Cancellation at the required dimension-bounded evaluation objects

For twisted coefficients the preceding lemmas give ${\beta }_X={ϵ}_X{\gamma }_X$, with both ${\beta }_X$ and ${\gamma }_X$ independently equivalences. Thus ${ϵ}_X={\beta }_X{\gamma }_X^{-1}$ is an equivalence. The coefficient-trivialization lemma conjugates this map to the displayed constant-coefficient map. This is cancellation of two actual equivalences, not cancellation from $\beta$ alone.

The finite-coefficient descent theorem Étale derived cdh locality on the actual qcqs site makes $B$ ordinary cdh-local on the entire prime-field site. The primitive root is needed only on the slice in step 1.1; it does not conjugate $B_3$ to $B$ globally on that site. The source localization is an exact functor on the stable derived categories. Apply it to ${\tau }^{\le 3}B\to B\to {\tau }^{>3}B$ and evaluate at $X$; the first arrow is the displayed equivalence. Its cofiber is therefore zero. This reasoning proves the claim at the listed evaluation objects without a dimension-bounded site/cofinality assertion.