Proposition: Relative classical closure

The rule (eq:relative-s) is nilpotent and tangent to the declared compact ghost space. For the sharp classical construction, and for the prepared density construction when $U\subset \{t>\epsilon \}$,

The auxiliary constraints are preserved by the corresponding coordinate pullback law.

Proof. The support of $c^{\nu }{\partial }_{\nu }{c}^{\mu }$ remains compact in $U$. For a scalar,

the first-derivative terms cancel and the second-derivative term vanishes by odd antisymmetry. The tensor calculation is the same identity ${{\mathcal{L}}_c}^2=\frac{1}{2}{\mathcal{L}}_{[c,c]}$, with $[c,c{]}^{\mu }=2c^{\nu }{\partial }_{\nu }{c}^{\mu }$. It proves nilpotence on $q$ and $c$.

Cauchy uniqueness gives $s_{\mathrm{rel}}{\mathsf{X}}^{\mu }=\kappa c^{\nu }{\partial }_{\nu }{\mathsf{X}}^{\mu }$. The inverse-coordinate and scalar pullbacks therefore cancel in the relational composition. This also preserves the coordinate equations and their fixed data, proving the auxiliary statement.