Nonresonant Quasi-Periodic Output Averaging

Integrate the mechanism at the output of a recurrent model, S4-style state-space model, or neural ODE used for long-horizon prediction. Add a fixed phase state z in [0,1)^d, update z <- (z + alpha) modulo one at every forecast step, and concatenate sine and cosine features of each phase coordinate to the model input. Maintain the last N outputs or run N phase-shifted evaluations, then return their weighted average. Start with d=1 or d=2 and choose alpha from fractional parts of irrational numbers such as square root of 2 and square root of 3. Construct weights offline by solving the convex quadratic program that minimizes the sum over k in a calibration set K of the squared magnitude of H_N(k), subject to the weights summing to one; add a small smoothness penalty and initially constrain the weights to be nonnegative. Estimate K from an FFT of calibration residuals, while H_N(k) is computed exactly from the selected weights and alpha. The first cheap experiment should use a synthetic torus signal with slowly decaying Fourier coefficients, comparing uniform, Hann, and optimized windows over 1,000 initial phases. Repeat with a small GRU forecasting a sinusoid plus colored noise at horizons 32, 64, 128, and 256. The prediction is that optimized-window error follows log E_N approximately equal to negative c N raised to sigma, whereas uniform averaging is algebraic. Fit both models and require the stretched-exponential model to predict held-out horizons within 20 percent. Also require the optimized response magnitude for calibration modes to be at least ten times smaller than the uniform-window response.