Laguerre Memory Convolution

Integrate this at the long convolution operation of a causal SSM/RNN or temporal convolution block, not at attention logits. Inputs are a batch of sequences $u\in\mathbb{R}^{B\times T\times d}$, a truncation length $L\leq T$, a basis size $K\ll L$, and a fixed or learned decay parameter $a\in(0,1)$. Precompute a basis matrix $B\in\mathbb{R}^{L\times K}$ by generating the first $L$ impulse-response samples $\ell_k[n]$ of the rational transfer functions $L_k(z)$; generate each response by filtering a discrete impulse through the stable denominator $(1-a z^{-1})^{k+1}$, and optionally orthonormalize the finite-window columns because truncation can make them non-orthogonal. The trainable parameters are coefficient vectors $c\in\mathbb{R}^{K\times d}$, or $C\in\mathbb{R}^{K\times d_{in}\times d_{out}}$ for channel mixing. At each forward pass form the kernel $h=Bc$ and compute causal convolution by zero-padding $u$ and $h$ to at least $T+L-1$, applying real FFTs, multiplying spectra, inverse FFTing, and cropping the first $T$ positions. Pseudocode is: B = laguerre_basis(L,K,a); h = B @ c; U = rfft(pad(u)); H = rfft(pad(h)); y = irfft(U * H)[:T]. Estimate $a$ by a small grid such as $\{0.8,0.9,0.95,0.98,0.995\}$, or learn $a=\sigma(\alpha)$ with clipping below one. Compute $\kappa(B)=\sigma_{max}(B)/\sigma_{min}(B)$ periodically and add a penalty on excessive conditioning if needed. The first cheap experiment should use a synthetic damped-system prediction task or 1D audio forecasting with $T=4096$, comparing a length-1024 direct convolution and a standard SSM baseline against $K\in\{16,32,64\}$. Measure validation error, parameter count, peak memory, and wall-clock throughput. Success means at least 4x fewer kernel parameters with no more than 1% relative error increase, or lower error at equal parameter count; also test sequences twice as long as those used in training and check that the learned impulse response remains bounded and does not develop unstable long-lag oscillations.