Cohomological Quotient RNN

1. Integration point: implement a two-state recurrent cell. The base state is \(x_{t+1}=f_\theta(x_t,\xi_t)\), while the auxiliary state is \(y_{t+1}=a y_t+\rho_\theta(x_t,\xi_t)\). Parameterize \(a=\tanh(\alpha)\) or use \(a=\sigma(\alpha)\) for one-sided contraction, and optionally clip \(|a|\leq a_{\max}<1\). Use a per-step objective \(\varphi_t=g_\theta(x_t,\xi_t)+b y_t\). 2. Pseudocode: first estimate or compute the invariant graph by running a detached burn-in trajectory, h_x = fixed_point_or_long_burnin(f, rho, x); then compute psi_t = g(x_t) + b*rho(x_t)/(1-a) and train with loss = mean(psi_t) rather than mean(phi_t). For exact finite-horizon equivalence, add the boundary correction +(u(x_0,y_0)-u(x_T,y_T))/T, where u=b*(y-h_x)/(1-a). Backpropagate through the quotient loss, and use the full loss only for validation. 3. Computed versus estimated: \(a\), \(b\), and \(\rho\) are model quantities; \(h\) can be estimated by burn-in because contraction gives geometric convergence, or represented by a small auxiliary network trained to satisfy \(h(f(x))\approx ah(x)+\rho(x)\). Estimate the empirical contraction rate from \(\|y_{t+k}-\tilde y_{t+k}\|/\|y_t-\tilde y_t\|\), and estimate the coboundary residual from phi - psi - u + u_next. 4. First cheap experiment: compare a standard GRU or linear SSM against this split cell on sequential MNIST or Copy Memory with identical parameter counts and a synthetic objective containing \(b y_t\). Sweep \(a\in\{0.2,0.5,0.8,0.95,0.99\}\) and horizons \(T\in\{32,128,512\}\). The falsifiable signatures are: paired fiber trajectories must contract with slope \(\log|a|\) on a log-distance plot; the cumulative full-minus-quotient objective must remain bounded while its time-average decays as \(1/T\); and the residual of the cohomological identity should decrease with burn-in as \(|a|^t\). At \(a\to1\), convergence and boundary-error constants should deteriorate sharply, providing a measurable stability boundary rather than a benchmark-only comparison.