Theorem: The selected conormal matching is not clean

The fiber product (eq:fiberproduct) is a smooth manifold of dimension fifteen. Its linearized matching space has dimension seventeen at every retained point. Hence the matching is not clean.

Proof. The copied smoothness and codimension of $N={\tilde{\mathcal{N}}}_{\mathrm{att}}$ give

A tangent vector to the product of the four separate conormal relations satisfies linearized base matching precisely when its common base component $u\in T_{w}W$ obeys

The seven frequency variations are unrestricted in this linearized base matching.

Define

The copied cone rank is two. The four attachment differentials are independent of the cone differentials, so the full differential has rank six. Therefore

The linearized matching space has dimension $10+7=17$.

The actual tangent space is $T_{w}N\times T_{\theta }\Theta$, of dimension $8+7=15$. It is strictly smaller than the linearized matching space. This violates the tangent-space equality in the definition of clean matching.