Algebraic Pinch-Curve Spectral Layer

Integrate this as a replacement for the dense spectral kernel in a 3D Fourier neural operator, spectral CNN, or volumetric U-Net block. The input is a feature tensor $x\in\mathbb R^{B\times C\times H\times W\times D}$. First compute its FFT, producing $\widehat x(k)$ on the discrete frequency grid. Parameterize a plane constraint $P_1(k)=a_xk_x+a_yk_y+a_zk_z$ with trainable vector $a$, and parameterize a second polynomial as $P_2(k)=b_xk_x+b_yk_y+b_zk_z+c_3(k)$, where $c_3(k)$ is a homogeneous cubic with trainable coefficients. To avoid scale degeneracy, normalize $a$ and normalize the cubic coefficient vector after every optimizer step. For every frequency, compute $r(k)=P_1(k)^2+P_2(k)^2$ and the curve gate $G(k)=\exp(-r(k)/\tau)$; use a small positive floor $G_{\min}$ if complete suppression is harmful. Apply a channel-mixing matrix $W(k)$, shared across frequencies or represented by a small radial MLP, as $\widehat y(k)=G(k)W(k)\widehat x(k)$. Pseudocode: K=fft_freq_grid(); P1=dot(K,a); P2=dot(K,b)+cubic(K,C); r=P12+P22; G=exp(-r/tau); Y=G[...,None,None]*matmul(W,Xhat); y=ifft(Y). Symmetrize numerically with r=0.5*(r+r.flip_frequency) to enforce inversion protection. The paper supplies the algebraic construction; estimate $\tau$ empirically from the median of $r(k)$ and optionally regularize coefficients with $\|a\|_2=1$. Begin with a small 3D Burgers, Navier-Stokes, or synthetic anisotropic advection benchmark, comparing a standard FNO with the same channel width and a dense spectral kernel. Measure validation error, parameter count, and FFT-block throughput. Success is lower error at equal parameter count, or at least a 5x reduction in spectral parameters with no loss in accuracy; inspect learned frequency responses to verify concentration around a stable curve rather than collapse to one frequency.