Fractional Jacobian topology loss

Integrate this at the output of a neural implicit field, segmentation network, or convolutional network that predicts two channels a_theta and b_theta interpreted as an S1 phase. Normalize them to u_theta=(a_theta,b_theta)/sqrt(a_theta^2+b_theta^2+epsilon), with epsilon such as 1e-6. Add the fractional term to the training loss. On a periodic image or voxel grid, apply an FFT to both normalized channels, construct integer frequency vectors q, multiply each Fourier coefficient by |q|^(1+s), sum squared magnitudes, and divide by the number of grid points. For nonperiodic data, approximate the equivalent Gagliardo seminorm using sampled pairs: sum_{i,j} ||u_i-u_j||^2/(||x_i-x_j||^(n+2 alpha)+epsilon), where alpha=(1+s)/2. Compute the Jacobian constraint on each 2D coordinate plane. Let phi=atan2(b_theta,a_theta), calculate wrapped phase differences around every elementary plaquette, sum them, and divide by 2*pi to obtain a soft winding estimate. Compare this winding to a target defect mask w_p_target using the squared penalty in the loss. Use the soft estimate during training and optionally use straight-through rounding after a warmup. Pseudocode: predict a,b; normalize u; compute FFT fractional energy; compute wrapped edge differences and plaquette windings; form task_loss + lambda_frac*E_frac + lambda_J*E_J; backpropagate and update theta. The paper supplies the fractional exponent and Jacobian-current interpretation; the FFT estimator and differentiable winding surrogate are engineering approximations. First test on 64x64 synthetic images containing circles, holes, and vortex pairs. Compare a U-Net or SIREN using this loss against the same model with gradient or total-variation regularization at matched parameters and FLOPs. Measure defect recall, winding-current error, boundary Hausdorff distance, and task loss. Success means fewer erased or spurious holes at equal task accuracy while fractional energy suppresses high-frequency noise.