The distinct conditional quantum expansion

For independent smooth fields $h,\phi$, set $v_0[h]=v[h,0]$. The bare geometric coefficients are

Linearity in $j$ of (eq:coordinates) gives

Substituting the solution expansion in the bare expression therefore reproduces Theorem thm:descent. Round and square superscripts denote different expansions.

Suppose, conditionally, that a chosen quantum construction has compatible retarded products $R_1,R_2$, interaction $\kappa V_1+{\kappa }^2{V}_2$, and the convention

Then the coefficient at ${\kappa }^2$ is algebraically

This follows simply by collecting coupling powers.

Equation (eq:fourterms) does not construct any of its renormalized products. In particular, the last term requires a two-vertex and three-insertion construction; the second contains the nonlocal coordinate kernel. Their existence is not implied by an external smooth-metric response. The retarded-product framework DF does not discharge these new domain and extension obligations by notation alone.

Neither ${{\mathcal{O}}_f}^{[2]}$ nor (eq:fourterms) is identified with V's separately prescribed Newton–wave cubic form CheungV. No identity with the inherited characteristic quantum coefficient is asserted.