Nucleation-Controlled Attractor Switching
Implementation & benchmark of arXiv:2608.05251 — Nucleation beyond Equilibrium: Fronts Control Invasion in Bistable Ecosystems
Source paper: Nucleation beyond Equilibrium: Fronts Control Invasion in Bistable Ecosystems arXiv:2608.05251 ⓘ · analyzed Aug 31, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Use the critical-droplet mechanism to control noise injection and perturbation-based switching in bistable recurrent networks or diffusion samplers. Instead of applying uniform noise, estimate front speed and interface cost, then create the smallest spatially localized perturbation expected to exceed the critical droplet size and trigger deterministic growth toward the target attractor.
Formulas
Mathematical statement
Consider the noisy vector dynamics \(d\mathbf{h}=\mathbf{F}_{\theta}(\mathbf{h})dt+\sqrt{2TG(\mathbf{h})}d\mathbf{W}\), where \(\mathbf{h}\) is the hidden state, \(\mathbf{F}_{\theta}\) is the deterministic drift, \(T\) is noise intensity, \(G\) is noise covariance, and \(\mathbf{W}\) is a Wiener process. Let \(v\) be the deterministic invasion speed of the target attractor into the current attractor, and let \(\kappa\) be an empirically estimated effective interfacial cost. The nucleation action is \(\Phi(R)=S_d\kappa R^{d-1}-V_dvR^d\), with critical radius \(R_c=(d-1)\kappa/v\). Under a coarse-grained Gaussian fluctuation approximation, the successful nucleation probability scales as \(P_{\mathrm{nuc}}\asymp\exp[-\Phi(R_c)/T_{\mathrm{eff}}]\), where \(T_{\mathrm{eff}}\) is estimated from hidden-state innovation variance. Thus larger front speed reduces the barrier while larger interface cost increases it.
Implementation notes
Integrate the controller into a spatial ConvGRU, neural cellular automaton, or iterative energy-based sampler whenever the hidden state is trapped in an undesired attractor. Maintain a soft mask identifying cells closer to the current and target prototypes. Every K recurrent steps, run a detached calibration rollout: insert target-state circular patches with several radii R, evolve for H steps without gradients, and fit v from the change in target-mask area or interface position. Estimate R_c as the smallest tested radius whose expected area increases; compute kappa=v*R_c/(d-1). Estimate T_eff as the median per-cell variance of recent hidden-state innovations. If the target patch is subcritical, enlarge it to R_c or inject localized Gaussian noise only inside that patch; if it is supercritical, stop injecting noise and allow deterministic dynamics to amplify it. Use a controller loss for failed switching and an L2 penalty on perturbation magnitude. The first experiment should use a 32x32 bistable denoising task with images deliberately initialized in the wrong attractor, comparing uniform Gaussian noise, simulated annealing, and this controller. Measure switching probability over repeated seeds as a function of patch radius and noise level. The predicted signature is that the 50-percent switching radius tracks R_c within 20 percent and that plotting log switching probability against \(\Phi(R_c)/T_{\mathrm{eff}}\) is approximately linear. At fixed kappa and T_eff, doubling v should reduce the barrier by a factor of approximately \(2^{-(d-1)}\).
Verification
Stage 1 · Toy mechanism gate: Failed ✗
Stage 2 · Mechanism transferred to benchmark: Not tested
Stage 2 · Practical benchmark result: Not run
Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works
Stage 1 — Mechanism check agent confidence 9/10
Built a reproducible Allen–Cahn bistable lattice MVP with front-speed calibration, early-area critical-radius estimation, localized stochastic perturbations, and algebraic checks. The formula maximization matched the predicted Rc to 0.004%, and the explicit d=2 prediction that doubling v halves both Rc and the barrier was exactly reproduced; measured vRc across the two usable dynamical sweeps varied by about 8%, within the intended 20% signal. The stochastic experiment produced zero switching for both localized and matched-total-variance uniform noise at all tested temperatures, so the promised exponential nucleation signature and practical controller win were not observed.
- Agent confidence
- 9/10
- Baseline
- Matched-total-variance uniform noise: switching probability 0/40 at T = 0.005, 0.01, 0.02, 0.04, 0.08.
- Idea
- Localized target patch: switching probability 0/40 at every tested T; deterministic calibration gave Rc=4.44 at v=0.245 and Rc=2.80 at v=0.420, with vRc=1.087 and 1.177. Exact algebra Rc relative error=3.97e-5; explicit barrier ratio at 2v=0.5 predicted and observed.
Limitations: This is a toy Allen–Cahn lattice rather than a ConvGRU/NCA or trained denoising task. The stochastic range was too low relative to the empirical barrier, giving saturated zero probabilities; no meaningful log-probability fit, repeated controller calibration overhead, learned prototypes, simulated annealing baseline, FLOP/speed comparison, or 32x32 image benchmark was tested. The inverse-speed dynamical trend had only two finite critical-radius points because stronger biases were already supercritical at the smallest tested radius.
How to run: python3 nucleation_experiment.py
Verdict computed by deterministic test code from paired-seed statistics — not by the language model.
Artifacts
- nucleation_experiment.py 4.3 KB View Raw
- report.md 1.9 KB View
- results.json 4.6 KB View Raw