D4: an admitted cancellation cochain is not a completion

This proof uses the actual auxiliary-row kernel. It does not borrow the wavefront estimate of ${\mathcal{C}}_{3,\mathrm{D}}$ for a different functional. No new retarded application is needed; zero such applications is within the structural bound “at most one.”

Under (eq:UCP), the copied remainder and (eq:C-cancel) give the algebraic identity

That identity is not released as an integrated completion. The family ${\mathcal{C}}_{\xi }$ depends on the parameter. No single functional whose variation is this family has been constructed.

In fact, for the representation forced by the frozen class,

for every $F$. Thus a nonzero ${\bar{\mathcal{C}}}_{\xi }$ cannot be such a variation. Its cocycle is

which the next section detects.