O.2: the classical κ 2 master 2(S 0,S 2)+(S 1,S 1)=0 on S inv

{(a) Full ${\ell }_4$ — Constructed.} XVI-P's p23 stages, re-executed from the pinned zip, reproduce the ${\ell }_{2,3,4}$ exports byte-identically. The ${\ell }_4$ export (3360 terms) equals, term by term, an independent reconstruction from the $\Gamma \Gamma$ Lagrangian $\sqrt{-g}{g}^{mn}({{\Gamma }^a}_{mb}{{\Gamma }^b}_{na}-{{\Gamma }^a}_{mn}{{\Gamma }^b}_{ab})$. That reconstruction uses exact $ϵ$-grading, $g^{-1}$ and $\sqrt{-g}$ to second order in $H$, and a sparse ring over $\mathbb{Q}$. Its only normalisation, $-2$, is fixed by XVI-O's conformal probe and coincides with XVI-O's convention ${\ell }_4=-2[{ϵ}^4](\sqrt{-q}R)$, since $L_{\Gamma \Gamma }=\sqrt{-g}R+\partial B$. A one-term mutation is detected.

{(b) Explicit $O({\kappa }^2)$ antibracket expansion — Constructed.} With $S_{\mathrm{EH}}={\sum }_n{\kappa }^{n-2}\int {\ell }_n$, $s=s_0+\kappa s_1$ ($s_2=0$, derived), and $S_{af}\ni \int h^{\ast mn}(sh{)}_{mn}$, the field projection is

Its ${\kappa }^2$ coefficient is $2\int W_2$ with $W_2=\sum E_{mn}({\ell }_4)(s_{0}h{)}_{mn}+E_{mn}({\ell }_3)(s_{1}h{)}_{mn}$. This is the component form of $2(S_0,S_2)+(S_1,S_1)$. $W_2$ is linear in $c$, so $\int W_2=0$ for all $c$ iff $E_{c^{\mu }}(W_2)=0$. On an independent XVI-Q jet engine, $W_2$ has 15428 terms and $E_{c^{\mu }}(W_2)=0$ exactly for $\mu =\mathrm{0..3}$; the ${\kappa }^0$ and ${\kappa }^1$ regressions also vanish. Doubling one ${\ell }_4$ term, or dropping $s_1$, breaks the result.

(c) Ghost/antifield sector — Constructed (re-executed evidence). In XVI-P's exact P13 master (re-executed; content-identical to p23), every ${\kappa }^2$ cell vanishes: af0 ghost/gauge fixing (transformed), and af1 and af2 in both invariant and $\Psi$-transformed variables. $s_2$ is derived zero in all sectors, from the affine $g$, the linear Lie derivative, and the ghost law; it is not invented. This answers XVI-O's “absent under inherited $s$”. $s$ is nilpotent exactly. Seven mutations each produce residuals. Caveat: Q01 has no second, independent engine for this sector.

. O.2 at ${\kappa }^2$ is Constructed in XVI-Q; XVI-O's own record is unchanged (read-only). The scope is classical, $S_{\mathrm{inv}}$, $m=0$: not quantum, not the prepared $S_D$, not the insertion.