Projected Scalar Green Functions and Schur Reduction of PT-Symmetric Quaternionic Principal-Symbol Deformations

We study the Green function of a real scalar field ϕ+ on a fixed real Lorentzian spacetime (M,g). Quaternionic data enter only through a fixed auxiliary two-component representation of the scalar principal-symbol deformation Gμν=gμνI+δGμν, which is not a spacetime metric. The grading Θ selects the retained component. Strict projection discards the complementary component, whereas exact Schur reduction retains the complementary block contribution before elimination. For δGHμν=Eμνe2, the faithful representation of e2 produces off-diagonal mixing BW and the exact retained operator Deff,+=D+++BD−−−1B. When the complementary block has a local separation scale M, the effective field theory (EFT) correction begins with δDeff,+=−M−2B2, whose frozen principal symbol is −[Eμνkμkν]2/M2. Under the stated local microscopic PT assumptions, this retained correction is PT-symmetric. It enters the standard scalar propagator, resolvent, and one-loop expansion. No quaternionic spacetime geometry, full interacting quaternionic quantum field theory, all-order renormalization theorem, or observational result is derived.