Theorem: Exact prescription compatibility

The subtraction, positivity tests, and tail constructions above do not alter:

1. the original slice-elliptic equation (eq:originalrods);

2. its decaying spatial boundary condition on the domains where that condition is proved;

3. the clock calibration or the matter-scalar lapse normalization;

4. the distinction between a bulk causal inverse and the spatial Newton inverse.

Theorems thm:dipole and thm:GreenTail prove compatibility on their actual domains. Theorem thm:IRfailure identifies contractions for which that compatibility cannot be extended without an additional infrared choice.

Proof. Every rod expression is obtained from $\Delta {\mathsf{G}}_{\mathrm{sp}}=\mathrm{id}$ and the unchanged source $\mathcal{C}$ derived from the induced metric. No lapse multiplier is introduced. Uniqueness in Theorem thm:dipole enforces the same decaying boundary condition.

The local parametrix acts on the propagating field operator $\mathcal{B}$, not on the rod equation. The fixed classical pullback uses the already frozen clock calibration. A choice of smooth covariance correction changes a free contraction but not the defining classical elliptic equation.

The affine obstruction is precisely a failure to meet that unchanged equation with its unchanged boundary condition. It is not repaired by assigning a retarded spacetime inverse to the rods.