Theorem: Remainder release

For the inherited coefficient and parameter space:

1. The companion has the switching display (eq:companion-display), sampling support (eq:sampling-support), and no ramp-layer contribution. It vanishes under the copied condition (CP).

2. On the complete gauge-fixed free shell, the remaining Einstein row is the compact auxiliary pairing ${\mathfrak{B}}_{\xi }$. It vanishes when $D^{\ast }b=0$ near the parameter support.

3. It does not vanish universally on that shell under (CP). The admissible witness of Theorem thm:obstruction has the strictly negative value (eq:main-value). Its carrier class is nonzero.

4. Adding the scalar-sourced slot once leaves this metric variation unchanged.

Thus unqualified invariance of the named coefficient is Proved obstructed. Existence of a further cancelling completion is not decided.

Proof. Proposition prop:companion-support, Lemma lem:shell, Proposition prop:subclass, Theorem thm:obstruction, Proposition prop:aux-class, and Section sec:scalar-slot.