Marginal Fractional Coupling Layer

1. Integration point: add the operator to a residual sequence model, SSM, or deep network whose states are indexed by token position or layer depth. For hidden states H with shape [batch, length, channels], compute an FFT over the length dimension, multiply each frequency by |k|, inverse FFT, and use either a residual layer H_new=H+g(MLP(H))-alpha q or a regularizer lambda_frac E_1(H) added to the task loss. Use a real FFT, set the zero-mode multiplier to zero, and normalize frequencies by the Nyquist frequency. 2. Pseudocode: h_hat=rfft(H); q_hat=abs(freq)*h_hat; q=irfft(q_hat); H_new=H+beta*MLP(H)-alpha*q; loss=task_loss+lambda_frac*mean(abs(freq)*abs(h_hat)^2). Choose alpha initially as 0.9*2/lambda_max and optionally estimate the realized Jacobian gain with power iteration. 3. Computed from the paper: the |k| multiplier, fractional energy, algebraic-correlation prediction, and twist-response scaling. Estimated empirically: eta from a linear regression of log C(r) on log r over the middle 50 percent of distances, and the largest effective Jacobian eigenvalue. 4. First cheap experiment: compare a small residual MLP or SSM on Copy, Adding, and Long Range Arena ListOps at lengths 128, 256, 512, and 1024 against no coupling and first- or second-difference penalties. The predicted signature is an approximately length-independent intermediate-distance power law C(r) proportional to r^{-eta}, rather than exponential decay. Sweep alpha and verify that divergence begins near alpha_c=2/lambda_max within 20 percent after accounting for nonlinear gain. Also apply a phase ramp H_j to exp(i theta j/L)H_j and fit the response versus L; the prediction is a power law L^{-eta}, not a size-independent stiffness.