Scope and the inference that needs justification

X-F constructs three fixed-width tree symbols and proves their direct background identity through ${\kappa }^2$ at classical order XF. It also states that its gravitational switch acts on sources, not on action densities. The latter distinction matters when a quantum retarded map is requested.

The proposed continuation contains three logically separate tasks:

A tree kernel's admission proves none of the later tasks by itself.

Local covariance of renormalized products is also different from comparison of two interacting splittings with the free input fields held fixed. The latter comparison requires an explicit quantum transport or perturbative-agreement statement. The additional normalization conditions in HW, and the separate background transport analysis in TTZ, illustrate this distinction. Neither paper is used as an automatic completion theorem for the present switched nonlocal observable.

We give complete proofs of the finite algebraic and classical compatibility statements below. Conditional quantum statements are labelled as such.