Proposition: Scope of the finite coefficient

Equation (eq:finite-coefficient) is a defined coefficient of the constructed finite-width free algebra, independent of its inverse-coordinate or integrated-kernel presentation. It does not constitute a positive state on the full indefinite gravitational algebra.

Proof. Each propagated smooth test is smooth on the compact observation region, and the integral is finite. The finite-width product theorem justifies the contractions. Actual-functional presentation equality and linearity give presentation independence.

The tensor $P$ is indefinite, so positivity of ${\omega }_H(F^{\ast }{\star }_{H}F)$ for arbitrary graviton polynomials has not been asserted and does not follow from scalar vacuum positivity.