Relativistic Scaling and Magnetization-Current Feedback in Stern Gerlach-Modified Pair-Plasma Reconnection: SpinPIC2D Validation and Nonlinear Regimes

We investigate the relativistic scaling and electromagnetic feedback of Stern--Gerlach (SG) force driven spin transport in pair-plasma reconnection with SpinPIC2D. The model advances relativistic proper momentum and magnetic BMT spin precession, applies the SG force, deposits spin magnetization, and includes $\mathbf J_M=\nabla\times\mathbf M$ in Ampere's law. In a weak-seed scan at fixed $\gamma_{\rm tr}=2$, the normalized global magnetic flux-growth remains near the classical control-run for $\Xi\le0.1$, is $0.016$ at $\Xi=0.4$, and reaches approximately $0.12$ and $0.25$ at $\Xi=0.7$ and 1, respectively over $3\le t/\tau_{\rm sp}\le7$. Because $\partial/\partial y=0$ in the 2.5-D geometry, the direct $y$-directed SG term vanishes and the enhancement is indirect: sheet-normal SG sorting restructures branch-resolved electron and positron velocity distributions which results in changes in pressure moments and generates a layered magnetization current. Along a fixed-$\chi_{\rm sim}$ family, increasing $\gamma_{\rm tr}$ reduces $\Xi$ and suppresses branch sorting, whereas the matched-control flux-growth enhancement remains positive for $\gamma_{\rm tr}=2,3,5$. Retuning $\chi_{\rm sim}$ to hold $\Xi=1$ does not preserve nonlinear similarity: both the coupling and $J_{M,y}$ increase with $\gamma_{\rm tr}$, and the $\gamma_{\rm tr}=5$ case develops a multi-X-line state. Thus $\Xi$ orders the onset of SG-modified reconnection, while the nonlinear response also depends on the absolute spin coupling and magnetization-current amplitude.