What is evaluated

Fix a finite exterior algebra ${\Lambda }_N$ with a specified ordered basis. A coefficient extraction means a specified real linear map

for example the real or imaginary part of one basis coefficient. It is not a state, positivity functional, or ghost measurement.

There are two classes of classical report tasks:

1. a coefficient of the integrated classical master residual

   $${{\mathcal{R}}_{\mathrm{ME}}}^{[2]}=\frac{1}{2}({\mathcal{S}}^{[2]},{\mathcal{S}}^{[2]})$$

   evaluated on supplied compact field and antifield profiles supported in a common $A⋐U$;

2. a compactly smeared coefficient of

   $$s^2{\Phi }^A\text{or}({\mathcal{Q}}^{[2]}{)}^{2}F$$

   through order two, evaluated on supplied local profiles, optionally satisfying the free equations near the smearing support.

For (R1), one integrates the complete variational bracket, not an arbitrarily weighted representative of its divergence class. On compact evaluation profiles its divergence current is compact, so its integral is zero. Equivalently, one may insert a cutoff equal to one near the support of that current; no derivative of the cutoff then contributes.

For (R2), the phrase “free-shell projection” means evaluation of the already defined off-shell square on supplied free configurations. It does not mean that $s$ descends to the free quotient. Theorem thm:nondescent excludes that inference.