The AHSS first differential is the cellular coboundary

The differential is $d_1=j\circ k=jk$ in the exact-couple indexing: a class in $E_1^{p,q}=h^{p+q}(X^p,X^{p-1})$ is first sent by the pair map $k$ to $h^{p+q}(X^p)$, and then by the connecting map $j$ of the next pair to $h^{p+q+1}(X^{p+1},X^p)=E_1^{p+1,q}$.

Suppose $p\ge 1$. Restricting along the characteristic map ${\phi }_{\tau }:(D^{p+1},\mathrm{\partial }{D}^{p+1})\to (X^{p+1},X^p)$ and collapsing the complement of a $p$-cell $\sigma$ exhibits the $(\tau ,\sigma )$ component of $d_1$ as the map induced by $c_{\sigma }\circ \pi \circ {\phi }_{\tau }{\mathrm{\mid }}_{\mathrm{\partial }{D}^{p+1}}:S^p\to S^p$. The disk-boundary connector agrees with positive suspension in these oriented coordinates: cap the oriented disk with the cone on its outward-oriented boundary. The cone has orientation $dt$ followed by the boundary orientation, so its base boundary is the negative of the disk boundary. They glue to an oriented sphere, and collapse of the disk maps the cone to the positively oriented suspension. Thus no extra sign enters the component diagram. To type the characteristic pair map, first approximate its boundary cellularly in $X^p$, extend that boundary homotopy over the disk by [F7], and approximate the disk rel its now cellular boundary. All sources are finite, and the resulting map is homotopic through pair maps to the original; cellular representatives determine the same maps on the cofiber groups by homotopy invariance. For $p\ge 1$ the boundary vertex now maps into $X^0\subseteq X^{p-1}$, so the attaching-sphere composite after collapse is based. Its degree is unchanged. This is the attaching map followed by collapse onto the $\sigma$-sphere, as in the source diagram.

Suppose $p=0$ and orient a one-cell $e_{\tau }^1$ from its initial endpoint $v_{-}$ to its terminal endpoint $v_{+}$. Naturality reduces its component of $d_1$ to the boundary for $(D^1,S^0)$. The cofiber of $(S^0{)}_{+}\hookrightarrow (D^1{)}_{+}$ is a graph consisting of the oriented edge and a spoke from each endpoint to the common cone tip. Its oriented cycle traverses the edge from initial to terminal, then the terminal spoke forwards, then the initial spoke backwards. Collapsing the edge gives the map to $\mathrm{\Sigma }(S^0{)}_{+}$ with degrees $+1$ on the terminal circle and $-1$ on the initial circle. Wedge coordinates, [F3] in dimension one, and the normalized connector [F6] therefore send the endpoint coefficients to $a(v_{+})-a(v_{-})$; hence its coefficient at a vertex $v$ is ${\mathbf{1}}_{\{v_{+}=v\}}-{\mathbf{1}}_{\{v_{-}=v\}}$, including zero for a loop.

The cellular coboundary with coefficients $h^q(\ast )$ has, in the dual cell bases, the incidence entries $[e_{\tau }^{p+1}:e_{\sigma }^p]$ by duality of the cellular boundary.

For $p\ge 1$, the degree action [F3] says that the $(\tau ,\sigma )$ component of $d_1$ is multiplication by the degree of the map in step 1.2, namely the incidence number $[e_{\tau }^{p+1}:e_{\sigma }^p]$.

For $p=0$, step 1.3 gives the same incidence coefficient directly, including the negative coefficient at an initial endpoint that cannot be represented by a based self-map of $S^0$.

Steps 2.1 and 2.2 cover all $p\ge 0$ (for $p<0$ the source is zero, so the differential is zero even when $p=-1$ and the target need not vanish). Their component matrices agree with step 1.4, so $d_1$ is the cellular coboundary under [F1]. The second page is therefore the cohomology of the cellular cochain complex. For completeness, apply the choice-free consecutive-skeleton calculation in [F5]. The pair sequences give $H^p(X^p;G)=C^p\mathrm{/}\mathrm{im}{\delta }^{p-1}$ and identify $H^p(X^{p+1};G)$ with the kernel of the induced map to $C^{p+1}$; hence this is $\ker {\delta }^p\mathrm{/}\mathrm{im}{\delta }^{p-1}$. Higher cell attachments change neither adjacent group, so finitely many restrictions give $H^p(X;G)$. The same chase starts at $X^{-1}=\mathrm{\varnothing }$, and negative cohomology is zero. Every use of additivity here is over the finite set of cells, so there is no arbitrary family of representative or primitive choices. Taking $G=h^q(\ast )$ proves the asserted $E_2$ formula.

This proves the stated identification of $d_1$ and the resulting formula for the second page.