Certified Radical Coordinate Layer

(1) Integration point: implement a PyTorch or JAX module for scalar coordinates before an MLP or neural implicit field. Given x with shape B by 1, compute n positive quadratic features r_i(x)=sqrt(P_i(x)), concatenate them with normalized x, and feed the result to the MLP. This replaces Fourier features or a SIREN embedding rather than adding another loss to an existing sinusoidal network. Parameterize each quadratic as P_i(x)=u_i^2 x^2+v_i^2 x+(w_i^2+epsilon), with epsilon=10^{-4}, so the constant term is strictly positive. (2) Pseudocode: x=normalize(x); for i in 1..n: P=u[i]^2*x^2+v[i]^2*x+(w[i]^2+eps); r[i]=sqrt(P); h=MLP(concat(x,r)); y=head(h); loss=task_loss(y,target); backprop(loss); optimizer.step(). Initialize u, v, and w near zero with small random perturbations and use first-layer weight variance proportional to 1/n. (3) The mathematical result supplies the deterministic oscillation budget 2n; it does not require root counting during optimization. Empirically evaluate the learned scalar output on a dense ordered grid, count sign changes while ignoring values with absolute magnitude below 10^{-5}, and measure derivative variation with finite differences. Compare the observed count with the theoretical budget when the output is directly a radical sum; for an MLP after the embedding, report the count as an empirical diagnostic rather than claiming the theorem applies to the whole MLP. (4) First cheap experiment: fit noisy one-dimensional functions on [-1,1], including y=sin(2 pi x)+0.2 sin(30 pi x), using a parameter-matched ReLU MLP, Fourier features, SIREN, and the radical embedding with n equal to 4, 8, and 16. Measure train MSE, held-out MSE, extrapolation error, output sign changes, and sensitivity to label noise. A positive result is lower held-out or extrapolation error than the parameter-matched ReLU baseline, with substantially fewer spurious sign changes than Fourier features or SIREN and stable behavior across random seeds.