Brjuno Resonance Curriculum

Use a Fourier neural operator, periodic convolutional model, or diagonal state-space model whose hidden states are represented in Fourier modes. First estimate a reference transport vector ω: for synthetic advection data it is known; for real data fit it from phase shifts by least squares, and stop gradients through this estimate during the first experiment. At every training step, compute rₖ = |k·ω|/(|k| + ε) for every retained mode. Maintain geometrically shrinking thresholds σₙ = σ₀·2⁻ⁿ. Pseudocode: compute the FFT of the residual; mark a mode resonant when rₖ ≤ σₙ; multiply nonresonant updates by 1 and resonant-mode updates by gₙ = min(1, C/(1 + Bᴺ)); add the next resonance shell only when the exponentially weighted nonresonant residual falls below a tolerance. For a direct homological preconditioner, use correctionₖ = residualₖ/(i(k·ω) + δ) only on nonresonant modes and leave resonant modes unpreconditioned. The cone definition and small-divisor correction come from the paper; the finite budget, damping rule, and shell schedule are neural adaptations. The first cheap test should use a two-dimensional periodic advection equation or Kuramoto-style transport, comparing a Fourier neural operator with Adam against the resonance curriculum at equal parameter count. Sweep σ₀ and learning rate. The predicted signature is a sharp rise in gradient norm and rollout error when the active spectrum first contains modes with rₖ < σ; curriculum training should move this boundary to smaller rₖ. Runs with cumulative budget Bᴺ above a fixed threshold should show increasing spectral correction variance. Across seeds, the instability boundary should be reproducible within roughly 20%, while the curriculum should reduce variance specifically near the smallest-divisor modes rather than uniformly reducing all gradients.