Reduced Khovanov cohomology is invariant under based moves and basepoint transport

Under AC${}_{\omega }$ in [F1], apply [F3] to $(D,p)$ and $(D^{\mathrm{\prime }},p^{\mathrm{\prime }})$. Its moving-ball construction uses the compact isotopy-extension clause of [F2]. It gives a finite Reidemeister movie of their cut one-interval diagrams whose endpoints and short product collars remain fixed throughout. Closing the boundary collars by the same crossing-free cap recovers the two based diagrams. In each resolution this closure replaces the distinguished interval by the based circle and adds no crossing, so it preserves $n_{+},n_{-}$, cohomological degree and reduced quantum degree. The quotient/interval identification is the one proved in the defining reduced-complex item.

Every interior R1, R2 or R3 in the movie avoids the distinguished end collars. By [F4], its chain maps and contractions descend to the reduced quotient. A planar isotopy of the tangle identifies resolution factors and preserves every signed edge map; it fixes the distinguished interval and its degree-zero module. Thus each movie segment gives a bigraded chain homotopy equivalence of the reduced complexes. By [F5], the induced cohomology maps are mutually inverse, and finite composition gives the asserted isomorphisms in every $(i,\tilde{j})$.

Taking a movie with just one local move gives the based-move clause. Taking the same diagram with two marks gives the basepoint-transport clause by [F3]; orientation-preserving reparametrization of the single knot circle can carry either mark to the other. The relative supplier proves this case using an actual isotopy of cut arcs and fixed collars, so no relative-boundary statement for closed-link isotopies is being assumed. Together with step 2.1 this proves the full Statement. AC${}_{\omega }$ is used only through that supplier's isotopy extension and transversality, and the local algebra is choice-free.