Remark: Why this does not rescue clean composition

A nondegenerate quadratic form on these base-transverse directions does not make them fibers of ${\mathcal{F}}_{\mathrm{match}}\to \Gamma$ for a canonical relation $\Gamma$. They are absent from $T{\mathcal{F}}_{\mathrm{match}}$, not excess directions inside it.

A construction that integrates or resolves these base directions would have to specify a different representation, the map back to $W$, the amplitudes, and the resulting distributional theorem. None follows from the Hessian calculation. Such a construction is not silently substituted for the direct route.