C1 multiresolution spline features

(1) Integration point: modify the coordinate-input stage of a coordinate MLP, neural field, PDE surrogate, or diffusion spatial conditioner. Normalize each coordinate x to [-1,1]^d and concatenate outputs from R+1 trainable quadratic B-spline grids before the first linear layer. Choose grid spacings h_r=h_0/2^r, with the finest spacing selected from the expected minimum feature width rather than relying on training to discover it. (2) Pseudocode: initialize coefficient tables A[r] with small Gaussian values; for each batch of coordinates x, find the three nonzero neighboring knots per dimension at every level; evaluate B2 and, when derivative losses are used, its analytic first and second derivatives; compute S[r]=sum_k A[r,k] product_j B2(x_j/h_r-k_j); form z=concat(x,S[0],...,S[R]); predict y=MLP(z); compute the task loss; backpropagate through A and the MLP; update with AdamW. Optionally learn alpha[r] using softplus-constrained positive parameters. (3) Compute compact-support interpolation and derivatives exactly. Estimate only empirical design choices such as R, h_R, channel count, and whether raw coordinates are retained. Numerically check C1 continuity by finite-differencing gradients on both sides of random knot boundaries. (4) First cheap experiment: fit a 2D synthetic field containing a narrow Gaussian ridge and smooth background, comparing equal-memory Fourier features, a ReLU MLP, and the spline-augmented MLP. Measure target-field error, derivative error, loss versus optimizer steps, and wall-clock time. Success is lower narrow-ridge and derivative error or reaching a fixed error in fewer steps at equal parameter count and memory.