Gauge-invariant vortex charge loss

Integrate this loss into a neural field, U-Net, or Fourier neural operator that predicts complex coefficient fields $(\Phi,\Psi,\mathfrak a,\mathfrak b,\mu)$ on a 2D grid, or predicts a complex PDE solution from which $N$ is derived. At each training step, select one or more closed pixel loops around annotated or expected vortices. Use automatic differentiation or Sobel finite differences to evaluate $\partial$ and $\bar\partial$; explicitly compute $L\Phi=\bar\partial\Phi-\mu\partial\Phi$ and $L\Psi=\bar\partial\Psi-\mu\partial\Psi$, then construct $N=\Phi\mathfrak b-\Psi\mathfrak a-(\Phi L\Psi-\Psi L\Phi)$. Sample $K$ ordered points on each loop, calculate $u_k=N_k/(|N_k|+\varepsilon)$, and use atan2 on adjacent complex products to obtain wrapped phase increments. If the target charge is $n_\star$, add $\lambda_c\mathcal L_{\mathrm{charge}}$ to the ordinary pixelwise, supervised, or PDE residual loss. The boundary term with margin $m$ keeps $N$ away from zero on the loop; this is necessary because winding is invariant only while the loop contains no zero. Use $\varepsilon=10^{-6}$, gradient clipping, and a warm-up in which $\lambda_c$ increases from zero. Compute the rounded integer $\hat n$ as a detached evaluation metric, while optimizing the differentiable surrogate. The first experiment should use synthetic fields $N(r,\theta)=A(r,\theta)e^{in\theta}$ with random positive amplitudes, positive spatial gauge factors $q(x)$, coordinate warps, and known charges $n\in\{-2,-1,0,1,2\}$. Train a small CNN with and without the loss. Measure charge classification accuracy, vortex localization, pixel error, and robustness under amplitude rescaling. Success is fewer spurious charge changes under perturbations and higher out-of-distribution charge accuracy at comparable reconstruction loss and FLOPs.