Spherical harmonic spectrum regularizer

Integrate this at the representation layer immediately after producing a batch of vectors $z_i\in\mathbb R^3$ or after projecting arbitrary embeddings to three dimensions with a learned linear map. Normalize with $u_i=z_i/(\|z_i\|+\epsilon)$, convert each $u_i$ to $(\theta_i,\phi_i)$, and evaluate real or complex spherical harmonics up to degree $L$ using a differentiable library. Pseudocode is: u = normalize(z); a[l,m] = sum_i(c[i] * conj(Y(l,m,u[i]))); P[l] = mean_m(abs(a[l,m])**2); Lspec = lam * sum_l(w[l]*(P[l]-Ptarget[l])**2); loss = task_loss + Lspec + mu*repulsion(u); backprop(loss). For an MVP, use $L=8$, equal weights $c_i=1$, and compute the harmonics directly; later replace the dense evaluation by a nonuniform spherical-harmonic transform when $N$ and $L$ are large. The paper supplies the density-to-harmonic construction and the target-spectrum optimization principle; the neural-network gradient and batching are estimated by automatic differentiation. First test a 3-layer MLP on CIFAR-10 or a contrastive image encoder on CIFAR-10 with a 3D hyperspherical prototype layer. Compare no regularizer, cosine repulsion, and spectral regularization with $P_\ell^\star=0$ for $1\leq\ell\leq4$. The quantitative prediction is that the measured batch spectrum satisfies $P_\ell/P_0$ substantially below the unregularized baseline for low degrees and approaches the target with error decreasing as optimization proceeds. Sweep $\lambda$ and predict a stability boundary: when the regularizer gradient norm becomes comparable to the task gradient, approximately $\lambda\|\nabla\mathcal L_{\mathrm{spec}}\|/\|\nabla\mathcal L_{\mathrm{task}}\|\approx1$, training should begin to oscillate or classification loss should increase. Also measure angular-count variance in spherical caps; low-mode suppression should reduce large-cap variance relative to random embeddings.