Doubled-angle orientation order pooling

Integrate this at one or more early or mid CNN stages, or before converting patch features into transformer tokens. First create B orientation-energy maps e_b(x)≥0, using fixed steerable convolutions, learned oriented kernels followed by softplus, or an existing filter bank; the bin centers are θ_b=bπ/B. Compute per spatial location: (1) r=Σ_b e_b; (2) z_R=Σ_b e_b cos(2θ_b); (3) z_I=Σ_b e_b sin(2θ_b); and (4) q=sqrt(z_R²+z_I²)/(r+ε). Concatenate [z_R/(r+ε), z_I/(r+ε), q] to the ordinary feature map, or replace the original orientation-bin channels with these three channels followed by a 1x1 projection. If an explicit angle is needed, calculate φ=(1/2)atan2(z_I,z_R) only when q>q_min, and use a zero angle plus a confidence mask otherwise. The mathematical quantities z_R, z_I, and q are computed exactly from the learned or fixed energies; no circular statistics need to be estimated. Optionally add an equivariance check by rotating an input by α and verifying that (z_R,z_I) rotates by 2α. The first experiment should use CIFAR-10 or MNIST-C with synthetic line and bar images and a small ResNet baseline. Train baseline angle-bin pooling, ordinary scalar averaging, and doubled-angle pooling at equal parameter count. Measure classification accuracy under line reversal, random 180-degree orientation flips, and mixtures of opposing orientations, plus calibration of q. Success is higher accuracy and lower orientation error on reversed and canceling stimuli without materially increasing FLOPs.