# Characteristic-Region Gain Controller

- ID: 982
- Canonical URL: https://synthcore.org/idea/982/characteristic-region-gain-controller
- API JSON: https://synthcore.org/api/idea/982.json
- API Markdown: https://synthcore.org/api/idea/982.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.15833](https://arxiv.org/abs/2607.15833)
- Category: dynamics
- Solves: stability, memory
- ML areas: rnn, ssm, optimizer, training-dynamics
- Math tags: control-theory, dynamical-systems, spectral-theory, numerical-analysis
- Ratings: usefulness 7/10; difficulty 5/10; novelty 6/10

## Idea description

Use the q-fractional characteristic equation as an online trust-region controller for recurrent gain or residual-memory strength. Instead of allowing the recurrent Jacobian to cross the unit-circle boundary, estimate the dominant characteristic root and rescale the feedback gain whenever it approaches modulus one.

## Mathematical statement

For the linearized layer x_{t+1}=Ax_t+Cu_t with fractional memory u_t=sum from k=0 to infinity of w_k x_{t-k}, the characteristic function is F_g(z)=det[zI-A-gC W_{q,alpha}(z)], where g is a scalar feedback gain. The stability boundary is F_g(exp(i theta))=0 for some theta in [0,2pi), and stability requires the largest characteristic-root modulus rho(g) to be less than one. Estimate rho(g) from a K-delay state augmentation. A conservative controller maintains a margin delta>0 by setting g_new=g min(1,(1-delta)/(rho_hat+epsilon)), where rho_hat is the estimated dominant root modulus and epsilon prevents division by zero. This transfers the paper's complex stability geometry into an implementable recurrent-gain constraint.

## Key formulas

- $$F_g(z)=\det\!\left[zI-A-gC W_{q,\alpha}(z)\right],\qquad \rho(g)=\max\{|z|:F_g(z)=0\}$$
- $$\text{stable region: }\rho(g)<1,\qquad \text{boundary: }F_g(e^{i\theta})=0\text{ for some }\theta\in[0,2\pi)$$
- $$g_{t+1}=g_t\min\!\left(1,\frac{1-\delta}{\widehat\rho_t+\varepsilon}\right),\qquad \widehat\rho_t=\text{estimated dominant characteristic-root modulus}$$
- $$\left\|\frac{\partial \mathcal L}{\partial h_t}\right\|\approx C\,\widehat\rho^{\,T-t}\quad\text{when the linearized dynamics have dominant modulus }\widehat\rho<1$$

## Implementation notes

Attach the controller to the recurrent feedback matrix C, the residual coefficient gamma, or a scalar gain multiplying the memory branch. Every N training steps, freeze the current A,C,q,alpha, form the K-delay augmented Jacobian corresponding to the recurrence, and use an eigensolver or 10-20 power iterations to estimate the dominant modulus rho_hat. If rho_hat exceeds 1-delta, rescale C or gamma according to the displayed controller; if it is far below the target margin, permit a slow increase using g_new=min(g_max,g*(1+eta_gain)). A more accurate but slower alternative evaluates W_q(exp(i theta)) on a frequency grid and performs bisection over g to locate the first unit-circle intersection. The q-kernel and characteristic equation come from the paper; K, delta, N, gain caps, and the root estimator are empirical approximations. The first cheap test should use tanh RNNs on sequential MNIST and Copy-Memory, comparing SGD, gradient clipping, spectral normalization, and this controller at matched initial gains. Record loss, gradient norm, gain, and rho_hat. The falsifiable prediction is a sharp degradation or exploding-gradient transition near rho_hat=1. Runs maintained at rho_hat<=1-delta should show geometric gradient decay with fitted slope close to log(rho_hat), while unconstrained runs should diverge after crossing one. A gain sweep should yield an observed stable/unstable threshold within approximately 20% of the characteristic-root prediction, and q<1 should make this threshold less sensitive to memory truncation K than q=1.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: inconclusive
- Verdict: Built a finite-delay fractional-memory companion-matrix implementation with q-binomial weights, dominant-root estimation, and the proposed conservative gain controller. The mechanism manifested: the predicted boundary g*=1/sum(w)=0.09021 matched the swept transition 0.09021, rho crossed one between 0.9g* and 1.1g*, and measured gradient log-slope differed from log(rho) by 3.1e-5. Starting 35% above the boundary, control reduced rho from 1.0232 to 0.9533 and reduced final squared state norm from 4.39 uncontrolled to 8.37e-4 controlled.

### Mechanism check

- Verdict: Built a finite-delay fractional-memory companion-matrix implementation with q-binomial weights, dominant-root estimation, and the proposed conservative gain controller. The mechanism manifested: the predicted boundary g*=1/sum(w)=0.09021 matched the swept transition 0.09021, rho crossed one between 0.9g* and 1.1g*, and measured gradient log-slope differed from log(rho) by 3.1e-5. Starting 35% above the boundary, control reduced rho from 1.0232 to 0.9533 and reduced final squared state norm from 4.39 uncontrolled to 8.37e-4 controlled.
- Confidence: 8/10
- Limitations: Only a scalar positive-feedback finite K=30 toy system was tested, not a trained tanh RNN, sequential MNIST, Copy-Memory, SGD, gradient clipping, or spectral normalization. The q-fractional infinite-memory characteristic equation was approximated by a truncated companion matrix; noisy or delayed root estimation was not evaluated.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.1834
- Paired wins: 6/8
- Benchmark verdict: no significant win

## Artifacts

- [bench_report.json](https://synthcore.org/code/409/bench_report.json)
- [bench_run.py](https://synthcore.org/code/409/bench_run.py)
- [experiment.py](https://synthcore.org/code/409/experiment.py)
- [report.md](https://synthcore.org/code/409/report.md)
- [report_bench_2026-08-31T113007.md](https://synthcore.org/code/409/report_bench_2026-08-31T113007.md)
- [results.json](https://synthcore.org/code/409/results.json)
- [Download all files as ZIP](https://synthcore.org/download/409)

## Disclaimer

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