Hadamard fractal Fourier encoding

Integrate this at the input embedding of a coordinate MLP, NeRF, or implicit image representation. Normalize each scalar coordinate to the interval containing the target support. Choose a small base \(N\), spatial digit set \(\mathcal D\), and frequency digit set \(\mathcal L\) with equal cardinality \(q\); enumerate small integer candidates until \(\|H^*H-I_q\|_F<10^{-6}\), where \(H_{d,l}=q^{-1/2}\exp(2\pi i dl/N)\). For depth \(J\), generate frequencies by the recursion \(\Lambda_0=\{0\}\) and \(\Lambda_{j+1}=\{\lambda+lN^j:\lambda\in\Lambda_j,\ l\in\mathcal L\}\). For each selected multiplier \(t\), produce real features \([\cos(2\pi t\lambda x),\sin(2\pi t\lambda x)]\) and concatenate them before the MLP. Pseudocode: initialize an empty list; for each \(j=0,\ldots,J-1\), append \(\lambda+lN^j\) for all current \(\lambda\) and \(l\); then evaluate the trigonometric features. The Hadamard test and digit recursion are computed exactly from the paper. Estimate practical conditioning using \(G=\Phi^T\Phi/B\), where \(\Phi_{n,k}=\cos(2\pi\lambda_kx_n)\) and corresponding sine columns. Start with 1D fractal regression using a four-layer ReLU MLP, comparing against dyadic positional encoding and Gaussian random Fourier features at equal feature count. Track validation MSE, the condition number of \(G\), early loss descent at equal FLOPs, and high-frequency reconstruction error. A positive result is a lower condition number plus faster loss reduction or 1--2 dB higher PSNR at equal parameter count, especially when training coordinates are sampled from a clustered distribution approximating \(\mu\).