△ Mechanism confirmed, baseline not beaten 2026

Adaptive Ballistic-to-Diffusive Propagation Schedule

Implementation & benchmark of arXiv:2607.22240 — Fermions on a 1D lattice: localized sources and sinks with dephasing

Usefulness7/10

Difficulty5/10

Novelty8/10

Training Stability Accuracy Scalability Scheduler Ssm Rnn Training dynamics Regularization Dynamical systems Pde Spectral theory Control theory

Source paper: Fermions on a 1D lattice: localized sources and sinks with dephasing arXiv:2607.22240 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Use dephasing as a depth- or time-dependent control variable rather than a fixed regularizer: early layers retain coherent transport for feature discrimination, while later layers increase dephasing to eliminate unstable high-frequency oscillations. The schedule is selected from an observable spectral or correlation ratio, giving a falsifiable switch point instead of tuning noise blindly.

Formulas

$$\omega(k)=-2J\cos k,\qquad v_g(k)=\frac{d\omega}{dk}=2J\sin k$$

$$\widehat n_k(t+\Delta t)\approx e^{-D_{\gamma_t}k^2\Delta t}\widehat n_k(t),\qquad D_{\gamma_t}\simeq\frac{2J^2}{\gamma_t}$$

$$r_t=\frac{\|C_t-\operatorname{diag}(C_t)\|_F}{\|\operatorname{diag}(C_t)\|_F+\varepsilon},\qquad \gamma_{t+1}=\operatorname{clip}\left(\gamma_t\exp[\alpha(r_t-r_*)],\gamma_{\min},\gamma_{\max}\right)$$

$$\\gamma_{\\mathrm{switch}}\\sim4J\\quad\\text{or}\quad D_\\gamma k_*^2\\sim J|\\sin k_*|$$

Mathematical statement

For a nearest-neighbor lattice, coherent propagation has dispersion omega(k)=-2J cos(k), while strong dephasing produces diffusion with decay rate D_gamma k squared and D_gamma approximately 2J squared divided by gamma. Let k_max be the largest reliably represented feature frequency and let r_t be the ratio of off-diagonal correlation energy to diagonal energy at layer t. A practical controller increases gamma_t when r_t exceeds a target. The predicted crossover occurs when gamma_t is comparable to the coherent bandwidth, approximately 4J, or when dephasing and coherent transport rates are equal for a selected frequency k_star.

Implementation notes

Add the controller to a recurrent stack, deep graph network, or neural ODE solver. Give every propagation block a positive hopping scale J_t and dephasing parameter gamma_t, parameterized as exponentials of unconstrained scalars. After each block, estimate r_t from an explicit correlation state or from a cheap proxy: neighboring-feature cross-covariance divided by feature variance. Update gamma_t with gamma_t multiplied by exp(alpha times (r_t-r_target)), clip it to a prescribed range, and use the updated value in the next block. A no-correlation implementation can estimate r_t from high-frequency discrete Fourier energy of the hidden sequence. Initially stop gradients through the controller and compare fixed gamma, linearly annealed gamma, and feedback-controlled schedules. The paper contributes the coherent-to-diffusive mechanism and its diffusion limit; the feedback law and hidden-state proxy are engineering adaptations. Start with a 32-layer residual SSM on Sequential MNIST and a long-horizon copy task, using matched parameter counts. Record Jacobian singular values, high-frequency energy, validation loss, and effective receptive-field width. The prediction is a sharp reduction in high-frequency energy once gamma crosses approximately 4J, followed by diffusive width growth with variance approximately 2D_gamma t. The controlled model should keep the largest layer Jacobian below one more reliably than the fixed-ballistic baseline, with the transition location varying by no more than 20 percent across seeds.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

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 finite-lattice dephasing simulator, feedback controller, and matched recurrent propagation mini-experiment. The core mechanism manifested: ballistic MSD exponent was 2.00 at gamma=0, while diffusive MSD slopes matched 4J^2/gamma within 2.7% at gamma=0.5 and below 0.1% for gamma>=1; correlation and high-frequency energy also decreased with dephasing. The crossover was broad rather than a sharp switch near 4J, and adaptive control did not beat fixed ballistic propagation on the small classification task, so this is mechanism confirmation rather than an observed neural-performance win.

Agent confidence: 8/10

Baseline: Fixed ballistic: loss 0.09199984, accuracy 0.97265625. Fixed diffusive: loss 0.09874058, accuracy 0.96875.
Idea: Adaptive feedback: loss 0.09449144, accuracy 0.970703125; gamma changed from 0.5 to final 1.65498. Toy MSD slopes for gamma=[0.5,1,2,4,8,16] were [7.7801,3.9960,2.0000,1.0000,0.5000,0.2500] versus predicted [8,4,2,1,0.5,0.25].

Limitations: The neural experiment was a tiny synthetic pulse-position task, not Sequential MNIST or a long-horizon copy task; it used one seed, a short 16-layer stack, and did not measure Jacobian singular values or effective receptive-field width. The high-frequency observable was a population FFT proxy, and the finite-time crossover estimate was not a rigorous sharp transition measurement.

How to run: python3 adaptive_diffusion.py && /home/maxwelhelp/main/bin/python3 mini_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.22240) Sep 2, 2026

No significant difference vs. the tuned baseline (+1.0%, p=0.7946, wins 4 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0005
Idea mean: 0.0005
Effect (Δ): 0 (+1.0%; negative = idea better)
Wins: 4 / 8 paired seeds
p-value: 0.7946 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Confirmed ✓
Practical effect: No effect
Baseline tuning: swept over 12 configs

Limitations:

Only the built-in dynamics task was tested, using 400 training examples, 200 test examples, and 18 epochs; Jacobian spectra, receptive-field width, Sequential MNIST, and copy tasks were not tested.

How to run: python3 bench_adaptive.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

The adaptive schedule was implemented and evaluated on the registered dynamics track with matched GRU systems, an equal-budget tuned baseline sweep, and 8 paired seeds. The mechanism signature was confirmed, but the adaptive system was not better: its MSE was slightly higher and the paired permutation test was non-significant.

Baseline: best fixed GRU lr=0.01 gamma=0.0; mean MSE 0.0004684777286456665
Idea: adaptive GRU lr=0.01 gamma0=0.5 target=0.12 alpha=0.5 depth=4 gmax=1.0; mean MSE 0.00047325076775450725

Benchmark result