D4: an admitted cancellation cochain is not a completion

A kernel calculation is useful because it isolates what the obstruction does not say.

Let the inherited auxiliary kernel be denoted $b_{\xi }$, with the normalization

Fiber components are understood. Define the new metric–scalar cochain

This is a functional of the allowed observable slots. It does not make $b=D\gamma$ an off-shell relation in the inherited carrier.