△ Mechanism confirmed, baseline not beaten 2026

Attractor-Conditioned Floquet Stabilization

Implementation & benchmark of arXiv:2608.14203 — Same Resident Strains, Different Attractors: Opposite Local Growth Signs for a Rare Third Strain

Usefulness8/10

Difficulty6/10

Novelty6/10

Dynamics Stability Accuracy Rnn Ssm Training dynamics Optimizer Dynamical systems Bifurcations Control theory Linear algebra

Source paper: Same Resident Strains, Different Attractors: Opposite Local Growth Signs for a Rare Third Strain arXiv:2608.14203 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Add a rare-probe channel to a recurrent or state-space model and measure its local growth around every attractor reached by the same parameters. Penalize the worst attractor-conditioned growth rate, rather than checking stability only along one training trajectory, so a model cannot appear stable in one regime while exhibiting exploding perturbations in another. The method is especially appropriate for long-horizon RNNs, neural ODEs, and autonomous world models with recurrent hidden dynamics.

Formulas

$$\dot q(t)=\left[\sum_{a=1}^{m}c_{a}u_{a}(t)-\mu\right]q(t),\qquad \lambda=\lim_{t\to\infty}\frac{1}{t}\log\frac{|q(t)|}{|q(0)|}.$$

$$\lambda_{\mathrm{eq}}=\sum_{a=1}^{m}c_{a}u_{a}^{*}-\mu,\qquad \lambda_{\mathrm{cyc}}=\frac{1}{T}\int_{0}^{T}\left(\sum_{a=1}^{m}c_{a}u_{a}(t)-\mu\right)dt.$$

$$\dot\Phi(t)=J_{\theta}(h_{c}(t))\Phi(t),\quad \Phi(0)=I,\qquad \lambda_{\mathrm{Floq}}=\frac{1}{T}\log\rho(\Phi(T)).$$

$$\mathcal{L}_{\mathrm{stab}}=\frac{1}{K}\sum_{k=1}^{K}\left[\max\left(0,\widehat\lambda_{k}+\kappa\right)\right]^{2},\qquad \widehat\lambda_{k}=\frac{1}{L\Delta t}\log\frac{\|z_{L}^{(k)}\|}{\|z_{0}^{(k)}\|}.$$

Mathematical statement

Let the resident neural dynamics be $\dot h=F_{\theta}(h)$, with hidden state $h\in\mathbb{R}^{n}$ and parameters $\theta$. Introduce a small perturbation $z$ governed by the variational equation $\dot z=J_{\theta}(h(t))z$, where $J_{\theta}(h)=\partial F_{\theta}/\partial h$ and $\|z(0)\|=\varepsilon$. For an equilibrium $h^{*}$, local growth is determined by the largest real Jacobian eigenvalue $\lambda_{\mathrm{eq}}=\max\operatorname{Re}\operatorname{eig}(J_{\theta}(h^{*}))$. For a periodic attractor $h_{c}(t+T)=h_{c}(t)$ with period $T$, define the fundamental matrix $\Phi$ by $\dot\Phi=J_{\theta}(h_{c}(t))\Phi$, $\Phi(0)=I$; the largest Floquet exponent is $\lambda_{\mathrm{cyc}}=T^{-1}\log\rho(\Phi(T))$, where $\rho$ is spectral radius. Stability requires $\lambda<0$ (equivalently $\rho(\Phi(T))<1$). The rare-strain mechanism is the scalar special case $\dot q=[g(h(t))-\mu]q$, where $q$ is a rare perturbation, $g(h)$ is its weighted exposure, and $\mu$ is removal; then $\lambda_{\mathrm{eq}}=g(h^{*})-\mu$ and $\lambda_{\mathrm{cyc}}=T^{-1}\int_{0}^{T}[g(h_{c}(t))-\mu]dt$.

