The derived couple is exact

At $D_{p,q}^{\mathrm{\prime }}$, let $a$ lie locally in $\ker j^{\mathrm{\prime }}$ and write $a=ix$ with $x$ at $D_{p-1,q+1}$. The condition $[jx]=0$ says locally $jx=dz=jkz$ for $z$ at $E_{p,q+1}$. Thus $x-kz\in \ker j$ and locally $x-kz=iy$ for $y$ at $D_{p-2,q+2}$. It follows that $a=ix=i^{2}y$, since $ik=0$. This belongs to $\mathrm{im}{i}^{\mathrm{\prime }}$ because $iy$ is in $D_{p-1,q+1}^{\mathrm{\prime }}$. Conversely a local element of $\mathrm{im}{i}^{\mathrm{\prime }}$ has the form $i^{2}y$ and $j^{\mathrm{\prime }}(i^{2}y)=[jiy]=0$. These two local containments descend to $\ker j_{p,q}^{\mathrm{\prime }}=\mathrm{im}{i}_{p-1,q+1}^{\mathrm{\prime }}$.

At $E_{p,q}^{\mathrm{\prime }}$, represent a local class in $\ker k^{\mathrm{\prime }}$ by a $d$-cycle $e$. Since $D_{p-1,q}^{\mathrm{\prime }}\hookrightarrow D_{p-1,q}$ is monic, $k^{\mathrm{\prime }}[e]=0$ implies $ke=0$ in $D$. Original exactness gives locally $e=jx$ for $x$ at $D_{p+r-1,q-r+1}$. Then $[e]=j^{\mathrm{\prime }}(ix)$, with $ix$ at $D_{p+r,q-r}^{\mathrm{\prime }}$. Conversely $k^{\mathrm{\prime }}{j}^{\mathrm{\prime }}(ix)=kjx=0$. Thus $\ker k_{p,q}^{\mathrm{\prime }}=\mathrm{im}{j}_{p+r,q-r}^{\mathrm{\prime }}$ after descent.

At $D_{p,q}^{\mathrm{\prime }}$, let $a$ lie in $\ker i^{\mathrm{\prime }}$. Its image in $D_{p,q}$ satisfies $ia=0$, so locally $a=ke$ for $e$ at $E_{p+1,q}$. Since $a$ already lies in $D^{\mathrm{\prime }}=\mathrm{im}i$, we have $ja=0$; therefore $de=jke=ja=0$. The class $[e]$ is defined and $k^{\mathrm{\prime }}[e]=a$. Conversely $i^{\mathrm{\prime }}{k}^{\mathrm{\prime }}[e]=ike=0$ for every cycle class. Descending gives $\ker i_{p,q}^{\mathrm{\prime }}=\mathrm{im}{k}_{p+1,q}^{\mathrm{\prime }}$.

These are exactly the three equalities required for page $r+1$, with the degrees supplied by [F1]. The arguments include zero kernels, images and homology objects: the reverse containments are zero-composite identities and never require a nonzero witness. At $r=1$ the shifted indices give the first derived couple; every larger page is covered by the same printed formulas. No section, global lift or axiom of choice is used.