Counterterm domain and solve

1 [corrected in p20; the p19 text is retracted]. p19 reported that full closure of the MS-bar counterterm fails with the derived ghost-equation factor $+1$ and needs $-1$. That was a Grassmann-orientation bookkeeping error: the route-G densities are built in the order (c,cb), whereas the P15 $\Sigma$ coordinates are (cb,c)-oriented, and $V[c.cb\#k]=-V[(\mathrm{cb},\mathrm{c})\#k]$. With the measured ratio $-1$ applied, the derived factor $+1$ gives full closure (p20 erratum note). The relations below are stated in the (cb,c) orientation:

• The ghost-equation operator $2iD$, verified by P15 as $\Sigma =2iD[G_{Kc}]$, gives $2iDV[c.K\#1]=-V[(\mathrm{cb},\mathrm{c})\#0]$ and $2iDV[c.K\#2]=-V[(\mathrm{cb},\mathrm{c})\#1]$.

• Applied to the c.K pole, this reproduces the P15 $\Sigma$ pole $(-3/8,1/4)$ in (cb,c) coordinates, i.e.\ $(+3/8,-1/4)$ in the route-G (c,cb) coordinates.

1: finding about the p14r contract (contract not modified). The p14r counterterm-domain certificate counts Im ${\sigma }_0$ 3, quotient 8, $D\cap \ker$ 5 and rank $P_{ch}{\sigma }_0$ 4. These are artefacts of a real/imaginary split over $\mathbb{Q}$: the densities carry an explicit $i$ for odd derivative counts, so an $i$-multiple counts as independent. Over $\mathbb{Q}(i)$ the counts are 6, 5, 6 and 3 (the real dimensions under a compatible real structure), and Route C agrees. With the split, p19 reproduces the p14r counts (control).

and mutations. All controls are true. Eight mutations each change the outcome:

• drop D03; drop D01; flip the D03 phase; double D03; trace operator instead of $s_{0}h$; de Donder $q^{2}P$ instead of $E$; drop the D03 evanescent part (naive 4D). Each gives OBSTRUCTION (nonlocal residual).

• D13 with $G_t$ dropped gives NONZERO but CANCELLABLE (rank 3).

Recorded blind spots of the projected identity: dropping D11 (transverse by itself), dropping D13 ($A_{T_1}=A_{E_1}=0$) and flipping the K.c row sign (each row vanishes separately). In p20, the full-closure test detects the last two, and the D11 normalisation pin covers the first. Removing D11 from the inventory makes the inventory incomplete, and projected_defect_computed is then not earned.