# Holonomy-Attractor Recurrent Cell - ID: 2795 - Canonical URL: https://synthcore.org/idea/2795/holonomy-attractor-recurrent-cell - API JSON: https://synthcore.org/api/idea/2795.json - API Markdown: https://synthcore.org/api/idea/2795.md - Verification status: unverified - Source: [arXiv:2608.28894](https://arxiv.org/abs/2608.28894) - Category: dynamics - Solves: stability, accuracy, generalization - ML areas: rnn, ssm, world-model, training-dynamics, regularization - Math tags: dynamical-systems, geometry, linear-algebra, control-theory - Ratings: usefulness 5/10; difficulty 6/10; novelty 8/10 ## Idea 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. ## Mathematical statement 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. ## Key formulas - $$M(s,t)=\begin{pmatrix}1&t&s+\frac{1}{2}t^{2}\\0&1&t\\0&0&1\end{pmatrix},\qquad (s,t)\in\mathbb{R}^{2}.$$ - $$\omega(T(\gamma_n))\to+\infty\quad\Longrightarrow\quad L(\gamma_n)[v]\to\begin{cases}[X],&[v]\in\mathcal{D}^{\geq}_{\omega},\\[-X],&[v]\in\mathcal{D}^{\leq}_{\omega}.\end{cases}$$ - $$\mathcal{L}_{\mathrm{proj}}=\frac{1}{K}\sum_{k=1}^{K}\left(1-\left|\hat{h}_{k}^{\mathsf{T}}\hat{X}\right|\right)+\lambda_{\mathrm{grp}}\left\|\widetilde{M}_{k+1}\widetilde{M}_{k}-\widetilde{M_{k+1}M_{k}}\right\|_{F}^{2}.$$ - $$\hat{h}_{k}=\frac{J_{k}u}{\|J_{k}u\|_{2}},\qquad \hat{X}=\frac{X}{\|X\|_{2}},\qquad J_{k}=\frac{\partial h_{k}}{\partial h_{k-1}}.$$ ## Implementation notes 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. ## Disclaimer AI-generated research hypothesis, automatically tested. Not peer-reviewed.