Blow-up analysis and compactness for critical Dirac equations on $\mathbb S^n$

We study blow-up and compactness for the critical nonlinear Dirac equation \[ D_{g_0}ψ_i = H_i|ψ_i|^{\frac{2}{n-1}}ψ_i \qquad \text{on }\mathbb S^n,\quad n\ge 3, \] where $g_0$ is the round metric and $H_i\to H>0$ in $C^0(\mathbb S^n)$. Under a first-threshold bound on the critical mass, we prove th