Denjoy Affine-Orbit Memory

1. Integration point: implement this as a recurrent memory branch alongside a GRU, vanilla RNN, or state-space model. Maintain a scalar phase θₜ and update θₜ₊₁ = (θₜ + α) mod 1. Use either a differentiable soft assignment over N cells or a hard nearest-cell assignment. Each cell stores an affine payload map Aₙh + bₙ, and its output is combined with the main network through a learned gate. For a finite prototype, choose N cells and α = √2 − 1 modulo 1; use periodic indexing n mod N. 2. Pseudocode: parameterize positive widths as lₙ = softplus(uₙ) + ε, normalize them, compute sₙ = lₙ₊₁/lₙ, identify the active cell n from θₜ, update the phase coordinate zₜ₊₁ = aₙ₊₁ + sₙ(zₜ − aₙ), update the payload hₜ₊₁ = Aₙhₜ + bₙ, and advance θ. Add λ Σₙ(log sₙ)² if bounded distortion is desired. 3. Computed directly: widths, local slopes, total width, and telescoping endpoint ratios. Estimated empirically: task loss, gradient norms, payload Jacobian products, and delayed-retrieval accuracy. 4. First cheap experiment: compare this module with a matched-parameter GRU on sequential MNIST and a delayed-copy task with delays from 10 to 10,000. Test uniform widths, fixed nonuniform widths, and learned widths. The falsifiable prediction is that the measured product of phase slopes after m steps equals lₙ₊ₘ/lₙ within 1% away from lookup boundaries. In the finite N-cell version, the phase product after exactly N steps must equal 1. Phase instability should appear only when the payload Jacobian product exceeds 1, not merely from iterating the phase branch.