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 that every blowing-up sequence has, after passing to a subsequence, a unique blow-up point $p$ satisfying $H(p)=\max_{\mathbb S^n}H$. The rescaled solutions converge to a ground-state Dirac bubble that carries the entire limiting critical mass. Under local uniform $C^1$ bounds on $H_i$, we use the Kelvin transform and a second rescaling to prove that $p$ is an isolated simple blow-up point. We also derive sharp pointwise and gradient estimates in cylindrical coordinates and identify the normalized Green limit. If the coefficients converge in $C^2$ near $p$, the Pohozaev identity yields $\operatorname{Hess}_{g_0}H(p)=0$. Consequently, if a positive $H\in C^2(\mathbb S^n)$ satisfies $Δ_{g_0}H(p)<0$ at every global maximum, then solutions with critical mass bounded by a level slightly above the first threshold are uniformly bounded in $L^\infty(\mathbb S^n)$. This bound remains uniform under small $C^2$ perturbations of $H$.