Fractional-to-Local RG Residual Block

1. Integration point: use this block in a graph neural network, neural operator, vision state-space model, or transformer token mixer. Construct the graph or grid Laplacian L once; for large n, estimate L^{\sigma/2}x with Lanczos/Krylov methods or approximate it by a Chebyshev polynomial. Parameterize \sigma=\sigma_{\min}+(2-\sigma_{\min})\operatorname{sigmoid}(a), so the default branch remains below the predicted crossover. Mix the fractional branch with an ordinary local branch using q=\operatorname{sigmoid}(b(\sigma-2)) or, for a strictly continuous model, q=\operatorname{sigmoid}(b(\sigma-2)/ au). 2. Pseudocode: compute y_frac=frac_laplacian(x,sigma); compute y_local=Lx; set y=(1-q)y_frac+q y_local; update x=x+alpha*MLP(x)-beta*h*y. Estimate lambda_max(L^{sigma/2}) by 10-step power iteration and clip h to 0.9*2/lambda_max. Add a weak penalty gamma*(sigma-2)^2 only if unconstrained training collapses to an extreme exponent. 3. Computed from the paper: the fractional kinetic operator, canonical dimension epsilon'=epsilon-3delta, long-range validity condition, and crossover sigma*=2. Estimated empirically: sigma, gate q, spectral radius, and task loss. 4. First cheap experiment: train a 6-layer graph network on Cora or a 2-D reaction-diffusion next-step prediction task, comparing GCN, fixed sigma=1.5, fixed sigma=2, and the learnable crossover block. Sweep h across the predicted linear boundary h_c=2/lambda_max(L^{sigma/2}); measure whether exploding activations begin near h_c and whether the learned sigma or gate changes rapidly within |sigma-2|<0.1. The quantitative architectural prediction is continuous behavior for sigma<2 followed by a slope change in validation loss, learned spectral response, or gate occupancy at sigma approximately 2; the measured transition should lie within 20% of sigma=2. For a purely linear block, the norm decay should satisfy mode-wise |x_k^{(i)}|=|1-h lambda_i^{sigma/2}|^k|x_0^{(i)}| below h_c.