Abstract

We distinguish restriction of classical evaluations from restriction compatible with the inherited quantum product. One nonempty open set $W$ is fixed before parameters are selected. On the complete gauge-fixed free shell, impose $D^{\ast }b=0$ on $W$. Under a uniform causal plateau hypothesis, XII-F's auxiliary remainder vanishes on this subclass, including for the full displayed second-order coefficient. This is an evaluation identity, not an action on a quantum quotient.

We then test the coefficientwise restriction kernel. The constraint smearing $L_q=\int q^{\mu \nu }(D^{\ast }b{)}_{\mu \nu }$ is an auxiliary-field functional off shell. Its commutator with a metric trace smearing must therefore use the inherited $b\gamma$ causal block. We compute that commutator and construct compact smearings with a strictly nonzero scalar coefficient for every nonempty $W$. Consequently the restriction kernel is not a two-sided ideal. A quotient killing all these constraints while retaining the unchanged brackets forces $\hslash 1=0$.

An admissible smooth rank-one change of ordering independently fails to preserve coefficientwise vanishing on the subclass. We also display the free differential of the coefficient, retaining its coordinate ramp and scalar switching terms, and distinguish tangency from closure. The copied XII-F witness remains a classical boundary of the claim, not a quantum anomaly. A reduced algebra and an integrated cancelling completion remain unresolved. No new quantum domain or retarded product is released.