M2 (certificate P18\ D11\ M2.json)

M2 states that on $U={\mathbb{R}}^4\setminus \{0\}$ the inverse Fourier transform of ${P^{DR}}_{AB}$ equals $R_A{R}_B{{\Pi }^R}_{\mathrm{tr}}$, with ${{\Pi }^R}_{\mathrm{tr}}=(i{\kappa }^2/4)K{u^R}_{\mu }$, $K=\alpha D_2+\beta D_0$, $\alpha =A^2/60$, $\beta =A^2/18$ and $A=1/4{\pi }^2$.

• Route B (exact; primary). Write $P^{DR}={\kappa }^2{N}_{\epsilon }\hat{G}$ with $\hat{G}={\sum }_s{A}_s(d)S_s\mathrm{MB}$ and $\mathrm{MB}=1/\epsilon +2-L$. At $d\to 4$ its nonlocal part is $-(i{\kappa }^2/16{\pi }^2)L{\sum }_s{A}_s(4)S_s(p)$; everything else is polynomial. On the other side, $\mathcal{F}[u^R]=c_{\mathrm{FT}}\log (-p^2-i0)+$const, and $K\to \hat{K}(p)$ with ${\hat{P}}_{mn}=p_m{p}_n-{\eta }_{mn}{p}^2$. This operator is fourth order, so the sign convention of the transform drops out. M2 then holds iff, for each of the 100 component pairs, $X_{AB}={\sum }_s{A}_s(4)S_s{|}_{AB}=-4{\pi }^2{c}_{\mathrm{FT}}{A}^2{Y}_{AB}=-(i/4)Y_{AB}$, where $Y=R_A{R}_B({\hat{D}}_2/60+{\hat{D}}_0/18)$. This holds exactly on all 100 pairs. The difference then has a polynomial Fourier transform, so it is supported at $\{0\}$, and $\mu$ enters only through polynomials.

• {$c_{\mathrm{FT}}$, calibrated twice.} (a) From the library DR bubble, $\mathcal{F}[{D_F}^2]=-N_{\epsilon }\mathrm{MB}$ and ${D_F}^2=A^{2}u$. (b) As $-r$, with the XVI-O residue $r=-i{\pi }^2$. Both give $c_{\mathrm{FT}}=i{\pi }^2$.

• Route A (structural corroboration). M1 is re-run: the P integrand equals the Fourier convolution of the XVI-O product $T=VV(CC+CC)$ (100/100). The frozen XVI-O code, imported read-only, re-derives $\alpha ,\beta$ and verifies $T=Ku$ on $U$ for all 256 components. A regularised convolution agrees with the pointwise product on $U$ (cited: XVI-O lem:BV_rules_on_U, Constructed).

• Controls and mutations. D11 is transverse exactly in $d$; the FP and PV loop routes agree. Six mutations are each rejected: $\alpha \leftrightarrow \beta$, off-diagonal weight 1, flipped FT sign, doubled symmetry factor, D01 in place of D11, and $\beta =0$.

• Scope. The flag is qualified as D11_matching_verified_scope = “4D components”. M2 is shown on the 4D components only. The evanescent components are the 8 $\tilde{g}$-carrying structures, i.e.\ components with an index pair in the $\epsilon$-scalar directions. Their status is not_applicable_not_verified, because XVI-O is strictly 4D and has no counterpart. The evanescent structures of D11 have nonzero $L$-coefficients (8, recorded); they are $\epsilon$-scalar components outside the 4D statement.

• Consequence (M3/M4). M3 is now unconditional on the 4D components: $L_{\mathrm{match}}$ is supported at 0. It is transverse and homogeneous of degree four, so it lies in the 2-dimensional transverse h$\cdot$h subspace. Its coordinates need the XVI-O finite constant of $u^R$ and are not computed.