△ Mechanism confirmed, baseline not beaten 2026

Disorder-Controlled Basin Merging

Implementation & benchmark of arXiv:2607.16561 — Glauber dynamics phase transitions in athermal random field Blume-Capel and Blume-Emery-Grifitths models

Usefulness7/10

Difficulty6/10

Novelty7/10

Dynamics Stability Generalization Optimizer Training dynamics Rnn Regularization Statistical mechanics Dynamical systems Bifurcations Stochastic processes

Source paper: Glauber dynamics phase transitions in athermal random field Blume-Capel and Blume-Emery-Grifitths models arXiv:2607.16561 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Replace a continuously saturated recurrent state or optimizer momentum variable by a ternary state s in {-1, 0, +1} governed by a mean-field Blume-Emery-Griffiths energy, and use annealed random fields as a controllable disorder parameter. The system should exhibit multiple persistent attractors below a critical noise amplitude and substantially reduced initial-condition dependence above it. This creates a measurable noise schedule: increase disorder until independent runs converge to the same coarse state trajectory, then reduce it if task accuracy is harmed.

Formulas

$$\mathcal{H}(\{s_i\})=-\frac{J}{2N}\left(\sum_i s_i\right)^2-\frac{K}{2N}\left(\sum_i s_i^2\right)^2-\sum_i(H+h_i)s_i+\Delta\sum_i s_i^2$$

$$m=\frac{1}{N}\sum_{i=1}^N s_i,\qquad q=\frac{1}{N}\sum_{i=1}^N s_i^2,\qquad e_j(s)=-(Jm+H+\xi_j)s+\Delta s^2$$

$$p_j(s\mid m)=\frac{\exp[-\beta e_j(s)]}{\sum_{r\in\{-1,0,+1\}}\exp[-\beta e_j(r)]},\qquad \dot p_j(s)=\gamma\left(p_j^{\mathrm{eq}}(s)-p_j(s)\right)$$

$$\operatorname{Var}_{a,b}\left[\overline{m}^{(a)}-\overline{m}^{(b)}\right]\to 0\quad\text{for }R>R_c,\qquad R_c\approx\sqrt{\frac{2}{\pi}}\text{ in the RFIM-like regime}$$

Mathematical statement

The paper studies the spin-1 Hamiltonian with ternary variables s_i in {-1, 0, +1}. Define magnetization m = N^(-1) sum_i s_i and activity q = N^(-1) sum_i s_i^2. For a neural layer, use s_j in {-1, 0, +1}, local field u_j = J m + H + xi_j, and single-unit energy e_j(s) = -u_j s + Delta s^2. A finite-temperature Glauber relaxation uses p_j(s) proportional to exp(-beta e_j(s)), while the zero-temperature limit selects energy-lowering states. Here J is the ferromagnetic coupling, K is the biquadratic coupling, H is a uniform field, xi_j is a Gaussian random field with scale R, Delta is the crystal-field penalty, beta is inverse temperature, and gamma is the relaxation rate. The transferable signature is a critical disorder scale: below it, the deterministic mean-field flow has multiple attractors and depends on initialization; above it, random fields merge or destroy those attractors. In the random-field-Ising-like regime, the paper reports a crossover to a critical value R_c = sqrt(2/pi), subject to the paper's convention for whether R denotes variance or standard deviation.