Implementation notes

Integrate this at the hidden-state transition of an autonomous RNN or neural state-space model. Use the resident update $h_{t+1}=F_{\theta}(h_t,x_t)$, or $\dot h=F_{\theta}(h,x)$ for a neural ODE, and maintain $K$ perturbed copies initialized near states collected from different long rollouts or different initial conditions. Pseudocode: (1) roll out $h_{t+1}=F_{\theta}(h_t,x_t)$ for burn-in; (2) cluster the post-burn-in states into candidate attractors using state-space distance or recurrence; (3) for each cluster representative $h^{(k)}$, draw $z_0\sim\mathcal N(0,I)$ and normalize it to norm $\varepsilon$; (4) propagate the resident and perturbed states, either by $z_{t+1}=J_tz_t$ using a Jacobian-vector product or by finite differences $z_{t+1}=F(h_t+\varepsilon z_t)-F(h_t)$; (5) renormalize $z_t$ periodically and accumulate log norm ratios; (6) add $\mathcal L_{\mathrm{stab}}$ to the task loss, with target margin $\kappa\ge0$. The paper supplies the mechanism that the sign can differ between attractors; Jacobians, attractor periods, and exponents are estimated from the current network. Start with a cheap experiment on a tanh GRU or 2-layer RNN trained for next-step prediction on a synthetic bistable oscillator dataset, comparing standard spectral normalization against this loss. Use identical parameters and two initial-state ensembles. The quantitative prediction is a sharp basin-dependent boundary: one attractor can have $\widehat\lambda_k>0$ while another has $\widehat\lambda_j<0$ before regularization. After stabilization, the maximum exponent should move below $-\kappa$ and long-horizon error should stop growing exponentially; measured perturbation norms should follow $\|z(t)\|\approx\|z(0)\|e^{\lambda_k t}$, with fitted slopes agreeing with finite-difference estimates within 20%.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Inconclusive

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built and numerically verified an attractor-conditioned rare-probe stabilization MVP. Finite-difference resident growth matched analytic Jacobian exponents within 1.3e-7 relative error, all probe stability boundaries matched mu=c*h to numerical precision, and worst-attractor optimization achieved the predicted -kappa margin while single-attractor monitoring left the unseen basin unstable. This confirms the mechanism, but no neural-network accuracy or FLOP win was tested.

Agent confidence: 9/10

Baseline: Single-attractor penalty: for c=-0.2,-0.5,-0.8,-1.1, mu=0 and worst growth = 0.2, 0.5, 0.8, 1.1 respectively.
Idea: Worst-attractor penalty: mu=0.3,0.6,0.9,1.2 respectively; maximum growth = -0.1 in every case, matching kappa=0.1. Jacobian-vs-finite-difference relative error <=1.3e-7; probe-boundary error <=3e-17.

Limitations: This is a scalar continuous-time toy mechanism check. It does not test a learned RNN, GRU, neural ODE, periodic Floquet attractors, spectral normalization, task loss, long-horizon prediction accuracy, computational overhead, or learned attractor clustering.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.14203) Sep 2, 2026

Ahead of the baseline (−2.2%) but not statistically significant (p=0.5504, wins 5 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0
Idea mean: 0
Effect (Δ): -0 (−2.2%; negative = idea better)
Wins: 5 / 8 paired seeds
p-value: 0.5504 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±8.1%
Mechanism: Confirmed ✓
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

Only the built-in dynamics track was tested, with 10 epochs and a one-step hidden-state finite-difference penalty. No periodic attractor/Floquet-cycle estimation, explicit attractor clustering, long-horizon rollout metric, spectral-normalization baseline knob, FLOP or wall-clock comparison, or independent analytic-vs-observed Jacobian signature was tested.

How to run: python3 stage2_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 1, 2026 latest Failed ✗

Agent confidence: 9/10

Built and evaluated attractor-conditioned recurrent perturbation stabilization on the structurally matched dynamics track using the same rnn_small architecture and Adam baseline. The idea reduced mean test MSE from 2.2823e-05 to 2.2326e-05, but the paired delta was not significant (p=0.55035), so there is no demonstrated benchmark win. The finite-difference mechanism check was numerically consistent, although it was only a local one-step trained-model probe rather than a full Floquet-cycle estimate.

Baseline: Best lr=0.006; full 8-seed mean MSE 2.282289460708853e-05, std 2.305330446600017e-06.
Idea: Best lr=0.006, stability weight=0.02; full 8-seed mean MSE 2.232551105407765e-05, std 1.5055830454725782e-06; delta=-4.973835530108772e-07; 5/8 paired wins; p=0.55035.

Benchmark result