Signed spectral attention

(1) Integration point: implement a linear attention or graph-message-passing layer. Project each token or node to $x_i=W_xh_i$ and $y_j=W_yh_j$, optionally divide both by a learned bandwidth $\ell$; use the signed radial interaction $K(x_i-y_j)$ instead of forming softmax logits. The value tensor is $V=W_vH$. Keep a separate positive normalization, such as $n_i=\sum_j|\widehat K_M(x_i-y_j)|+\epsilon$, rather than dividing by a signed sum. (2) Pseudocode: sample $M$ frequencies $\Omega_m\sim q(\omega)=|\widehat K(\omega)|/C$ and set $s_m=\operatorname{sign}(\widehat K(\Omega_m))$; construct $Z_x=[\cos(X\Omega),\sin(X\Omega)]$ and $Z_y=[\cos(Y\Omega),\sin(Y\Omega)]$; compute $U=(C/M)Z_y^\top V$, replace it by $DU$ with $D=\operatorname{diag}(s_1,s_1,\ldots,s_M,s_M)$, and return $Z_x(DU)$. For cross-attention, use distinct query and key matrices but the same sampled frequencies. (3) What comes from the mathematics: the signed measure, the $C/M$ Monte Carlo scaling, the duplicated cosine/sine factorization, and the sign insertion are fixed by the formulas. Estimate $C$ and sample $q$ numerically for the chosen kernel; for Kuttner-Golubov kernels $K(u)=(1-\|u\|^\alpha)^\beta\mathbf{1}_{\{\|u\|\le1\}}$, obtain $\widehat K$ by numerical radial Fourier integration or a precomputed lookup table, then use rejection sampling with proposal $p_0$ and envelope $B$ accepting $\omega$ with probability $|\widehat K(\omega)|/(Bp_0(\omega))$. (4) First experiment: train a 6-layer, 8-head transformer on CIFAR-10 treated as 196 image patches, and compare standard softmax attention, ordinary positive RFF attention, and signed attention at equal feature width and measured FLOPs. Start with $M\in\{32,64,128\}$ and the parabolic kernel. Measure validation accuracy, wall-clock time, memory, and approximation error against exact pairwise aggregation on a small batch. Success is lower memory and superlinear sequence-length scaling with no more than 1 percentage point accuracy loss; a stronger result is faster loss descent or improved accuracy from the compact negative lobes at equal FLOPs.