Correction release (p20): erratum to p19

K1 (the p19 text is retracted). p19 stated that full ${\sigma }_0$ closure of the MS-bar counterterm fails with the derived ghost-equation factor $+1$, that it needs $-1$, and that this was unresolved. The statement is wrong; it came from a Grassmann-orientation bookkeeping error.

• The route-G densities are generated in the order (c,cb). The P15 ghost self-energy coordinates $\Sigma$ use the (cb,c) orientation, and $V[c.cb\#k]=-V[(\mathrm{cb},\mathrm{c})\#k]$. p20 measures this ratio (via pw_vertex) as $-1$ for $k=0,1$.

• In route-G (c,cb) coordinates, $\Sigma$ has pole $(+3/8,-1/4)$ and local part $(2/3,-5/12)$.

• The derived factor is $+1$ (the ghost equation $\Sigma =2iD[G_{Kc}]$ reproduces the P15 pole). With it, the complete MS-bar counterterm $S_{ct}=-(\text{h.h}+\text{c.K}+\text{c.cb poles})$ is ${\sigma }_0$-closed in every ghost $+1$ sector (h.c, c.b, c.cbs).

• Without the c.cb term, or with $\Sigma$ unconverted (the p19 error), ${\sigma }_0(S_{ct})$ has 88 entries in the c.b sector.

• Tree-level controls: $V_{S_0}[\mathrm{cb},\mathrm{c}]=2iD{V}_{S_0}[K,c]$, and ${\sigma }_0({S_0}^{Kc}+{S_0}^{\bar{c}c})=0$ (the ${S_0}^{Kc}$ part alone has a nonzero image).

• In quotient coordinates, the c.K-only data have c.cb components $(-3/8,1/4)$, which cancel against $\Sigma =(+3/8,-1/4)$. The full data including $\Sigma$ have quotient coordinates $0$.

redefined. The p19 flag (renormalised $P_{ch}{\sigma }_0(S_{ct})=0$, $x=0$) follows from the identity and locality, so it is not independent. In p20, projected_restoration_verified means full ${\sigma }_0$ closure of the complete one-loop ${\kappa }^2$ MS-bar $S_{ct}$ (including c.cb, $\Sigma$ held fixed) in all ghost $+1$ sectors, together with the projected identity. Scope: that counterterm only; not bulk symmetry, not O.1. The corrected computation closes, so the flag is earned. The p19 check is kept as a labelled route-consistency control, and it holds.

spots. The full-closure test acts as a mutation detector, with $\Sigma$ held fixed:

• drop D13 and flip K.c are detected (c.b, 88 entries);

• drop D01, drop D03, flip D03 and D03 $\times 2$ are detected (h.c, 280 entries);

• the p19 orientation error is detected;

• drop D11 is not detected: D11 is transverse, so its pole is closed by itself.