\partial }^{\gamma }$ as in $\mathcal{G}(C)=\sum C\bar{a}{\partial }^{\gamma }\delta$ and in the collapse ${{\partial }_w}^{\gamma }$). $\mathfrak{so}(1,3)$ acts by pull-back, $⟨X\cdot T,f⟩:=-⟨T,X\cdot f⟩=⟨T,(Xz{)}^{\mu }{\partial }_{\mu }f⟩$; for $T={\sum }_{\nu }{c}_{\nu }{\partial }_{\nu }\delta$ this gives $X\cdot {\partial }_{\nu }\delta =X^{\mu }{}_{\nu }{\partial }_{\mu }\delta$, i.e.\ $c\mapsto Xc$. On coordinate polynomials $p={\sum }_{\nu }{c}_{\nu }{z}^{\nu }$ the action (LC1) is $c\mapsto -X^{\mathsf{T}}c$. These differ for boosts. The intertwiner is the metric identification $z^{\mu }\leftrightarrow {\partial }^{\mu }\delta ={\eta }^{\mu \nu }{\partial }_{\nu }\delta$ (index raising by $\eta$, using $\eta X\eta =-X^{\mathsf{T}}$), extended multiplicatively: a coordinate polynomial $q(z)$ corresponds to $q({\partial }^{\cdot })\delta$, so the coefficient of ${\partial }_{\gamma }\delta$ equals $(-1{)}^{{\gamma }_1+{\gamma }_2+{\gamma }_3}$ times the coefficient of $z^{\gamma }$. Writing $P=\mathrm{diag}((-1{)}^{{\gamma }_1+{\gamma }_2+{\gamma }_3})$, the action on $W_N$ in the basis ${\partial }_{\gamma }\delta$ is ${{\rho }^{\delta }}_N(X)=P{{\rho }^{\mathrm{poly}}}_N(X)P$.

4. (LC3)

   A contact change $\Delta {\mathfrak{R}}_{\mathrm{L}}(u)={\sum }_{|\gamma |\le N}{C}_{\gamma }(u){\partial }_{\gamma }\delta$ is infinitesimally covariant iff $C\circ {\rho }_N(X)={{\rho }^{\delta }}_N(X)\circ C$ for all $X$. Homogeneous solutions are the Lorentz-equivariant contact assignments. The inhomogeneous correction of a concrete obstruction ${\mathrm{Obs}}_L(X,u)=X_{\ast }{\mathfrak{R}}_{\mathrm{L}}(u)-{\mathfrak{R}}_{\mathrm{L}}(X\cdot u)$ requires an explicit $\varpi$ and is Unresolved (a Euclidean-radial bump is not Lorentz-invariant; a boost obstruction is expected).

A linear divergence identity of an Euler tensor is not a Ward identity.

{Lorentz action on $B_N$ and on contact distributions.}