Two characters on one punctured Riemann surface

We develop an abstract framework for coupled period–realization of meromorphic $1$–forms on punctured Riemann surfaces. A configuration datum $C$ gives the combinatorics and determines a restricted character domain ${\mathrm{\Delta }}_C\subset \mathrm{H}\mathrm{o}\mathrm{m}({\mathrm{\Gamma }}_{g,n},\mathbb{C}{)}^2$ with a scale–fixed slice ${\mathrm{\Delta }}_C^{\mathrm{s}\mathrm{c}}$. Assuming Teichmüller–regularity, degeneration detection, and pushability, we prove that there is point in ${\mathrm{\Delta }}_C^{\mathrm{s}\mathrm{c}}$ which corresponds to a surface carrying two meromorphic differentials realizing any prescribed restricted pair. This abstracts the Weber–Wolf extremal–length minimization method while constructing minimal surfaces.