Thermodynamic Two-State Expert Gate

1. Integration point: insert the module after the hidden state of a GRU, S4/SSM, transformer world model, or recurrent dynamics predictor. Compute a scalar $q_t$ with a linear projection or a two-layer MLP, maintain two expert transition heads $f_0$ and $f_1$, and combine their outputs with the scalar gate $p_t$. For sequence prediction, use the same gate for a short block of $K$ steps or add a penalty on changes in $p_t$ so the regime is slower than token-level noise. 2. Pseudocode: compute q = q_net(h,x); compute p = sigmoid((q-q0+b)/tau); compute h0=f0(h,x) and h1=f1(h,x); set h_next=(1-p)*h0+p*h1; add the task loss, lambda_smooth*(p_next-p)**2, and a separation term that prevents both experts from becoming identical. Estimate pseudo-labels z_t by fitting a two-component Gaussian mixture to detached q_t values, or omit the calibration term initially. 3. Computed from the paper: the two-state mixture variance $p(1-p)$, the susceptibility peak at $p=1/2$, and the interpretation of excess fluctuations as regime interconversion. Estimated empirically: $q$, $q_0$, $\tau$, expert outputs, and transition labels. Fit $q_0$ to the median of the training gate logits and initialize $\tau$ from their interquartile range. 4. First cheap experiment: train a GRU world model on a synthetic switching linear dynamical system with two transition matrices, plus a small Lorenz-system dataset whose parameters switch between two regimes. Compare a single GRU, an unconstrained MoE, and this gate at equal parameter count. Sweep an externally supplied interpolation parameter controlling the regime mixture. The quantitative prediction is that forecast residual variance and gate susceptibility $\operatorname{Var}(p)/\tau$ peak near the balanced mixture $p\approx0.5$, with the peak location predicted by fitted $q_0$ within 20%; away from that point, the two-state model should reduce long-horizon error and gate autocorrelation should decay more slowly than token noise.