B3: a response functional is not a test function

There is also a type issue to resolve in a future B3 statement. For the original quadratic insertion ${\mathcal{B}}_f$, its metric derivative at a smooth scalar configuration is a linear covector in the metric slot, with scalar field degree one. A cubic vertex has a quadratic Euler source in the metric.

Their single rooted contraction is therefore a scalar-valued functional of field degree $(2,1)$, not an element of ${C_c}^{\infty }(M)$. This agrees with the distinction between classical response functionals and their functional derivative kernels FR.

A smooth compact test can instead be an intermediate rooted coefficient after fixing the smooth fields and transposing the propagator, possibly with source localization. Those are different types. A complete B3 theorem must state:

• the domain of fields and external tests;

• whether its codomain is a test space, a polynomial functional space, or a continuous dual pairing;

• which intermediate coefficient is claimed compact;

• the exact support statement for that coefficient.

This observation does not prove that the eventual map leaves a test space. The map and test-space identification are not yet fixed. Under the B0 stop rule, no new cubic retarded formula is supplied to fill that gap.