Batch C: D01, D03 (covariant route)

Finding C1. For rank-4 numerators, the p14r per-component R3 lift produces component sums quadratic in $\tilde{g}$ (D03: 10 of 100 components). After the component sums, their abstract value cannot be recovered, because the linear realization $\tilde{g}=(4-d)M$ is not faithful there. A finite component basis cannot carry a projector of non-integer trace.

Batch C therefore uses a covariant route that applies the same rules R1–R6 with free external indices only:

1. The exact covariant 4D numerator is fitted from the frozen export and verified as a polynomial identity on all 100 components. It is unique (upper external indices, off-diagonal weight 2).

2. The lift $\eta \to g_4$ is made.

3. Two independent loop routes are used: Feynman parameters with symmetric integration, and the frozen dred.reduce_tensor with the J library and S0. They agree exactly in $d$.

4. Then $\hat{g}=g_4-\tilde{g}$ on the free indices.

D13 is reproduced exactly by this route (control), and the rules are unchanged (no contract change).

Further controls:

• The integrands are regenerated from $S_{\Psi }$ (D01 from the ${\ell }_3$ density, D03 from the ghost vertices) and equal the export on 100 of 100 components.

• The results are exchange symmetric.

• The nonlocal $\log$ coefficient equals minus the pole.

• The 4D poles lie in the span of the five p14r $h.h$ counterterm directions.

Seven mutations are each rejected: ghost-loop sign, symmetry factor, premature $d=4$, dropped $\tilde{g}$, external index position, off-diagonal weight, and the component route with the linear realization.