Scope and the named construction

This manuscript begins where XI-A stopped: with a signed quadratic operator rather than a graviton two-point function. The construction is performed at the free background $(\eta ,0)$. A prescribed nonzero free scalar used in a diagnostic is an input to a coupling expansion, not a replacement background for the free Hessian.

There are three different objects:

They must not be identified merely because they involve derivatives of vector fields.

The polynomial algebra constructed below uses compactly smeared linear fields and their finite products. It has causal commutation relations, but no chosen Hadamard covariance or state. In particular, it is not the Wick algebra used in VII-C. This restricted algebraic construction suffices for the free multiplet and does not presume admission of a nonlocal relational composite.

The standard distinction between free gauge fixing and interacting BV quantization is respected BFR. We construct the former, including its free differential, without supplying an interacting master equation.