DCRP43-QC: Same-Parent Scalar Re-Root and Gauge-Quotient Collapse

Building on the quotient-correct same-parent audit task left by DCRP43, and drawing on the existing identities and constructions of DCRP30, 34, 35, 40, and 42, this round proves that DCRP30's same-parent re-root identity gives the exact transformation law for the planar shear scalar q and its horizontal gradient (i.e., the planar vorticity): the normalized planar-vorticity re-root multiplier is always e^{-S_0}, independent of γ. At the same time, it shows that the absolute scalar q=w-∂_zφ is not invariant under the residual-potential gauge, so DCRP43's reading of the original |r|^p turnover as a 'new same-parent scalar tax' does not hold and is corrected here. The new STOP (STOP-D43-QC) reduces the problem to a gauge-invariant question — whether the gauge-quotient scalar dynamics carries a nonzero transport-projection residue — left to DCRP44.