Abstract

On the identical power-law Einstein–scalar background and original-rod prescription displayed in XX-B, we construct a local bundle-valued subtraction parametrix with the actual action-Hessian normalization. Its difference from a normalized Feynman inverse is smooth near the diagonal. We prove uniform dipole estimates for the original Newton inverse acting on compact-source retarded or advanced responses, and give a precise sufficient decreasing-source condition for analogous one-sided covariance estimates.

The supplied free Hadamard class does not select a physical state or impose an infrared condition. We exhibit, on the same coupled operator, a smooth finite-rank modification whose value on a nonzero gauge-invariant tensor observable is negative. This excludes a positive physical state with that exact covariance, without excluding positive representatives elsewhere in the class. A separate polynomial-solution modification shows that the class does not even ensure that the decaying rod inverse is defined.

We also compute an intrinsic obstruction to a frequently overstated tail requirement: the native two-rod commutator has a nonzero inverse-distance spatial separation term. Thus an inverse-square bound cannot be demanded of both ordered two-rod covariances without changing the canonical data. This obstruction does not remove the one-rod term in the quadratic lapse.

The smooth subsidiary contraction defect is retained and its exact correction problem is written explicitly. No positive full physical state, unconditional one-sided native covariance-tail theorem, or nonlinear lapse contraction is released. Accordingly, the requested Q2 package remains incomplete; its boundary is proved and not concealed by a renamed closure.