Manuscript X-B@3 · X-B-v3.tex · section · 2d415d58

Product sheet and the off-shell insertion domain

Product sheet and the off-shell insertion domain · lines 554–593

{}1.12

Item	Declaration
Regulator	The same ramp $σ_{ε} = χ (t / ε)$ , fixed $ε > 0$ ; not a mollifier and no sharp limit
Field content	Only $ϕ$ is quantum; $h, k$ are classical and off shell
Observable	The bare coefficients $O^{(0)}, O^{(1)}, O^{(2)}$
Interaction	Only $V_{1, g}$ , with physical perturbative order $κ$ ; no gravitational vertex
Product	One map $R_{1} (V_{1, g}; F_{c})$ on specified off-shell linear insertion representatives
Quantum order	All positive- $ℏ$ corrections to this map vanish
Ward order	Completed coefficient at $κ^{1}$ only; no full quantum identity at $κ^{2}$
Gauge	Exactly the ramp algebra (eq:algebra)
Topology	Smooth background topology and compact-test topology; no $τ_{Σ}$
Normalization	$R_{1} (V_{1, g}; F_{f}) = ϕ (- L_{1, g} [h] E_{m}^{A} f) = \int f ρ_{1, g}$