DCRP68 / X72R51: Collapse of Axisymmetric Director Integrability

Building on the two zero-spin axisymmetric candidate modes (Type A, Type B) that DCRP67 screened out, this round adds a constraint the previous round left uncounted: S+R=∇V must genuinely be a Euclidean velocity gradient, which requires satisfying the first-order compatibility equation ∂kLij=∂jLik. A pointwise computation of Type A's first-order jet-system determinant gives (r²−9λ²)(r²+9λ²)²/256, showing rigidity everywhere except one resonance point that collapses on its own under uniform stretching; Type B's unique torsion, meanwhile, produces a nonzero Euclidean-frame curvature and so must vanish. Both modes are thereby closed, and both would force the vorticity direction to stay fixed in space, contradicting isotropic third-order covariance — the conclusion is that every surviving aligned/no-flip branch must carry positive-measure cofactor shape activity, which is handed to DCRP69 to test whether this activity can cancel against a pressure/transport phase lock.