Lemma: Persistence under a smooth frequency reduction

Smooth coordinate changes in the individual conormal parameterizations preserve the failure of clean matching.

Adding $k$ freely varying auxiliary parameters adds $k$ to both the actual and linearized matching dimensions. Deleting $k$ freely varying vertical parameters in a local product quotient subtracts $k$ from both dimensions. The two-dimensional tangent discrepancy persists.

In particular, a local quotient by the one-dimensional frequency redundancy in Theorem thm:projection does not convert this matching into a clean one.

Proof. A smooth reparameterization carries both the actual tangent space and the linearized matching space isomorphically to their new descriptions. Their equality or inequality is therefore invariant.

For a free auxiliary factor, both spaces acquire its full tangent space as a direct summand. For a local product quotient by free vertical parameters, both lose the same vertical summand. Their dimension difference remains two.

The frequency kernel in Theorem thm:projection is one-dimensional and vertical. In a local quotient chart, the two dimensions become fourteen and sixteen, respectively, and are still unequal.