Operational DRED rules (p14r)

D11 matching (defined, not verified).

M0, baseline. $S_{\mathrm{inv}}$, not $S_D$. $a_1=\frac{1}{2}\tau \eta -h$ is XVI-O's vertex (lem:d11-prefactor: $\Pi =(i{\kappa }^2/4)T$).

M1, exact integrand identity (computed). For all 100 pairs $(A,B)$ of independent components,

where:

• ${\tilde{C}}_{ab}(k)=k_a{k}_{b}i/(k^2+i0)$ is the Fourier transform of $C=-\partial \partial D_F$;

• $V$ is XVI-O's own V_tensor, imported from code/matter_loop/d11_tensor.py;

• $R_A$ is the $\eta$-raising map $h^{\mu \nu }={\eta }^{\mu a}{\eta }^{\nu b}{h}_{ab}$, with orbit multiplicity 2 for off-diagonal $A$;

• $\mathcal{P}={\kappa }^2/2$ is P's prefactor, which carries the symmetry factor $\frac{1}{2}$.

The identity matches ${{\Gamma }^{(2)}}_1=\frac{1}{2}\int h^{\mu \nu }{\Pi }_{\mu \nu \rho \sigma }{h}^{\rho \sigma }$ (XVI-O) with P's ordered extraction, and it is bound to the exported D11 integrand by its sha256.

M2, nonlocal agreement on $U= R^4{0}$ (defined only). $⟨{\mathcal{F}}^{-1}[{P^{\mathrm{DR}}}_{AB}],f⟩=⟨R_A{R}_B{{\Pi }^R}_{\mathrm{tr}},f⟩$ for $f\in \mathcal{D}(U)$, with ${{\Pi }^R}_{\mathrm{tr}}=(i{\kappa }^2/4)\mathcal{K}{u^R}_{\mu }$, $\alpha =A^2/60$ and $\beta =A^2/18$ (lem:BV_rules_on_U, all 256 components). This needs the evaluated amplitude, which is p15+ work.

M3, local difference (conditional on M2). $L_{\mathrm{match}}={\mathcal{F}}^{-1}[P^{\mathrm{DR}}]-RR{{\Pi }^R}_{\mathrm{tr}}$. The degrees are as follows. The scalar extension ${u^R}_{\mu }$ has scaling degree 4. The kernel $\mathcal{K}{u}_{\mu }^R$ (with $\mathcal{K}$ of order 4) has scaling degree at most 8. A contact term has $\mathrm{sd}({\partial }^{\alpha }\delta )=4+|\alpha |$. Kernel scaling degree at most eight; degree of divergence at most four; hence the local ambiguity has derivative order at most four.

If M2 holds, then $L_{\mathrm{match}}$ is supported at $\{0\}$ and lies in the bounded contact space of Lorentz-covariant even $hh$ structures of derivative order $\le 4$. This space is enumerated: order 0 (${\mu }^4$) has dimension 2 (transverse 0); order 2 (${\mu }^2{p}^2$) has dimension 4 (transverse 1, the linearised Einstein–Hilbert structure); order 4 has dimension 5 (transverse 2). Odd orders cannot occur with four indices built from $p$ and $\eta$.