Implementation notes

Integrate this at the recurrent-state update or as an optimizer-side auxiliary dynamical system. For each recurrent unit group, maintain ternary probabilities p_j(-1), p_j(0), and p_j(+1). Compute m = M^(-1) sum_j E[s_j] and q = M^(-1) sum_j E[s_j^2], draw a fresh Gaussian field xi_j from N(0, R^2), and update p_j by p_j <- (1 - gamma dt) p_j + gamma dt p_j^eq, where p_j^eq is the three-state softmax of -beta e_j(s). Use E[s_j] = p_j(+1) - p_j(-1) as the differentiable activation, or sample a ternary state with straight-through estimation. Pseudocode is: initialize probabilities from several seeds; compute m and q; draw xi; evaluate the three energies; apply a numerically stabilized softmax; relax the probabilities; propagate the activation through the network; backpropagate the task loss plus an optional entropy penalty. The Hamiltonian and critical-disorder mechanism come from the paper; the neural coupling values, relaxation rate, and exact measured threshold must be estimated empirically. First test a one-layer ternary RNN or a two-layer MLP on MNIST and sequential-MNIST. Sweep R over logarithmically spaced values and run 32 initializations per value. Measure pairwise distances between late-time m trajectories and the variance of final activations. The falsifiable prediction is a sharp reduction in between-seed variance above a threshold, with a candidate crossover near R_c = sqrt(2/pi) after matching the paper's field convention. Below the threshold, at least two long-lived clusters of final m should remain. Also test whether the same threshold predicts the point where exploding or seed-sensitive gradients disappear.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

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 9/10

Built a ternary mean-field disorder simulator with stabilized softmax relaxation, finite-N annealed random fields, basin-spread measurement, and a tanh baseline. The slope prediction matched 1/R to 8.3e-13 relative error, the observed crossover was R=0.7997 versus predicted Rc=sqrt(2/pi)=0.7979 (absolute error 0.00183), and the relaxed Jacobian matched theory to 5.1e-13. Initial-condition spread was large below Rc (MSD 1.88 at R=0.45) and essentially zero above it (5.6e-5 at R=0.8 and below numerical precision by R=0.9); the ternary finite-N check gave |m|≈0.835 at R=0.6 and m≈-0.001 at R=1.0, versus persistent ±0.815 for the saturated tanh baseline. This confirms the proposed mechanism in the RFIM-like toy regime, but does not establish a downstream neural-network accuracy or optimization win.

Agent confidence: 9/10

Baseline: Tanh mean-field coupling 1.4: final m from starts -1,+1 = [-0.81453, +0.81453], persistent two-basin behavior.
Idea: RF ternary/RFIM-like map: Rc predicted 0.79788, observed 0.79971; pairwise final-m MSD 1.8765 at R=0.45, 0.0000561 at R=0.8, ~0 at R>=0.9; ternary finite-N final m=0.8348 at R=0.6 and -0.00115 at R=1.0.

Limitations: Only the mean-field/RFIM-like toy regime was tested (K=Delta=H=0); no MNIST, RNN/MLP training, gradient-stability sweep, FLOP/speed measurement, 32-seed task experiment, or comparison against a trained optimizer was performed.

How to run: python3 disorder_basin.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.16561) Sep 2, 2026

Significantly WORSE than the tuned baseline (+1188.5%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: matched_ternary_rnn
Paired seeds: 8
Baseline mean: 0.0032
Idea mean: 0.0414
Effect (Δ): 0.0382 (+1188.5%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Confirmed ✓
Practical effect: Hurts
Baseline tuning: swept over 5 configs

Limitations:

Only the built-in dynamics track was tested; no sequence or tabular transfer, longer training budgets, alternative ternary temperatures/crystal fields, or discrete straight-through sampling was evaluated.

How to run: python3 stage2_disorder_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 a matched ternary disorder recurrent model on the structurally appropriate dynamics track. The RFIM critical-slope check matched Rc=sqrt(2/pi) to 6.16e-13 absolute error, and the trained-model signature confirmed reduced late magnetization above Rc, but the idea significantly worsened task MSE: paired delta +0.03819 with permutation p=0.0081.

Baseline: Best continuous tanh recurrent baseline, lr=0.01, weight_decay=0: mean test MSE 0.00321310 across 8 seeds.
Idea: Best ternary disorder recurrent model, lr=0.01, R=1.2: mean test MSE 0.04139929 across 8 seeds.
Idea wins: 0
Mechanism signature confirmed: true
Paired delta mean: 0.03818619274534285
Permutation p value: 0.0081

Benchmark result