Coxeter Folding Reversible Recurrence

Implement a recurrent block whose hidden state is an n-point complex polygon, optionally augmented with ordinary feature channels. Choose a periodic fold schedule j_1,...,j_L corresponding to a staircase; begin with a fixed alternating schedule and later learn a soft categorical distribution over schedules. At fold j, read the three neighboring points, compute the new point using the displayed rational expression, and leave every other point unchanged. Apply affine gauge normalization after each full Coxeter cycle because translation and complex scaling are nuisance symmetries. For memory-efficient training, do not save intermediate activations: during backward reconstruction, rerun the folds in reverse order, using the same involution. A small learned input-injection network may act once per cycle, followed by projection back onto the fold-generated manifold. The fold rule, quasi-periodic indexing, and reverse-order inverse are taken from the paper; learned schedules and feature readouts are adaptations. First run sequential MNIST or a synthetic polygon-dynamics task, comparing a GRU and a reversible coupling layer at equal hidden width. Measure activation memory, forward-reverse reconstruction error, gradient norm versus sequence length, and the largest finite-time Lyapunov exponent. The predicted signature is a sharp failure boundary when a fold denominator approaches zero; away from that boundary, e_T should remain near floating-point roundoff and memory should be nearly independent of sequence length. An integrable schedule should also show lower long-horizon Jacobian growth than a randomly ordered fold schedule.