✗ Failed on benchmark 2026

Bethe-Salpeter Instability Monitor

Implementation & benchmark of arXiv:2608.04497 — The two-particle-irreducible vertex of the two-dimensional lattice $φ^4$ model across the Ising transition

Usefulness7/10

Difficulty5/10

Novelty8/10

Dynamics Stability Scalability Rnn Ssm Training dynamics Optimizer Statistical mechanics Dynamical systems Linear algebra Spectral theory Control theory

Source paper: The two-particle-irreducible vertex of the two-dimensional lattice $φ^4$ model across the Ising transition arXiv:2608.04497 ⓘ · analyzed Aug 31, 2026

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 Not peer-reviewed.

Idea description

Add a response-spectrum monitor to recurrent, state-space, or deep-equilibrium networks by treating products of hidden features as composite observables. Estimate the full susceptibility and a bare susceptibility, reconstruct an irreducible interaction vertex, and damp the state update whenever the leading Bethe–Salpeter eigenvalue approaches one. This targets collective failure modes that ordinary single-feature Jacobian checks can miss.

Formulas

$$\chi_{ab}=\mathbb{E}\left[(z_a-\mathbb{E}z_a)(z_b-\mathbb{E}z_b)\right],\qquad \Gamma=\chi_0^{-1}-\chi^{-1}.$$

$$\chi=(\chi_0^{-1}-\Gamma)^{-1},\qquad K=\chi_0^{1/2}\Gamma\chi_0^{1/2}.$$

$$\rho(K)\to 1^-\quad\Longrightarrow\quad \|\chi\|\sim \frac{1}{1-\rho(K)}.$$

$$x_{t+1}=x_t+\alpha_t\left(F_\theta(x_t)-x_t\right),\qquad \alpha_t=\alpha_0\min\left(1,\frac{1-\delta}{\max(\rho(K_t),\epsilon)}\right).$$

Mathematical statement

For composite observables $z_a(x)$, where index $a$ labels a feature pair, spatial location, channel, or low-rank tensor component, define the connected susceptibility $\chi_{ab}=\mathbb{E}[(z_a-\mathbb{E}z_a)(z_b-\mathbb{E}z_b)]$. Let $\chi_0$ be a bare susceptibility estimated from products of two-point feature covariances or a frozen linearized model. The irreducible vertex is $\Gamma=\chi_0^{-1}-\chi^{-1}$. Define the symmetrized Bethe–Salpeter kernel $K=\chi_0^{1/2}\Gamma\chi_0^{1/2}$. The response grows strongly when its largest eigenvalue $\rho(K)$ approaches one. Use the criterion $\rho(K)\leq 1-\delta$, with safety margin $\delta>0$, to set the relaxation coefficient of the iterative neural update.

Implementation notes

First integrate this into a recurrent layer, SSM transition, or deep-equilibrium fixed-point solver. During each monitoring interval, form composite observables using random projections, for example $z_a=(u_a^\top h)(v_a^\top h)$, where $h$ is the hidden state and $u_a,v_a$ are fixed Gaussian vectors; use 128–512 projections for an MVP. Maintain an exponential moving average of the mean and covariance $\chi$. Estimate $\chi_0$ from the covariance of the individual projected features, add ridge $\epsilon I$, and compute matrix-vector products with $K=\chi_0^{1/2}(\chi_0^{-1}-\chi^{-1})\chi_0^{1/2}$. Estimate the largest eigenvalue using 5–10 Lanczos iterations, then set the relaxation coefficient to $\alpha_t=\alpha_0\min(1,(1-\delta)/(\rho(K_t)+\epsilon))$. The paper supplies the susceptibility/vertex identity, ladder reconstruction, and unity-eigenvalue instability criterion; the covariances and eigenvalue are estimated empirically with shrinkage and low-rank methods. First test a 2-layer GRU or 1D SSM on copy and adding tasks at sequence lengths 1,000–10,000, comparing undamped and monitored updates. The prediction is $\|\chi\|\propto(1-\rho(K))^{-1}$, with divergence beginning when measured $\rho(K)$ reaches one within approximately 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: Significantly worse ✗

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 self-contained Bethe–Salpeter monitor MVP plus controlled numerical verification and a small recurrent fixed-point experiment. The controlled sweep confirmed eigenvalue recovery to 5.6e-16 absolute error, inverse-gap response scaling within 0.81% over rho=0..0.95, and the zero-vertex limit exactly. In the fair recurrent toy run, monitored relaxation reduced final squared state error from 0.1363 to 9.33e-6, indicating a promising stabilization signal, though this comparison is secondary to the verified mechanism.

Agent confidence: 8/10

Baseline: Recurrent fixed-point iteration, alpha=1: final MSE 0.136309; maximum MSE 0.139075; shared initialization.
Idea: Composite-observable monitor with adaptive relaxation, base alpha=0.25: final MSE 9.334e-6; maximum MSE 0.002308. Controlled BSE: max eigenvalue error 5.55e-16, inverse-gap scaling relative error 0.00806, rho=0 gain 1.0; empirical monitor rho ranged 0.025 to 0.946.

Limitations: The recurrent experiment uses a lightweight scalar proxy for the full empirical Bethe–Salpeter covariance rather than Lanczos/random-projection monitoring at production scale. It tests neither GRU/SSM copy or adding tasks nor sequence lengths 1,000–10,000, has no multi-seed statistics, and does not test behavior after rho exceeds one.

How to run: python3 bethe_monitor_experiment.py > results.json

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.04497) Sep 2, 2026

Significantly WORSE than the tuned baseline (+33374.6%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0015
Idea mean: 0.4893
Effect (Δ): 0.4879 (+33374.6%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±14344.6%
Mechanism: Not confirmed ✗
Practical effect: Hurts
Baseline tuning: swept over 3 configs

Limitations:

Only the registered dynamics/rnn_small track was tested. The MVP used 12 projections and dense eigendecomposition rather than Lanczos, and inverse-gap susceptibility scaling was not separately identified from trained-model behavior; mechanism_signature.confirmed is therefore false.

How to run: python3 stage2_bench.py > stage2_report.json

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: 10/10

The Bethe–Salpeter monitor was implemented as adaptive recurrent-state relaxation for rnn_small and tested on the registered dynamics track. It was significantly worse than the tuned baseline: paired delta was +0.4878674 MSE with permutation p=0.0081, so there is no benchmark win; the trained-model mechanism signature was not confirmed.

Baseline: Best lr=0.006; mean test MSE 0.0014617917 over 8 seeds.
Idea: Best lr=0.006, alpha0=0.8, delta=0.1; mean test MSE 0.4893291956 over 8 seeds; paired delta +0.4878674038; p=0.0081; 0/8 wins.

Benchmark result