# Compactified Burst Controller

- ID: 900
- Canonical URL: https://synthcore.org/idea/900/compactified-burst-controller
- API JSON: https://synthcore.org/api/idea/900.json
- API Markdown: https://synthcore.org/api/idea/900.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.14269](https://arxiv.org/abs/2607.14269)
- Category: dynamics
- Solves: stability, speedup
- ML areas: rnn, ssm, training-dynamics, inference-speedup
- Math tags: dynamical-systems, control-theory, bifurcations, numerical-analysis
- Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10

## Idea description

Use the paper's distinction between radial attraction and tangential instability at infinity to detect impending hidden-state bursts before they cause numerical failure. When the state approaches a radially growing directional equilibrium, temporarily add radial damping or switch to a bounded fallback update, then restore the original dynamics after angular ejection.

## Mathematical statement

For x = r u and a homogeneous leading field F_n, define a(u) = u transpose F_n(u) and G(u) = F_n(u) - a(u)u. A candidate equilibrium at infinity satisfies G(u_star) = 0. Positive a(u_star) causes radial growth because dr/dt is approximately r^n a(u_star). Tangential instability is diagnosed from the projected Jacobian J_T = P D G(u_star) P on the tangent space, where P = I - u_star u_star transpose. If the largest real part of an eigenvalue of J_T is positive, the direction is angularly unstable: the state can grow while nearly aligned with the direction and later be ejected, producing a large but transient burst. A radial controller subtracts kappa(r)u from the vector field, reducing radial growth without directly suppressing angular motion.

## Key formulas

- dr/dt = r^n a(u) + O(r^(n-1)), and du/dtau = G(u), with d tau = r^(n-1) dt.
- G(u_star) = 0, alpha_star = a(u_star) > 0, and max Re eig(P D G(u_star) P restricted to the tangent space) > 0.
- F_control(x) = F(x) - kappa(r)u, where u = x/r and kappa(r) = k softplus((r - r_c)/delta).
- d||x||/dt is approximately r^n a(u) - kappa(r).

## Implementation notes

1. Integration point: insert the controller inside every neural ODE function evaluation or recurrent hidden-state update, before advancing the solver. It is intended for long-horizon RNN and SSM rollouts, DEQ iterations, and polynomial residual blocks where rare norm bursts cause NaNs or excessive adaptive solver steps. 2. Pseudocode: compute r = norm(h) plus epsilon and u = h/r; evaluate the estimated leading field v = F_n(u); compute a = dot(u,v) and q = norm(v - a u). Maintain an exponential moving average of a. If r exceeds r_c, the averaged radial growth exceeds a margin, and q is below an angular-equilibrium threshold, replace dh/dt = F(h) by dh/dt = F(h) - kappa(r)u. Estimate angular instability every K steps using finite differences or automatic differentiation of G. 3. The paper supplies the radial and tangential infinity mechanism; r_c, the growth margin, q threshold, damping gain k, and transition width delta are calibrated on non-divergent training rollouts. 4. First cheap experiment: compare vanilla updates, global gradient clipping, and this controller on a cubic RNN trained on the copy and addition tasks, evaluating at ten times the training horizon. Sweep the cubic gain lambda. Prediction: uncontrolled bursts begin when averaged a becomes positive while q is initially small; controller interventions occur before the norm maximum, reduce solver failures, and preserve trajectories with negative a. The measured critical lambda for burst onset should agree with the infinity-atlas prediction within 20 percent.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: Built a readable 2D homogeneous cubic burst simulator with infinity-atlas diagnostics and gated softplus radial damping. The core math manifested: finite-difference a(u*) and tangent eigenvalue matched predictions to 1e-12, radial threshold-time scaling matched the cubic blow-up law within 8e-5 relative error, and angular slopes matched beta within 1.25%, including the stable negative-beta case. The predicted finite-horizon onset boundary alpha=0.8203 was observed between 0.8 and 0.9; in burst regimes the controller prevented numerical failure and held max radius near 2.2-3.0 versus vanilla failure or large excursions.

### Mechanism check

- Verdict: Built a readable 2D homogeneous cubic burst simulator with infinity-atlas diagnostics and gated softplus radial damping. The core math manifested: finite-difference a(u*) and tangent eigenvalue matched predictions to 1e-12, radial threshold-time scaling matched the cubic blow-up law within 8e-5 relative error, and angular slopes matched beta within 1.25%, including the stable negative-beta case. The predicted finite-horizon onset boundary alpha=0.8203 was observed between 0.8 and 0.9; in burst regimes the controller prevented numerical failure and held max radius near 2.2-3.0 versus vanilla failure or large excursions.
- Confidence: 9/10
- Limitations: This is a deterministic 2D toy cubic vector field, not a trained RNN, copy/addition task, adaptive ODE solver, or FLOP/speed benchmark. The controller uses an instantaneous angular-equilibrium gate rather than implementing EMA growth estimates or periodic Jacobian estimation, and the burst prevention result is not evidence of end-to-end neural-network training gains.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.01475
- Paired wins: 1/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_report.json](https://synthcore.org/code/374/bench_report.json)
- [burst_controller_experiment.py](https://synthcore.org/code/374/burst_controller_experiment.py)
- [compactified_bench.py](https://synthcore.org/code/374/compactified_bench.py)
- [report.md](https://synthcore.org/code/374/report.md)
- [report_bench_2026-08-31T095644.md](https://synthcore.org/code/374/report_bench_2026-08-31T095644.md)
- [results.json](https://synthcore.org/code/374/results.json)
- [Download all files as ZIP](https://synthcore.org/download/374)

## Disclaimer

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