Proposition: Contractions forced by exact quantum elimination

Suppose an auxiliary Wick construction is defined on the displayed linear constraints and preserves their ordinary graph ideal as a quantum ideal, with no nonzero scalar identified with zero. Then its contractions must satisfy

as identities of their admitted smeared pairings.

For the sharp zero-Cauchy prescription and the fixed de Donder matrix,

Consequently an independent central $w$-sector cannot implement exact elimination while retaining the old $hh$ contractions.

Proof. For a linear field $h(b)$,

The first term belongs to the ordinary graph ideal. If that ideal is also a quantum ideal, the scalar contraction must vanish. This gives the first identity in (eq:forced-covariance). Reverse products give the transposed identity. Applying the same test to an auxiliary linear field gives the last identity.

The tensor contraction $C_x{H}_{hh}{C_y}^{\mathsf{T}}$ is a constant multiple of ${\eta }^{\mu \nu }\square {\Delta }_0^{+}$, hence zero before the two Cauchy solution maps are applied. This yields $H_{ww}=0$.

The mixed component is

In the present $+---$ convention its mode formula is

It is nonzero and has the reflected branch. Thus assigning zero mixed contractions to an independent central coordinate would violate the first identity.