# Vortex-Criticality Controller for Phase RNNs

- ID: 2252
- Canonical URL: https://synthcore.org/idea/2252/vortex-criticality-controller-for-phase-rnns
- API JSON: https://synthcore.org/api/idea/2252.json
- API Markdown: https://synthcore.org/api/idea/2252.md
- Verification status: unverified
- Source: [arXiv:2608.13658](https://arxiv.org/abs/2608.13658)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: rnn, ssm, training-dynamics, regularization, optimizer
- Math tags: dynamical-systems, statistical-mechanics, bifurcations, control-theory, geometry
- Ratings: usefulness 6/10; difficulty 6/10; novelty 8/10

## Idea description

Represent recurrent hidden states as compact phases and monitor spacetime vortices, defined by wrapped phase differences around elementary space-time plaquettes. Add a feedback controller that increases relaxation toward the homogeneous phase when vortex activity becomes supercritical, while allowing larger recurrent gain when the system is excessively quiescent. This creates a falsifiable operating regime: useful computation should occur near, but below, the defect-proliferation transition rather than at arbitrary recurrent gain.

## Mathematical statement

Let \(\theta_i^t\in(-\pi,\pi]\) be the phase at hidden-unit or spatial index \(i\) and time \(t\), and let \([x]_{2\pi}\in(-\pi,\pi]\) denote principal-branch reduction. The paper's generic deterministic dynamics are \(\mathcal D_t\theta(i,t)=\mathcal K_i[\mathcal D_x\theta]\), where \(\mathcal D_x\theta(i,t)=[\theta_{i+1}^t-\theta_i^t]_{2\pi}\). For a space-time plaquette \(p=(i,t)\), define the integer vortex charge \(q_{i,t}=(2\pi)^{-1}\sum_{\ell\in\partial p}s_\ell[\mathcal D_\mu\theta]_{2\pi}\), with \(\mu=x,t\), oriented signs \(s_\ell\in\{+1,-1\}\), and \(q_{i,t}\in\mathbb Z\). A homogeneous state has \(\mathcal D_x\theta=0\), suppressing vortex nucleation. In the neural implementation, use relaxation parameter \(\alpha\in[0,1]\), recurrent gain \(g\), and defect density \(\rho=\langle\mathbf 1[q_{i,t}\neq0]\rangle\). The controller targets \(\rho<\rho_{\rm max}\) by adapting \(g\) or \(\alpha\). The predicted active-phase signature is \(\rho(g)\propto(g-g_c)^{\beta_{\rm DP}}\) above a critical gain \(g_c\), with \(\rho\to0\) below \(g_c\); finite-size scaling can be tested using \((g-g_c)L^{1/\nu_\perp}T^{1/\nu_\parallel}\), with exponents fitted empirically.

## Key formulas

- $$\mathcal{D}_{t}\theta(i,t)=\mathcal{K}_{i}\left[\mathcal{D}_{x}\theta\right],\qquad \mathcal{D}_{x}\theta(i,t)=\left[\theta_{i+1}^{t}-\theta_i^{t}\right]_{2\pi}.$$
- $$q_{i,t}=\frac{1}{2\pi}\sum_{\ell\in\partial p_{i,t}}s_\ell\left[\mathcal{D}_{\mu}\theta\right]_{2\pi},\qquad \rho=\frac{1}{|\mathcal P|}\sum_{(i,t)\in\mathcal P}\mathbf 1\{q_{i,t}\neq0\}.$$
- $$\theta_i^{t+1}=\operatorname{Arg}\!\left((1-\alpha)e^{\mathrm i\theta_i^t}+\alpha e^{\mathrm i(\theta_i^t+g h_i^t+\kappa\Delta\theta_i^t)}\right),\qquad \Delta\theta_i^t=\left[\theta_{i+1}^t-2\theta_i^t+\theta_{i-1}^t\right]_{2\pi},$$
- $$\rho(g)\sim A(g-g_c)^{\beta_{\rm DP}}\quad(g>g_c),\qquad \rho(g)=0\quad(g<g_c)\ \text{in the infinite-size limit}.$$

## Implementation notes

Integrate this as a recurrent layer with a phase-valued hidden state \(\theta^t\in\mathbb R^N\), arranged on a one-dimensional ring. Encode the input with \(h_i^t=(W_xx^t)_i+(W_h[\cos\theta^t,\sin\theta^t])_i\), and update the compact state using the Arg rule in the formulas. The recurrent gain \(g\) controls drive, \(\alpha\) controls relaxation, and \(\kappa\) supplies local spatial coupling. Read out from concatenated sine and cosine features. After each sequence, calculate wrapped spatial and temporal link differences, sum the four oriented links around each space-time plaquette, round the result divided by \(2\pi\) to obtain \(q_{i,t}\), and compute \(\rho\). Adapt the gain with \(g\leftarrow\operatorname{clip}(g+\eta_g(\rho-\rho_{\rm target}),g_{\min},g_{\max})\), or add \(\lambda_\rho\max(0,\rho-\rho_{\rm target})^2\) to the loss. Use stop-gradient through the controller initially. The paper supplies compact topology, vortex charge, and the competition between homogeneous relaxation and defect proliferation; estimate \(g_c\), the power-law exponent, and useful \(\rho_{\rm target}\) empirically. First test on vanilla tanh RNN, uncontrolled phase RNN, and controlled phase RNN using sequential MNIST or the adding task at lengths 100, 500, and 1000. Sweep at least 20 gains and measure loss, gradient norms, memory accuracy, and \(\rho\). The quantitative prediction is a reproducible crossover at \(g_c\): below it, defect density and long-horizon instability decay toward zero; above it, \(\rho\) grows approximately as \((g-g_c)^{\beta_{\rm DP}}\). A stronger test is finite-size scaling collapse across ring sizes; absence of a threshold or absence of improved memory near the subcritical boundary falsifies the transfer.

## Disclaimer

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