Overlap-Field Coverage Regularizer

Integrate this at the data-loader or augmentation layer, rather than inside the backbone. Choose d=2 or d=4 and map each training example or augmentation center c_i into [0,1)^d using a fixed random projection followed by a sigmoid, or use a learned projection with a modulo operation. Associate each augmentation with radius rho_i, using a decreasing schedule such as rho_i=rho_max(1-i/N)^q. At each training step, draw M uniform probe points z_j on the torus and compute H_N(z_j)=sum_i u_{rho_i}(z_j-c_i). For d=2, calculate u_r exactly from the disk-intersection formula. If s is the wrapped torus distance between centers, then u_r(s)=0 for s>=2r and u_r(s)=[2r^2 arccos(s/(2r))-(s/2)sqrt(4r^2-s^2)] divided by the torus volume otherwise. For d>2, tabulate the ball-intersection volume as a function of s and r using numerical integration, then interpolate during training. Use L_cov=(1/M)sum_j exp(-H_N(z_j)) to penalize uncovered probes. Estimate center uniformity with random Fourier-feature MMD against uniform torus samples, or use a repulsive periodic kernel. The paper supplies the overlap field and exponential-integral structure; finite-N truncation, differentiable embeddings, and MMD are adaptations. First test on CIFAR-10 with a small ResNet and compare ordinary random augmentation against the regularizer at equal augmentation count and FLOPs. Log mean and worst-decile H_N, validation accuracy, calibration, and rare-class accuracy. Success means higher worst-decile coverage and improved validation accuracy or accuracy at reduced labeled-data fraction without center collapse.