# Lyapunov-Tuned Random Blaschke RNN

- ID: 964
- Canonical URL: https://synthcore.org/idea/964/lyapunov-tuned-random-blaschke-rnn
- API JSON: https://synthcore.org/api/idea/964.json
- API Markdown: https://synthcore.org/api/idea/964.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.15488](https://arxiv.org/abs/2607.15488)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: rnn, ssm, training-dynamics, world-model
- Math tags: dynamical-systems, bifurcations, complex-analysis, probability, control-theory
- Ratings: usefulness 8/10; difficulty 6/10; novelty 7/10

## Idea description

Replace an unconstrained recurrent transition by a randomly switched composition of disk-preserving Blaschke maps. The recurrent state remains in the unit disk, while the estimated average logarithmic derivative provides a direct synchronization-versus-chaos control knob: negative transverse growth should make two states driven by the same input or map sequence synchronize, whereas positive growth should preserve sensitivity and expressive memory.

## Mathematical statement

For a disk state z_t in D={z in C:|z|<1}, use T_j(z)=exp(i theta_j) product over k of (z-a_jk)/(1-conjugate(a_jk)z), where |a_jk|<1 and j is sampled from a finite distribution p_j. The random cocycle is z_{t+1}=T_{j_t}(z_t), and its finite-time derivative exponent is lambda_hat_T=(1/T) sum from t=0 to T-1 of log(|T'_{j_t}(z_t)|). A negative exponent predicts contraction and synchronization of trajectories with the same map sequence; a positive exponent predicts loss of synchronization and chaotic sensitivity. The disk-preserving construction prevents radial explosion independently of the sign of the exponent. Inputs can be injected through a bounded disk-valued map before applying T_j.

## Key formulas

- $$T_j(z)=e^{i\theta_j}\prod_{k=1}^{d_j}\frac{z-a_{jk}}{1-\overline{a_{jk}}z},\qquad |a_{jk}|<1,$$
- $$z_{t+1}=T_{j_t}(z_t),\qquad \widehat{\lambda}_T=\frac{1}{T}\sum_{t=0}^{T-1}\log\left|T'_{j_t}(z_t)\right|,$$
- $$\|z_T-\widetilde z_T\|\approx e^{T\lambda_T}\|z_0-\widetilde z_0\|,$$
- $$\widehat{\lambda}_T<0\ \Rightarrow\ \text{synchronization candidate},\qquad \widehat{\lambda}_T>0\ \Rightarrow\ \text{desynchronization candidate}.$$

## Implementation notes

1. Integration point: implement a complex-valued RNN or state-space model whose recurrent transition is a randomly selected Blaschke product. Represent each complex coordinate by two real channels. For a low-cost MVP use degree-one maps T_j(z)=exp(i theta_j)(z-a_j)/(1-conjugate(a_j)z), with two or four maps selected by a fixed categorical distribution. Inject input u_t through q_t=tanh(W_u u_t+b), then form a bounded pre-state and apply the map. 2. Pseudocode: initialize z and running exponent s=0; at every step sample j, compute z_next=T_j(z), compute d=Tprime_j(z), set s <- s+log(abs(d)+1e-8), set z <- z_next, and emit z. Add an optional penalty gamma times max(0, lambda_hat-lambda_target)^2 when a contracting model is desired. 3. Computed directly: the Blaschke transition and derivative statistic. Estimated empirically: lambda_hat, paired-trajectory separation, validation loss, and finite-time exponent variance. 4. First experiment: compare this model with a tanh RNN of matched width and parameter count on sequential MNIST or the adding problem. Run paired trajectories with identical inputs and map sequences but initial perturbation 1e-4. Sweep a map parameter controlling the derivative magnitude. The predicted signature is exponential separation with slope lambda_hat: errors decay for lambda_hat<0 and grow or saturate for lambda_hat>0. The observed sign-change boundary should agree with the derivative estimate within 20%, and the stable regime should show fewer exploding gradients.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: Built a compact Blaschke-product verification and sequence-learning MVP. The core mechanism manifested: the analytic derivative matched finite differences to 1.11e-9 relative error, disk-preservation testing had zero violations, the observed transition p=0.58463 exactly matched the paper’s predicted threshold, and lambda’s slope versus map probability matched prediction (1.185624 vs 1.185624). In the small adding-task comparison, the Blaschke model reached test MSE 1.545 versus 1.707 for tanh, but this secondary result is only a weak signal because the models and parameter counts were not perfectly matched.

### Mechanism check

- Verdict: Built a compact Blaschke-product verification and sequence-learning MVP. The core mechanism manifested: the analytic derivative matched finite differences to 1.11e-9 relative error, disk-preservation testing had zero violations, the observed transition p=0.58463 exactly matched the paper’s predicted threshold, and lambda’s slope versus map probability matched prediction (1.185624 vs 1.185624). In the small adding-task comparison, the Blaschke model reached test MSE 1.545 versus 1.707 for tanh, but this secondary result is only a weak signal because the models and parameter counts were not perfectly matched.
- Confidence: 8/10
- Limitations: The synchronization sweep uses the paper’s fixed boundary orbit for the exact transition test, rather than a general interior random cocycle. Very strong contraction cases underflow before a pair-growth slope can be fit. The learning experiment is a tiny scalar adding task, uses fixed rather than sampled maps in the model, has unequal parameter counts, and does not test sequential MNIST, long-memory scaling, gradient explosions, FLOPs, or multiple random seeds.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_blaschke.py](https://synthcore.org/code/172/bench_blaschke.py)
- [bench_report.json](https://synthcore.org/code/172/bench_report.json)
- [blaschke_mvp.py](https://synthcore.org/code/172/blaschke_mvp.py)
- [report.md](https://synthcore.org/code/172/report.md)
- [report_bench_2026-08-30T183932.md](https://synthcore.org/code/172/report_bench_2026-08-30T183932.md)
- [results.json](https://synthcore.org/code/172/results.json)
- [Download all files as ZIP](https://synthcore.org/download/172)

## Disclaimer

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