= Abelian Ward splitting by Laurent series
{c}
{title2=$f_x(\zeta)=\sum_{n\in\mathbb Z}f_n(x)\zeta^n=h_1-h_0$}
On a <twistor line> use $\pi_{0'}=\zeta,\pi_{1'}=1$ and $\omega^A=x^{A0'}\zeta+x^{A1'}$. For a degree-zero <holomorphic> patching logarithm, its <Laurent series> splits as $h_0=-\sum_{n\ge0}f_n\zeta^n$, $h_1=\sum_{n<0}f_n\zeta^n$. The <twistor incidence relation> implies $\partial_{A0'}f_n=\partial_{A1'}f_{n-1}$. Hence $(-\partial_{A0'}+\zeta\partial_{A1'})h_\alpha=\partial_{A1'}f_{-1}$ on both patches. The <gauge potential> is $A_{A0'}=-\partial_{A1'}f_{-1}$, $A_{A1'}=0$. Exponentiating the two additive factors produces the linewise <holomorphic> splitting of $e^f$.
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