{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace or augment a low-dimensional recurrent transition with affine maps whose linear parts belong to a structured unipotent holonomy family, and train the cell so that positive accumulated translation produces a controlled projective attractor. This creates a measurable two-basin long-horizon behavior: hidden-state perturbation directions should align with a learned direction X or its antipode according to the sign of a scalar functional, rather than exhibiting unconstrained rotation or exploding anisotropy.",
 "formulas_latex": [
  "$$M(s,t)=\\begin{pmatrix}1\u0026t\u0026s+\\frac{1}{2}t^{2}\\\\0\u00261\u0026t\\\\0\u00260\u00261\\end{pmatrix},\\qquad (s,t)\\in\\mathbb{R}^{2}.$$",
  "$$\\omega(T(\\gamma_n))\\to+\\infty\\quad\\Longrightarrow\\quad L(\\gamma_n)[v]\\to\\begin{cases}[X],\u0026[v]\\in\\mathcal{D}^{\\geq}_{\\omega},\\\\[-X],\u0026[v]\\in\\mathcal{D}^{\\leq}_{\\omega}.\u0000\\end{cases}$$",
  "$$\\mathcal{L}_{\\mathrm{proj}}=\\frac{1}{K}\\sum_{k=1}^{K}\\left(1-\\left|\\hat{h}_{k}^{\\mathsf{T}}\\hat{X}\\right|\\right)+\\lambda_{\\mathrm{grp}}\\left\\|\\widetilde{M}_{k+1}\\widetilde{M}_{k}-\\widetilde{M_{k+1}M_{k}}\\right\\|_{F}^{2}.$$",
  "$$\\hat{h}_{k}=\\frac{J_{k}u}{\\|J_{k}u\\|_{2}},\\qquad \\hat{X}=\\frac{X}{\\|X\\|_{2}},\\qquad J_{k}=\\frac{\\partial h_{k}}{\\partial h_{k-1}}.$$"
 ],
 "id": 2795,
 "implementation": "Use this mechanism in a 2D or 8D latent RNN or state-space model, initially with a 2D diagnostic subspace. Parameterize the recurrent transition in homogeneous coordinates. Given hidden state h, input x, and unconstrained scalars s and t produced by a small input network, construct M(s,t), apply its affine action to h, add a bounded input residual, and use a sigmoid gate to interpolate with the previous hidden state. For higher-dimensional states, apply the same cell to disjoint two-dimensional planes or to a learned two-dimensional projection P h, while retaining an orthogonal residual branch for expressivity. Pseudocode is: compute s and t; build M(s,t); compute the affine update h_bar; set h_new equal to (1 minus gate) times h plus gate times tanh(h_bar); accumulate products of the linear parts and translations using homogeneous matrix multiplication; compute a Jacobian-vector product J_k u; and add the projective loss. The group matrix and two-basin limit are taken from the paper, while X, omega, and finite-time convergence rates are estimated from trajectories. Start with sequential MNIST or a length-512 copy task using a 2D vanilla tanh RNN, the structured cell, and the structured cell without the projective loss. Perturb hidden states and measure angular error to X versus cumulative q_k=omega(T_{1:k}). The falsifiable prediction is monotone angular alignment in each sign basin as q_k grows, with failure when q_k changes sign or remains bounded. Also measure long-horizon perturbation variance and compare its decay against an exponential or power-law fit before evaluating task accuracy.",
 "math_summary": "An affine map is represented in homogeneous coordinates by z mapped to Mz, where z is the augmented state vector and M has a two-dimensional linear part and a two-dimensional translation. The paper's complete-affine holonomy family is the two-parameter unipotent group G1 with matrix M(s,t). For a loop or product indexed by gamma, L(gamma) is its linear part, T(gamma) is its translational part, and omega is a scalar linear functional on translations. If omega(T(gamma_n)) tends to positive infinity, the theorem states that L(gamma_n)[v] tends to [X] for every projective direction [v] in the nonnegative half-circle determined by omega, while it tends to [-X] for directions in the nonpositive half-circle. The neural adaptation estimates X and omega from the learned recurrent cell and penalizes violations of this projective alignment while retaining the affine transition for task inputs.",
 "math_tags": [
  "dynamical-systems",
  "geometry",
  "linear-algebra",
  "control-theory"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "world-model",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.28894",
  "arxiv_url": "https://arxiv.org/abs/2608.28894",
  "summary_what_math_gives_to_ml": "The paper provides a concrete projective-dynamics mechanism for complete affine torus tilings: affine holonomy elements with sufficiently large positive translational invariant drive almost every initial direction toward one of two distinguished projective directions, with the basin selected by the sign of a linear functional. The transferable asset is not the billiard geometry itself, but a parameterized family of structured affine recurrent maps with analytically predictable directional attraction. A neural recurrent architecture can use this as a long-horizon stabilization module or as a regularizer on products of hidden-state Jacobians. The mechanism is most plausible for low-dimensional latent state-space models, where projective directions and cumulative affine translation can be computed cheaply and the predicted angular-collapse behavior can be directly falsified.",
  "title": "Symplectic Tiling Billiards on Complete Affine Tori",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 8,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Holonomy-Attractor Recurrent Cell",
 "url": "https://synthcore.org/idea/2795/holonomy-attractor-recurrent-cell",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}