Analytic KL spatial adapter

(1) Integration point: implement a spatially varying additive adapter inside a neural field, CNN, or coordinate MLP. For an input coordinate $x\in\mathbb R^d$, compute an adapter scalar or channel vector $a(x)=W_zf_M(x)$, where $f_M(x)$ contains the first $M$ tensor-product Gaussian-Hermite modes; add $a(x)$ to the feature map or use it to modulate a convolutional or MLP weight. Store only $M\times C$ coefficients rather than one parameter per pixel or voxel. (2) Pseudocode: choose multi-indices $A_M$ by increasing total degree; initialize trainable $\log\sigma$, $\log\ell$, and optionally $\log s$; for every batch coordinate $x$, compute normalized Hermite functions using $h_0(u)=\pi^{-1/4}e^{-u^2/2}$, $h_1(u)=\sqrt2u h_0(u)$, and $h_{n+1}(u)=\sqrt{2/(n+1)}u h_n(u)-\sqrt{n/(n+1)}h_{n-1}(u)$; form $\phi_\alpha(x)=\prod_jh_{\alpha_j}(x_j;\rho)$; evaluate the analytic paper eigenvalue routine $\lambda_\alpha(\sigma,\ell,s)$; and return $a(x)=\sum_\alpha\sqrt{\lambda_\alpha}z_\alpha\phi_\alpha(x)$. (3) The analytic eigenpair expressions and derivatives with respect to $\sigma,\ell,s$ are computed by automatic differentiation; estimate $s$ by minimizing $\widehat E_M$ on a fixed coordinate batch, and enforce positivity with softplus. Do not numerically diagonalize a kernel matrix during training. (4) First experiment: fit a $64\times64$ image-coordinate MLP and a small CIFAR-10 convolutional model with a spatial bias field. Compare a dense per-pixel bias, learned Fourier features, and this KL adapter at equal trainable parameter count. Measure validation loss, high-frequency energy, parameter memory, and wall-clock time while sweeping $M\in\{8,16,32,64\}$. Success is lower validation error or comparable error with at least 4x fewer spatial parameters, smooth degradation as $M$ decreases, and no eigensolve cost when $\ell$ changes.