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 $G$, \[ \mathrm{i}\partial_t \psi = D\psi - \mathcal N(\psi), \qquad \psi(0)=\psi_0, \] 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, \[ \mathcal N(\psi)=\mathcal B\bigl(\Pi_+\psi,\Pi_-\psi\bigr), \] where $\Pi_\pm$ 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 \(s>-\frac18\). We also establish a blow-up alternative in $H_D^s(G)$ for the maximal forward lifespan.