We study the Cauchy problem for a mixed-sign quadratic Dirac equation on a noncompact N–star metric graph G, \( \textrm{i}\partial _t \psi = D\psi - \mathcal {N}(\psi ), \qquad \psi (0)=\psi _0, \) where \(\psi =(\psi _1,\psi _2)^{\mathsf T}:\mathbb {R}\times G\rightarrow \mathbb {C}^2\) 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 \(s>-\frac{1}{8}\) . We also establish a blow-up alternative in \(H_D^s(G)\) for the maximal forward lifespan.
