Spherical-design directional heads

Integrate this at the directional-feature or attention-head aggregation point. Generate or import a spherical t-design X={x_j in R^3} with N directions, normalize every x_j to unit length, and freeze these directions during training. For each token embedding h, use a learned projection q=A h in R^3. For direction j, compute the angular score z_j=q^T x_j, form a value v_j=V_j h or a shared equivariant branch v_j=V(h,x_j), and aggregate either by deterministic quadrature y=(1/N)sum_j phi(z_j,v_j) or by attention y=sum_j softmax_j(z)v_j. Pseudocode: input h; q=A(h); for j=1,...,N compute z_j=dot(q,x_j) and v_j=V(h,x_j); compute a_j=softmax(z)_j if using normalized attention; output y=sum_j a_j v_j, or output y=(1/N)sum_j phi(z_j,v_j) for the strict quadrature variant. If directions are made learnable, add L_iso=||N^{-1}sum_j x_jx_j^T-I_3/3||_F^2 and L_mean=||N^{-1}sum_j x_j||_2^2. The paper supplies the exact moment identities and the expected O(t^{-1}) error for smooth directional observables. Estimate actual approximation error by evaluating random spherical polynomials of degrees 1 through t and comparing design averages with dense sphere quadrature. First experiment: compare random directions, learned directions, and a fixed t=4 design in a small ViT on CIFAR-10 and on a synthetic 3D rotation-classification task, keeping FLOPs and parameter counts fixed. Success is lower seed-to-seed variance, better rotated-input accuracy, and equal accuracy with fewer directions or heads.