✗ Failed on benchmark 2026

Dissipative Softmax Latent Layer

Implementation & benchmark of arXiv:2608.26621 — When dissipative steady states admit thermodynamic occupation laws

Usefulness7/10

Difficulty6/10

Novelty8/10

Architecture Stability Accuracy Sample efficiency Vae Diffusion sampling Moe routing Stochastic processes Statistical mechanics Graph theory

Source paper: When dissipative steady states admit thermodynamic occupation laws arXiv:2608.26621 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Add a finite-state stochastic latent layer with conditional states i=1,...,K and an auxiliary reset state 0. The network predicts thermodynamic logits X_i, while transition rates are constructed so that the conditional stationary distribution approaches p_i=exp(X_i)/Z_C under rapid reset, even though the full latent graph retains directed probability currents. This creates a calibrated stochastic layer with controllable mixing and a separate mechanism for maintaining exploration.

Formulas

$$X_i=\beta\left(\Delta\mu-\mathcal{F}^{\mathrm{cost}}_i\right),\qquad Z_{\mathcal C}=1+\sum_{i=1}^{K}e^{X_i},\qquad p_i=\frac{e^{X_i}}{Z_{\mathcal C}}.$$

$$P_i=(1-P_0)p_i,\qquad \lim_{r/k\to\infty}P_i=p_i.$$

$$\log\frac{k_{ij}}{k_{ji}}=X_j-X_i,\qquad \sum_{(i,j)\in c}\log\frac{k_{ij}}{k_{ji}}=0\quad\text{for every conditional cycle }c.$$

$$\lVert\mathrm{d}X-\log(k_{ij}/k_{ji})\rVert=O(|J|),\qquad \sigma=O(J^2)\quad\text{near }J=0.$$

Mathematical statement

Let i,j in {1,...,K} denote conditional states, 0 denote the reset state, beta>0 be an inverse-temperature scale, Delta_mu be a maintained driving affinity, and F_i^cost be a neural-network-predicted cost. Define X_i=beta(Delta_mu-F_i^cost) and Z_C=1+sum_i exp(X_i), giving p_i=exp(X_i)/Z_C. If P_0 is the stationary probability of the reset state and P_i is the full-graph probability, sector separation gives P_i=(1-P_0)p_i. Construct conditional transition rates k_ij with log(k_ij/k_ji)=X_j-X_i, while reset transitions carry the dissipative cycle. If r is the reset rate and k is a typical conditional redistribution rate, the strong-reset limit r/k to infinity gives P_i to p_i. Near autonomous redistribution, the integrability defect is proportional to residual cycle current J, whereas entropy production sigma is proportional to J squared.

Implementation notes

Integrate this layer into a small VAE, discrete diffusion model, or mixture-of-experts router. Given hidden representation h, use an MLP to output costs F_i^cost(h), then set X_i=beta(Delta_mu-F_i^cost(h)). Use K conditional states plus reset state 0. For each conditional pair, let a_ij=softplus(g_ij(h))+epsilon and define k_ij=a_ij*exp((X_j-X_i)/2) and k_ji=a_ij*exp((X_i-X_j)/2); this enforces the exact conditional rate-ratio law. Add transitions from conditional states to 0 with rate r and reset-to-conditional transitions chosen to form a directed reset cycle. Sample by Gillespie simulation or uniformization, run a short burn-in, and pass the resulting state to the decoder or router. The paper-derived quantities are the occupation law, rate-ratio construction, sector factorization, and rapid-reset limit. Estimate P_0, mixing time, cycle currents, and conditional KL divergence from sampled trajectories. First test: train a small MNIST VAE with K=8 states against Gumbel-Softmax and categorical baselines, sweeping r/k over 0.1, 1, 10, and 100. The prediction is that KL(q_cond||p) decreases approximately linearly with k/r for sufficiently large r/k and that measured occupations approach p_i within 20 percent when r/k is at least 10, while the directed reset flux remains nonzero.

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 finite-state dissipative softmax CTMC with exact conditional rate-ratio construction, reset exchange, stationary solving, cycle-current measurement, and a routing mini-comparison. The mechanism manifested: conditional error scaled approximately as (r/k)^-0.92, KL scaled approximately as (r/k)^-1.84, and weak-affinity sweeps gave current scaling A^1.000 and entropy production scaling A^2.000. The dissipative router approached the direct-softmax NLL as r/k increased, but did not provide a meaningful loss win.

Agent confidence: 8/10

Baseline: Direct categorical softmax NLL = 1.697034
Idea: Stationary conditional NLL was 1.699703 at r/k=1, 1.697657 at r/k=10, and 1.696939 at r/k=100. Exact conditional rate-ratio error was 2.22e-16; occupation-error slope was -0.923 versus predicted -1; KL slope was -1.841 versus predicted -2; weak-affinity current slope was 1.000 and entropy-production slope was 2.000; current remained approximately 0.024 at large reset ratio.

Limitations: The requested MNIST VAE/Gumbel-Softmax training experiment was not implemented; only a fixed synthetic routing/NLL comparison was run. The construction verifies conditional occupation convergence q(i|conditional) toward p(i), while symmetric reset exchange keeps P0 near 1/2, so full-graph probabilities do not converge to p without an additional P0 design. No Gillespie trajectory sampling, GPU benchmark, mixing-time estimate, or learned MLP costs were tested.

How to run: python3 dissipative_softmax.py && /home/maxwelhelp/main/bin/python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.26621) Sep 2, 2026

Significantly WORSE than the tuned baseline (+31560.0%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Token_expert_sequence
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 0.0016
Idea mean: 0.4947
Effect (Δ): 0.4931 (+31560.0%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±1580.9%
Mechanism: Not confirmed ✗
Practical effect: Hurts
Baseline tuning: swept over 3 configs

Limitations:

The trained-model mechanism signature produced NaN diagnostics and therefore does not support the mathematical transfer claim. The official track uses classification error, which is coarse and saturated near zero for the baseline; no additional benchmark tracks were run after the structurally matched track showed a significant loss.

How to run: python3 official_bench.py

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

Stage 2 — Benchmark Sep 2, 2026 latest Failed ✗

Agent confidence: 10/10

The official registered bench run completed on token_expert_sequence, a structurally matched multi-token MoE-routing task, using 8 paired seeds and a tuned baseline sweep. The dissipative layer was significantly worse: test error increased by 0.493125 with permutation p=0.0081, so the harness verdict is idea worse (significant). The mechanism signature was not confirmed because its trained-model diagnostics returned NaN.

Baseline: mlp_tiny, lr=0.01, epochs=25: mean test err=0.0015625 over 8 seeds.
Idea: Dissipative reset/cycle readout, best tested r=0.5 at lr=0.01, epochs=25: mean test err=0.4946875 over 8 seeds; paired delta=+0.493125, p=0.0081.

Benchmark result