Dyadic Harish-Chandra spectral layer

Integrate the layer between an input feature tensor on a discretized homogeneous space and a pointwise channel mixer. Use features $f\in\mathbb R^{N\times C}$, a precomputed approximation to the symmetric-space transform $\mathcal F_{G/K}$, spectral coordinates $\lambda_q$, and dyadic bands $B_j(\lambda_q)$. Begin with a compact quotient or finite homogeneous graph where graph-Laplacian eigenvectors approximate spherical eigenfunctions. For every scale $j=0,\ldots,J$, parameterize $s_j(\lambda_q)$ with a small MLP of invariant coordinates, such as Casimir eigenvalues or $\|\lambda_q\|$. Weyl invariance can be imposed by tying values across Weyl-related coordinates, or by using a radial multiplier $s_j(\lambda)=g_j(\|\lambda\|)$. Compute $z=\mathcal Ff$, then set $z_j(q)=s_j(\lambda_q)B_j(\lambda_q)z(q)$, sum over $j$, apply $y=\mathcal F^{-1}(\sum_j z_j)$, and finish with a channel projection, residual connection, and nonlinearity. Add a smoothness penalty $L_{\mathrm{sym}}=\sum_j\sum_q|\Delta_\lambda s_j(\lambda_q)|^2$, where $\Delta_\lambda$ is a finite-difference spectral derivative. Add a scale penalty $L_{\mathrm{scale}}=\sum_j2^{-2j(a+r-M)}\|s_j\|_2^2$ to discourage violation of the paper's dyadic bound. Use known $a$ and $r$ for the chosen space, and start with $M=2$ or $3$. Estimate transform quality by reconstructing random signals and estimate equivariance error using known group actions. The first experiment should use a small homogeneous graph or spherical rotation benchmark, comparing this layer with a standard graph convolution and an unconstrained spectral filter at equal parameter count. Record validation accuracy, equivariance error $\|L(T_gf)-T_g(Lf)\|/\|Lf\|$, peak memory, and runtime. Success means lower equivariance error with equal or better accuracy, and graceful accuracy retention when the number of dyadic bands is reduced.