# Denjoy Affine-Orbit Memory

- ID: 767
- Canonical URL: https://synthcore.org/idea/767/denjoy-affine-orbit-memory
- API JSON: https://synthcore.org/api/idea/767.json
- API Markdown: https://synthcore.org/api/idea/767.md
- Verification status: unverified
- Source: [arXiv:2607.11748](https://arxiv.org/abs/2607.11748)
- Category: dynamics
- Solves: stability, memory, accuracy
- ML areas: rnn, ssm, memory, training-dynamics
- Math tags: dynamical-systems, geometry, linear-algebra
- Ratings: usefulness 5/10; difficulty 6/10; novelty 8/10

## Idea description

Add a bounded phase variable and a bank of local affine transport maps to an RNN or state-space model. The phase follows an irrational rotation, while the hidden state is transported through cells whose widths determine local gains, giving a controllable memory mechanism with analytically known distortion rather than an unconstrained recurrent Jacobian.

## Mathematical statement

The construction replaces the rotation-orbit point xₙ = {nα} by an interval Iₙ of length lₙ > 0, where α is irrational and L = Σₙ lₙ is finite. The lifted map sends Iₙ to Iₙ₊₁. Let Iₙ = [aₙ, aₙ + lₙ] and define the affine phase map F(z) = aₙ₊₁ + sₙ(z − aₙ), with slope sₙ = lₙ₊₁/lₙ. For a payload state h in Rᵈ, use a cell-dependent affine update h⁺ = Aₙh + bₙ. The phase update is θ⁺ = θ + α modulo 1, and a lookup maps θ to the active cell. The key identity is that the product of phase slopes over m steps telescopes: Πₖ₌₀ᵐ⁻¹ sₙ₊ₖ = lₙ₊ₘ/lₙ. Therefore, long-horizon phase transport has no hidden gain beyond the ratio of endpoint widths. Equal widths produce unit slopes, while bounded width ratios produce bounded phase distortion.

## Key formulas

- L = Σₙ∈ℤ lₙ < ∞,    xₙ = {nα},    α ∉ ℚ
- p(t) = sup{x ∈ [0,1): Σ_{n: {nα} ∈ [0,x)} lₙ ≤ t},    p(Iₙ) = xₙ
- F(z) = aₙ₊₁ + (lₙ₊₁/lₙ)(z − aₙ),    z ∈ Iₙ = [aₙ, aₙ + lₙ]
- Πₖ₌₀ᵐ⁻¹ (lₙ₊ₖ₊₁/lₙ₊ₖ) = lₙ₊ₘ/lₙ

## Implementation notes

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.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.