Theorem: Failure of coefficientwise ordering preservation

For the admissible difference (eq:rankone),

Consequently

The corresponding descended vanishing subspaces are not preserved by ${\bar{\alpha }}_d$ either.

Proof. Let $\ell =L_q^{\prime }$ be its field-independent linear derivative kernel. Then

The factor $\hslash /2$ in (eq:ordering-map) cancels the factor $2$. A second self-contraction would require a fourth derivative, which is zero. This proves (eq:ordering-square) exactly, not merely to first order.

Although ${L_q}^2$ vanishes on ${\mathcal{C}}_W$, the $\hslash$ coefficient of its transport is the nonzero constant $C^2$. Coefficientwise vanishing therefore fails. On-shell descent gives the same conclusion because that constant is not in ${\mathcal{N}}_{\mathrm{sol}}$.