Cellular boundary is the incidence degree matrix

Project the three-skeleton connecting map defining $d_n$ to the summand of a fixed $(n-1)$-cell.

For $n\ge 2$, first pass from $X^{n-1}$ to $X^{n-1}\mathrm{/}{X}^{n-2}$ as required by the relative target of the cellular boundary, and then collapse all summand spheres except that of $e_{\beta }^{n-1}$. Equivalently, collapse the entire complement $X^{n-1}\setminus e_{\beta }^{n-1}$, including $X^{n-2}$. Naturality of the connecting homomorphism for the characteristic disk identifies the coefficient with the composite from its oriented boundary sphere to this quotient sphere. This is precisely the reduced-homology endomorphism in Incidence number of two CW cells, so its degree is $[e_{\alpha }^n:e_{\beta }^{n-1}]$.

For $n=1$, use a characteristic interval compatible with the chosen edge generator. Its connecting homomorphism is the singular boundary $[v_{+}]-[v_{-}]$. For each chosen vertex generator $e_v={\epsilon }_v[v]$, the identity $[v]={\epsilon }_v{e}_v$ gives coefficient ${\epsilon }_v({\mathbf{1}}_{\{v_{+}=v\}}-{\mathbf{1}}_{\{v_{-}=v\}})$, exactly the incidence number of Incidence number of two CW cells. This includes negatively chosen zero-cell generators and gives zero when both endpoints coincide. The boundary is an element of a direct sum, so only finitely many coefficients are nonzero; equality of all its projections therefore proves the formula for every $n\ge 1$; separately, $d_0=0$ by definition.