✗ Failed on benchmark 2026

Worst-Case Switched Residual Stability Certificate

Implementation & benchmark of arXiv:2607.12189 — Stability and Bifurcations of Planar Switched Linear and Homogeneous Systems

Usefulness8/10

Difficulty6/10

Novelty7/10

Dynamics Stability Accuracy Optimizer Mlp Rnn Training dynamics Dynamical systems Control theory Linear algebra Spectral theory

Source paper: Stability and Bifurcations of Planar Switched Linear and Homogeneous Systems arXiv:2607.12189 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Model a residual network, recurrent update, or optimizer as a switched linearized system in which each layer type, token, data batch, or optimizer regime selects a matrix mode. Constrain the worst-case product growth over admissible switches, rather than merely constraining every individual Jacobian, so arbitrary mode sequences remain contractive.

Formulas

eₖ₊₁ = Mₛₖ eₖ, Mₚ = I + hAₚ, eₖ = Mₛₖ₋₁ ... Mₛ₀ e₀

ρ̂(M) = limsupₖ→∞ max₍ₚ₀,...,ₚₖ₋₁₎ ||Mₚₖ₋₁ ... Mₚ₀||^(1/k)

ρ̂(M) < 1 implies ||eₖ|| ≤ C qᵏ ||e₀|| for constants C > 0 and 0 < q < 1

ρ̂(M) = 1 is the predicted switching-stability boundary; λsw = 0 is the equivalent continuous-time boundary

Mathematical statement

For modes p in {1,...,m}, the switched perturbation dynamics are eₖ₊₁ = Mₛₖ eₖ, where Mₚ is the linearized update matrix and sₖ is the active switching signal. The joint spectral radius is ρ̂(M) = limsupₖ→∞ max over mode sequences of ||Mₛₖ₋₁ ... Mₛ₀||^(1/k). Uniform exponential stability under arbitrary switching requires ρ̂(M) < 1, while ρ̂(M) = 1 is the stability bifurcation boundary. For a continuous-time system, the analogous worst-case exponent is λsw = limsupT→∞ T⁻¹ maxσ log ||Φσ(T)||, where Φσ(T) is the state-transition matrix under switching; stability requires λsw < 0.

Implementation notes

Integrate the certificate at the Jacobian level of a residual MLP, RNN, or optimizer. For a block Fₚ(x) = x + h fₚ(x), compute the local Jacobian Mₚ(x) = I + h times the Jacobian of fₚ with respect to x. Modes can represent layer types, attention regimes, optimizer states, or batch-dependent activation regimes. During calibration, collect representative mode Jacobians using automatic differentiation. Estimate worst-case products with beam search: initialize B = {I}; for each horizon r, form every product MₚB for p in the mode set and B in the current beam; score each product by log spectral norm; retain the top K products; record gᵣ = max log norm divided by r and ρ̂ᵣ = exp(gᵣ). Pseudocode: B=[I]; for r=1,...,R: C={M[p] @ B0 for p in modes and B0 in B}; B=top_K(C, score=log_norm); rho[r]=exp(max_score/r). Exact matrices may be replaced by Jacobian-vector products and power iteration. The first experiment should use a two-dimensional residual network with two noncommuting matrices, followed by a small CIFAR-10 ResNet with two block modes. Sweep residual scale h or a global scale s and compare against individual spectral normalization. The quantitative prediction is a transition from decaying to growing perturbations when measured ρ̂R crosses one; increasing beam horizon should move the estimated threshold by less than 20 percent.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: No measurable effect

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 an exact 2D two-mode switched residual stability verifier plus a finite-width beam-search certificate in switched_stability.py. Scaling all modes by s scaled the estimated joint growth rate by s exactly, and the observed decay-to-growth transition occurred at the predicted finite-horizon boundary s≈0.8865. The horizon-estimated boundary moved from 0.8507 at r=4 to 0.8865 at r=16, a 4.2% change within the predicted 20% tolerance; the switched-product criterion allowed a 1.186x larger scale than the individual spectral-norm rule.

Agent confidence: 9/10

Baseline: Individual spectral-norm constraint: maximum mode norm=1.33757, allowable global scale boundary=0.74763.
Idea: Finite-horizon switched certificate: rho_hat_16=1.12804 at unit scale, boundary=0.88649. At 0.99 times the boundary, log growth per step=-0.01005 and product norm after 16 steps=0.85146; at 1.01 times the boundary, log growth per step=+0.00995 and product norm=1.17258. Beam width 16 matched exact enumeration at horizons 4, 8, 12, and 16.

Limitations: Only a 2D linear toy system was tested. No residual MLP, RNN, optimizer, automatic-differentiation Jacobian collection, CIFAR-10 model, training-loss comparison, FLOP or speed measurement, or larger-mode beam approximation was evaluated. The joint spectral radius estimates are finite-horizon quantities, and exact enumeration is practical here only because there are two modes and horizon 16.

How to run: python3 switched_stability.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.12189) Sep 2, 2026

No significant difference vs. the tuned baseline (+0.0%, p=0.7508, wins 2 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0038
Idea mean: 0.0038
Effect (Δ): 0 (+0.0%; negative = idea better)
Wins: 2 / 8 paired seeds
p-value: 0.7508 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±0.0%
Mechanism: Not confirmed ✗
Practical effect: No effect
Baseline tuning: swept over 6 configs

Limitations:

Only the registered built-in dynamics track was tested. The intervention used a representative local recurrent Jacobian and a conservative single-mode spectral penalty rather than a fully differentiable worst-case product over all sampled modes; the mechanism signature used four trained models and a short hidden-state probe rather than long-horizon rollout stability.

How to run: python3 bench_switched.py

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

Stage 2 — Benchmark Aug 30, 2026 latest Failed ✗

Agent confidence: 9/10

The matched dynamics/rnn_small benchmark completed with the mandated baseline sweep, shared learning-rate search space, idea sweep, and eight paired seeds. Baseline test MSE was 0.0038480483 versus 0.0038480595 for the idea, paired delta +1.12e-08 with permutation p=0.75075, so there was no significant win. The trained-model mechanism signature was also not confirmed: predicted rho=0.8613 but observed hidden-state ratio=1.4026.

Baseline: test MSE mean 0.003848048327199649, std 0.0019871782432351525, best lr=0.003 wd=0.0001
Idea: test MSE mean 0.003848059532174375, std 0.0019871702370856054, selected lr=0.003 wd=0.0001 lam=0.001; delta +1.1204974725842476e-08, p=0.75075

Benchmark result