Low-regularity well-posedness for a mixed-sign quadratic Dirac equation on $N$-star metric graphs

We study the Cauchy problem for a mixed-sign quadratic Dirac equation on a noncompact $N$-star metric graph $G$, $$\mathrm{i}{\mathrm{\partial }}_t\psi =D\psi -\mathcal{N}(\psi ),\psi (0)={\psi }_0,$$ where $\psi =({\psi }_1,{\psi }_2{)}^{\mathsf{T}}:\mathbb{R}\times G\to {\mathbb{C}}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. The nonlinearity acts edgewise and is given by a bilinear interaction between the positive and negative spectral parts, $$\mathcal{N}(\psi )=\mathcal{B}({\mathrm{\Pi }}_{+}\psi ,{\mathrm{\Pi }}_{-}\psi ),$$ where ${\mathrm{\Pi }}_{\pm }$ are the spectral projections of $D$ and $\mathcal{B}$ is a fixed bilinear map on ${\mathbb{C}}^2$ applied componentwise on each edge. This is a model quadratic interaction tailored to the mixed-sign Bourgain-space mechanism, rather than a general nonlinear Dirac equation on graphs. Using Bourgain-type spaces associated with the spectral resolution of $D$ and a mixed-sign bilinear estimate on $N$-star graphs, we prove local well-posedness in the operator Sobolev space $H_D^s(G)$ for $s>-\frac{1}{8}$. We also establish a blow-up alternative in $H_D^s(G)$ for the maximal forward lifespan.