The contraction count

For a bosonic symmetric Hessian, only the corresponding symmetric part of a two-point matrix contributes to the complete scalar contraction. Whether one is using a Wick contraction, a state expectation, or a Feynman time-ordering must nevertheless be specified. These are not interchangeable normalization conventions.

Equation (eq:prospective-contraction) identifies the first free scalar contraction to be studied. It is not an evaluated quantum family, because $H$ has not been chosen. It does not enumerate interacting diagrams, and a field-degree count alone does not justify calling it the complete one-loop gravity observable.