Pair-Hydrodynamic Long-Memory State Space

First, insert the module between the input encoder and prediction head of a long-horizon forecasting model. Choose M low-frequency wave numbers k_m equal to 2 pi m divided by L and maintain complex latent states z_m; impose z_(-m) equal to the complex conjugate of z_m when real outputs are required. Add a low-rank pair bank q_m equal to z_m z_(-m), or a small set of q_(m,n) equal to z_m z_n for multichannel data. Pseudocode is: initialize z_m to zero; for each input u_t, update z_m to exp(-D k_m squared Delta t) times z_m plus B_m u_t; compute q_m equal to z_m z_(-m); concatenate z, q, and encoder features; predict y_hat_t equal to C z plus P q plus R h_t. Parameterize D as softplus(d) and use the exact exponential update, which guarantees decay for every positive time step. The paper-derived quantities are the diffusion eigenvalues, pair eigenvalues, and hydrodynamic pair construction. Estimate D, B, readout coefficients, and overlap weights w(k) from data. In a cheap experiment, train on synthetic one-dimensional diffusion data with the target y_t equal to the spatial integral of x(r,t) squared. Compare a diagonal SSM, a single-mode SSM, and the pair model at equal parameter count. Fit the late-time impulse response to A times t to the power minus alpha. The prediction is alpha approximately equal to one half in one dimension when w(0) is nonzero, with A proportional to w(0) D to the power minus one half. Removing pair modes should eliminate this algebraic tail or replace it with excessively rapid decay. Changing D by a factor of four should change the predicted tail amplitude by a factor of approximately two.