Manuscript XVI-P@p23 · XVI-P.tex · section · 9b0037a7

Gate 1: mixed scalar Noether (retained)

Massless scalar matter-loop subsector (o70-sync) › Gate 1: mixed scalar Noether (retained) · lines 1667–1685

Coupling-stripped $S_{ϕ, i} = \frac{1}{2} \int a_{i}^{μν} \partial_{μ} ϕ \partial_{ν} ϕ$ ( $a_{0} = η$ ). With BRST ghost normalization $a = \frac{1}{2}$ (distinct from D11 coefficients $α, β$ ) and $s q = κa L_{c} q$ , the densitized identities

s_{0} a_{1} = a D_{c} η, s_{0} a_{2} + s_{1} a_{1} = a D_{c} a_{1}

are verified by jet-polynomial substitution (executable). The boxed density identity

s_{0} L_{2} + s_{1} L_{1} = a \partial_{ρ} (c^{ρ} L_{1})

follows and integrates under compact support. This is the mixed metric–scalar Noether cancellation (Gate 1), not an independent scalar BV master equation. Separate: (i) invariant baseline $S_{inv}$ ; (ii) prepared-action defect $(S_{D}, S_{D})$ ; (iii) insertion-class obstruction (XVI-N thm:massless-obstruction).

Displayed formulas in this passage

unlabeled · line 1672
$s_{0} a_{1} = a D_{c} η, s_{0} a_{2} + s_{1} a_{1} = a D_{c} a_{1}$
unlabeled · line 1677
$s_{0} L_{2} + s_{1} L_{1} = a \partial_{ρ} (c^{ρ} L_{1})$