Corollary: The last equation is not a redundant regular constraint

At $N$, the differential of $q_R{|}_{S_6}$ vanishes, but its zero set has codimension two in $S_6$.

In particular, the copied linear dependence of cone conormals does not permit the equation $q_R=0$ to be deleted while retaining the same base incidence.

Proof. Equation (eq:qR-on-S) is a positive quadratic expression in the relative angular variables. Its first derivative vanishes on the angular diagonal. That diagonal has codimension two in $S^2\times S^2$.

Deleting $q_R=0$ leaves all of $S_6$, of dimension ten, rather than $N$, of dimension eight.