Cohomological Jacobian Flattening

Integrate this into the transition function of an invertible RNN, state-space model, neural ODE discretization, or normalizing-flow-like recurrent map. Let the state be $x_t\in\mathbb{R}^d$ and define $x_{t+1}=F_\theta(x_t,a_t)$; initially omit the action/input from the Jacobian calculation or treat it as fixed. Use a small MLP $u_\phi(x)$ to represent the scalar potential and a learned scalar $c$. For each minibatch of states $x_i$, compute $y_i=F_\theta(x_i,a_i)$, the Jacobian log-volume $\ell_i=\log|\det DF_\theta(x_i,a_i)|$, and the residual $r_i=\ell_i-u_\phi(y_i)+u_\phi(x_i)-c$. Add $\lambda\mathcal{L}_{\mathrm{coh}}=\lambda B^{-1}\sum_i r_i^2$ to the task loss. For small $d$, compute the determinant by autograd Jacobian construction and slogdet; for larger $d$, use an architecture with triangular Jacobian, or estimate the trace of $\log|DF|$ only when a reliable matrix-log estimator is available. Alternate ordinary gradient updates of $(\theta,\phi,c)$, and constrain the gauge of $u_\phi$ by subtracting its minibatch mean. The mathematical quantity is the one-step coboundary residual; empirically estimate whether it telescopes by measuring $R_k(x)=\log|\det DF_\theta^k(x)|-[u_\phi(F_\theta^k(x))-u_\phi(x)+kc]$ over unrolled trajectories. Also compare against the stronger baseline $\mathbb{E}[\ell_\theta(x)-c]^2$, which forces pointwise log-Jacobian constancy, and against no Jacobian regularizer. First test on a 4- or 8-dimensional invertible residual map trained to predict the next state of a synthetic chaotic system, using 32- or 64-step training and 512-step rollout evaluation. Success means lower variance of $R_k/k$, fewer exploding or collapsed trajectories, bounded gradient norms over long unrolls, and equal or better one-step prediction at the same task loss. A useful target is at least a 2x increase in stable rollout horizon or a 30% reduction in gradient variance relative to the pointwise baseline.