✗ Failed on benchmark 2026

Marginal-Stability Disorder Schedule

Implementation & benchmark of arXiv:2607.28781 — Disorder induced time crystal in athermal random field Ising model with non-reciprocal interactions

Usefulness7/10

Difficulty4/10

Novelty7/10

Training Stability Accuracy Memory Rnn Optimizer Scheduler Training dynamics Dynamical systems Bifurcations Linear algebra Statistical mechanics

Source paper: Disorder induced time crystal in athermal random field Ising model with non-reciprocal interactions arXiv:2607.28781 ⓘ · analyzed Aug 31, 2026

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 Not peer-reviewed.

Idea description

Use the disorder-controlled stability boundary as a training schedule. Start with strong damping so optimization is well behaved, then reduce the damping margin toward zero to create long-lived oscillatory state memory after the network has learned useful representations.

Formulas

$$r(\sigma)=\sqrt{\frac{2}{\pi}}\frac{J}{\sigma}-1,\qquad \omega(\sigma)=\sqrt{\frac{2}{\pi}}\frac{K}{\sigma},\qquad \tau(\sigma)=\frac{1}{|r(\sigma)|}$$

$$\sigma_q=\frac{\sqrt{2/\pi}\,J_q}{1-\epsilon_q},\qquad r_q=-\epsilon_q,\qquad 0<\epsilon_q<1$$

$$C(\ell)\approx C(0)\exp(r\ell\Delta t)\cos(\omega\ell\Delta t)$$

Mathematical statement

Let r be the real part of the linearized eigenvalues and omega their imaginary part: r=sqrt(2/pi) J/sigma-1 and omega=sqrt(2/pi) K/sigma. The origin is stable when r<0 and unstable when r>0. In the stable linear regime, an oscillatory correlation has envelope exp(r t) and characteristic decay time tau=1/absolute value of r. Choose a positive damping margin epsilon and set sigma=sqrt(2/pi) J/(1-epsilon), which forces r=-epsilon. Annealing epsilon toward zero increases correlation time while retaining bounded dynamics. In discrete time, also enforce the Euler stability condition |1+Delta t(r plus or minus i omega)|<1.

Implementation notes

Apply the schedule to sigma in the oscillatory recurrent cell, or equivalently divide the recurrent preactivation by a time-dependent gain. At optimizer step q, estimate the effective self-coupling J_q from the spectral norm or average diagonal gain of the self-coupling block, then set sigma_q=sqrt(2/pi) J_q divided by (1-epsilon_q), using stop-gradient through this schedule. Start with epsilon_init=0.3 and anneal exponentially or by cosine to epsilon_final=0.02 over T training steps. Pseudocode is: J_gain=estimate_gain(W_self); epsilon=epsilon_final+(epsilon_init-epsilon_final)*exp(-q/T); sigma=sqrt(2/pi)*J_gain/(1-epsilon); h=oscillatory_cell(h,u,J_gain,K,sigma). Clip sigma to a positive range and separately check the discrete Euler modulus because large K or Delta t can destabilize an otherwise stable continuous-time system. The paper-derived quantities are the eigenvalue formulas and the disorder-to-gain relation; empirical quantities are the effective Jacobian, actual autocorrelation, and gradient variance. First test on the copy-memory and adding tasks with a small RNN, comparing fixed sigma, spectral normalization, and the proposed schedule. The quantitative prediction is that the measured correlation decay time should track tau=1/epsilon during annealing, while the autocorrelation envelope follows exp(-epsilon times lag times Delta t). Crossing epsilon below zero or violating the Euler modulus should produce a sharp increase in hidden-state norm and gradient variance. Require measured decay rates to agree with the predicted value within 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: Inconclusive

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 a reproducible NumPy MVP implementing the disorder-controlled sigma schedule and a matched recurrent memory toy task. The mechanism was confirmed: measured damping matched r=-epsilon, correlation-time error stayed below 1.3% across epsilon=0.30 to 0.02, and the analytic Euler boundary matched numerical spectral-radius search to machine precision; negative epsilon produced a positive eigenvalue and an oversized Euler step produced spectral radius 1.554. The schedule did not produce a task-performance win in this tiny run (accuracy 0.690 vs 0.700 for fixed sigma), so the evidence supports the stability/memory mechanism but not improved learning.

Agent confidence: 9/10

Baseline: Fixed epsilon=0.15: final loss 0.69034, accuracy 0.7000, mean hidden norm 1.2308
Idea: Annealed epsilon 0.30->0.02: final loss 0.68972, accuracy 0.6898, mean hidden norm 1.3678; math tau relative error 0.01%-1.21%, Euler-boundary relative error <=2e-14

Limitations: Only a small NumPy memory task was tested; copy-memory and adding benchmarks, spectral-normalization baseline, multiple seeds, gradient-variance analysis, and larger learned RNNs were not tested. The oscillator verification uses the specified linearized model directly rather than estimating an empirical Jacobian of a nonlinear recurrent network.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.28781) Sep 2, 2026

Ahead of the baseline (−21.2%) but not statistically significant (p=0.6429, wins 6 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0066
Idea mean: 0.0052
Effect (Δ): -0.0014 (−21.2%; negative = idea better)
Wins: 6 / 8 paired seeds
p-value: 0.6429 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 0 configs

Limitations:

Only the built-in dynamics track was tested; the mechanism signature used a short hidden-state perturbation replay rather than a full nonlinear Jacobian/eigenvalue analysis. The README was initially unavailable through the file tool but was read successfully via shell; the benchmark harness itself was not modified.

How to run: python3 stage2_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: 10/10

Implemented and ran the required dynamics benchmark with 8 paired seeds, a tuned baseline sweep, and a three-configuration marginal-stability schedule sweep. The idea had lower mean MSE (0.005164 vs 0.006554; delta -0.001390; 6/8 wins), but the paired permutation test was not significant (p=0.64285), so there is no benchmark win. The trained-model mechanism signature also failed quantitatively: observed perturbation envelopes were far below predicted exp(-epsilon), with all seeds unconfirmed.

Baseline: rnn_small, best lr=0.003 and weight_decay=0.0001; full 8-seed MSE mean 0.0065538855, std 0.0036887285
Comparison: delta_mean=-0.0013896531, p_value=0.64285, verdict=no significant win
Idea: scheduled recurrent preactivation scaling; best 8-seed MSE mean 0.0051642324, std 0.0060285827; 6/8 paired wins
Mechanism: predicted envelope=0.9154676; observed ratios=0.0267-0.2331; all 8 signatures confirmed=false

Benchmark result