The fixed regularization

Fix the Euclidean quadratic form $|p{|}_{\mathrm{E}}^2={{\sum }_{\mu =0}}^3{p_{\mu }}^2$ in the given Minkowski coordinate chart. It is auxiliary regulator data, not a change of spacetime metric.

For $\tau >0$ define

The parameter $\tau$ is distinct from the fixed ramp width $\epsilon$. Only $\tau$ is sent to zero.

Every line uses the same $\tau$. Smooth external weights are not regularized. Differentiation commutes with $S_{\tau }$, so the inherited derivative placement is preserved.

The complete time-ordered graph factor is

with finite tensor sums taken exactly as in the selected pair. The grouping uses the full time-ordered lines. It does not expand the desired difference into separately undefined mixed products of $H$ and ${G_{\mathrm{act}}}^{\mathrm{A}}$.

For fixed $\tau$, (eq:A-reg) is a smooth function. The attempted off-collision prescription is its weighted distributional limit on ${X_{\Gamma }}^{\circ }$, followed by the collision extensions demanded at E2 and E3. Existence on all of ${X_{\Gamma }}^{\circ }$ is an obligation, not a definition by assertion.