What is missing at second dynamical order

The physical first-vertex correction has an additional factor $\kappa$. Its retained expansion is

Only $c_0,c_1$ enter the response part through this physical order.

For the classical second Euler coefficient write

The supplied classical response is

For $P_{m}u=0$, the handoff includes

These are classical statements.

The full classical second relational coefficient is

The last term is not supplied by the constructed single first-vertex map. Writing its classical formula does not define a normalized second-vertex map, an iterated causal product, or a second-order quantum identity.