✗ Failed on benchmark 2026

Carrier-Probed Hidden-State Training

Implementation & benchmark of arXiv:2607.24134 — Sensor-Limited Observability and Carrier-Induced Reachability of Low-Order Rotor-Coupled NVH in Production Electric Drives: A Magnetic Co-Energy, Gramian, and Active Projection Framework for Production-Signal Feasibility Analysis

Usefulness7/10

Difficulty5/10

Novelty7/10

Dynamics Accuracy Sample efficiency Stability Ssm Rnn World model Data augmentation Training dynamics Control theory Dynamical systems Linear algebra Optimization

Source paper: Sensor-Limited Observability and Carrier-Induced Reachability of Low-Order Rotor-Coupled NVH in Production Electric Drives: A Magnetic Co-Energy, Gramian, and Active Projection Framework for Production-Signal Feasibility Analysis arXiv:2607.24134 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

When a neural state-space model has latent directions that are invisible under normal inputs, add a small structured carrier to the input or hidden-state update during selected training windows. The carrier changes local measurement and transition projections, analogous to the paper's carrier-dependent measurement and force projections, and can reveal modes that passive training leaves unconstrained.

Formulas

$$u_t^{(m)}=u_t+a\,m_t,\qquad m_t\in\mathbb{R}^{m},\quad \|m_t\|_2\le 1,$$

$$W_o(m)=\sum_{t=0}^{H-1}\Phi_t(m)^T C_t(m)^TC_t(m)\Phi_t(m)\,\Delta t,$$

$$W_r(m)=\sum_{\tau=0}^{H-1}\Psi_{\tau+1}(m)B_\tau(m)B_\tau(m)^T\Psi_{\tau+1}(m)^T\,\Delta t,$$

$$m^*=\arg\max_m\left[\log\det(W_o(m)+\delta I)+\alpha\log\det(W_r(m)+\delta I)-\beta a^2\right].$$

Mathematical statement

Let the passive model be $z_{t+1}=f_\theta(z_t,u_t)$ and $y_t=h_\theta(z_t)$. Introduce a known carrier $m_t$ with amplitude $a$ and define $u_t^{(m)}=u_t+a m_t$. Around the carrier-driven trajectory, define $A_t(m)=\partial f_\theta/\partial z$, $B_t(m)=\partial f_\theta/\partial u$, and $C_t(m)=\partial h_\theta/\partial z$. The carrier is useful when it increases the smallest observability or reachability eigenvalue. Select it by maximizing a log-determinant information score penalized by carrier energy, where $\delta>0$ stabilizes singular matrices and $\alpha,\beta\ge0$ control reachability and excitation cost.

Implementation notes

Apply this to an SSM, GRU, or world model during training rather than necessarily at deployment. Maintain a small candidate library consisting of sinusoidal carriers at frequencies $\omega_k$, binary pseudorandom sequences, channel-wise sign patterns, and short orthogonal pulses. For each minibatch, run the model passively and with $u_t+a m_t$. Compute Jacobians $A_t,B_t,C_t$ using automatic differentiation or Jacobian-vector products, propagate finite-horizon transition matrices, and evaluate the log-determinant score. Select the highest-scoring carrier, train on that rollout, and add an output-consistency loss requiring the model to predict the carrier-induced change. Pseudocode is: sample candidates $m$; roll out $z^{(m)}$; estimate $A_t(m),B_t(m),C_t(m)$; accumulate $W_o(m)$ and $W_r(m)$; score each candidate; optimize $\mathcal{L}_{task}+\gamma\|\Delta y_{pred}-\Delta y_{actual}\|^2$ for the selected carrier. The paper supplies the active-projection and Gramian mechanism; candidate search and Jacobian estimation are empirical neural adaptations. Begin with a GRU on a synthetic system containing one observable fast mode and one passive-hidden slow mode, then test a neural ODE on Lorenz-63 with one latent coordinate masked from the output. The predicted signature is a carrier threshold: below amplitude $a_c$, $\lambda_{\min}(W_o)$ remains near zero; above it, the eigenvalue should increase approximately as $a^2$ for small perturbations because the carrier-induced output projection is first order in $a$ and the Gramian is quadratic. Test whether hidden-coordinate recovery begins at this transition.

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 8/10

Built a two-state nonlinear SSM with an observed fast mode and a passive-hidden slow mode, plus finite-horizon observability/reachability Gramian computation and carrier candidate scoring. The passive hidden Gramian eigenvalue was exactly 0, and the carrier-induced minimum observability eigenvalue followed a^1.999 versus the predicted a^2, with coefficient variation 0.095%. Under identical noise, passive q recovery had absolute error 0.8, while a=0.16 probing reduced it to 0.009; candidate scoring selected the carrier with the strongest hidden observability. The mechanism manifested in this toy setting, but this is not evidence yet for GRUs, neural ODEs, or general learned models.

Agent confidence: 8/10

Baseline: Passive a=0: lambda_min(Wo)=0.0; q_hat=0.2 versus true q=1.0, absolute error 0.8.
Idea: At a=0.16: lambda_min(Wo)=0.04309; fitted quadratic exponent=1.9989 (predicted 2.0), quadratic-ratio CV=0.00095; noisy q_hat=0.9318, absolute error 0.0682 in the first run and 0.9914/0.0086 error in the candidate recovery run. The practical observability threshold lambda_min(Wo)>1e-8 occurred at a=0.01; useful noisy recovery began around a=0.08–0.16.

Limitations: Only a hand-designed two-dimensional Euler SSM was tested; no GRU/neural ODE, minibatch training, output-consistency loss, automatic Jacobian implementation, FLOP/runtime comparison, or deployment impact was evaluated. The threshold rule is an arbitrary numerical criterion, and very small-amplitude recovery is unstable under noise despite the clean Gramian scaling.

How to run: python3 carrier_probe.py && /home/maxwelhelp/main/bin/python3 candidate_check.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.24134) Sep 2, 2026

Ahead of the baseline (−0.0%) but not statistically significant (p=0.459, wins 6 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0007
Idea mean: 0.0007
Effect (Δ): -0 (−0.0%; negative = idea better)
Wins: 6 / 8 paired seeds
p-value: 0.459 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Not confirmed ✗
Practical effect: No effect
Baseline tuning: swept over 3 configs

Limitations:

Only the built-in actuated-pendulum dynamics track was tested. The implementation used a fixed cosine carrier and simplified nominal consistency target rather than automatic Jacobian-based log-determinant candidate selection and propagated Gramians. Multiple architectures, deployment effects, and a full idea-side learning-rate sweep were not tested.

How to run: python3 carrier_bench.py

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

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 9/10

Implemented Carrier-Probed Hidden-State Training on the structurally matched dynamics track with the shared rnn_small architecture, carrier-perturbed controls, and an output-consistency loss. The best idea achieved MSE 0.000652743 versus baseline 0.000652883, but the paired permutation test was not significant (delta=-1.40e-7, p=0.459), and the mechanism signature was not confirmed because predicted and nominal carrier responses had near-zero correlation.

Baseline: dynamics/rnn_small, lr=0.01 after sweep; 8-seed test MSE mean=0.000652883, std=0.000169289
Idea: cosine carrier consistency loss, amplitude=0.30; 8-seed test MSE mean=0.000652743, std=0.000169230; paired delta=-1.3957e-7, 6/8 wins, permutation p=0.45895; signature confirmed=false

Benchmark result