Abstract

Phase I constructs the unprepared classical BV complex and its prepared and switch deformations through order $\kappa$, with reconciled odd-field antibracket couplings, the antifield rule $s{{\Phi }^{\star }}_A=(S,{{\Phi }^{\star }}_A)$, and the invariant baseline $S_{\mathrm{inv}}=S_D-V_{\sigma }-V_g$. Gate N-Ins uses the actual predecessor kernel $t_D$ of XVI-I, XVI-H, and XII-D as inherited by XVI-K. The plain divergence $D_{\mathrm{div}}$ is separated from the gauge-fixing divergence $D_{\mathrm{gf}}=D_{\mathrm{div}}\mathcal{P}$. The corrected root current is

not $E^A(D_{\mathrm{div}}\mathcal{P}\alpha )$. All values computed for the superseded vector are withdrawn.

For one massless special family, admissible smooth data give $H{|}_U=2m_0(e_0\otimes e_1+e_1\otimes e_0)$ and $C{|}_U=4m_0{p}^{\prime }(t-z)\mathrm{d}{x}^2$, with $m_0=\int S>0$, and hence $s_0{F}_t\ne 0$. A validated enclosure is $s_0{F}_t\in [3.3566,3.3637]\times 10^{-4}$. That nonvanishing is conditional on inherited reduction identities, re-checked pointwise and not re-proved, and on configuration-space admissibility. Lemma 8 is discharged. Route A is sufficient. Route B is not claimed discharged. Theorem 11 is a conditional obstruction of one regular, seed-preserving, antifield-linear completion of that one massless instance. It is not a universal completion. The certified small-mass interval is $0\le m<1.3715\times 10^{-4}$, with $\mathrm{Chi}\le 17843$. Arbitrary prescribed $m>0$ remains unresolved. N-Ins.4 remains unresolved as a universal antifield-linear completion. The prepared theory is not a BV complex. No ultraviolet completion, no loops, no graph closure, and no quantum coefficient are claimed. Numerical checks are corroboration, not certificates. No compilation is asserted. No symbolic execution is claimed.